Symmetries in Physics - uni-bielefeld.de · 2018-02-21 · IX Symmetries in classical ﬁeld theory...

Symmetries in Physics

Nicolas Borghini

version of February 21, 2018

Nicolas BorghiniUniversität Bielefeld, Fakultät für PhysikHomepage: http://www.physik.uni-bielefeld.de/~borghini/Email: borghini at physik.uni-bielefeld.de

http://www.physik.uni-bielefeld.de/~borghini/

Contents

I Basics of group theory • • • • • • • • • • • • • • • • • • • • • • • • • • 1I.1 Group structure 1

I.2 Examples of groups 3I.2.1 Finite groups 3

I.2.2 Infinite groups 9

I.3 Subgroups 9I.3.1 Definition 9

I.3.2 Generating set 10

I.3.3 Cosets of a subgroup 10

I.4 Conjugacy and conjugacy classes 11I.4.1 Equivalence relation 11

I.4.2 Conjugacy 12

I.4.3 Examples of conjugacy classes 12

I.5 Normal subgroups 14I.5.1 Definition 14

I.5.2 Quotient group 15

I.5.3 Examples 15

I.5.4 Further definitions and results 16

I.6 Direct product 16I.6.1 Cartesian product of groups 16

I.6.2 Inner direct product 17

I.7 Group homomorphisms 18I.7.1 Definitions 18

I.7.2 Properties of homomorphisms 18

I.7.3 Inner automorphisms 19

II Representation theory • • • • • • • • • • • • • • • • • • • • • • • • • •21II.1 Group action 21

II.2 Group representation: first definitions and results 21

II.3 Examples of group representations 22II.3.1 Trivial representation 22

II.3.2 Representations of the dihedral group D3 22

II.3.3 Three-dimensional rotations 24

II.3.4 Direct sum and tensor product of representations 25

II.4 Classifying representations 26II.4.1 Equivalent representations 26

II.4.2 Reducible and irreducible representations 27

II.5 Schur’s lemma 30

III Representations of finite groups • • • • • • • • • • • • • • • • • • • • •32III.1 Full reducibility of the representations 32

III.2 Orthogonality relations 34

III.3 Irreducible representations of a finite group 35III.3.1 First consequences of the orthogonality relations 35

III.3.2 Character theory 36

iv

III.4 Reduction of a representation 40III.4.1 Decomposition of a representation into irreducible representations 40

III.4.2 Regular representation 41

III.4.3 Reduction of a tensor-product representation 45

III.5 Representations of the symmetric group Sn 46III.5.1 Irreducible representations of Sn 46

III.5.2 Product of irreducible representations of the symmetric group 49

IV Physical applications • • • • • • • • • • • • • • • • • • • • • • • • • •52IV.1 Electric dipole of a molecule 52

IV.2 Molecular vibrations 52

IV.3 Crystal field splitting 53

V Lie groups and Lie algebras • • • • • • • • • • • • • • • • • • • • • • • •58V.1 Lie groups 58

V.1.1 Definition and first results 58

V.1.2 Examples 59

V.2 Lie algebras 63V.2.1 General definitions 63

V.2.2 Lie algebra associated to a Lie group 65

V.2.3 Lie algebras associated to the classical Lie groups 68

VI The groups SO(3) and SU(2) and their representations • • • • • • • • • • •69VI.1 The group SO(3) 69

VI.1.1 Rotations in three-dimensional space 69

VI.1.2 The group SO(3) and its Lie algebra so(3) 71

VI.2 The group SU(2) 74VI.2.1 The group SU(2) and its Lie algebra su(2) 74

VI.2.2 Relation between the Lie groups SU(2) and SO(3) 75

VI.3 Representations of the Lie groups SU(2) and SO(3) 76VI.3.1 Representations of SU(2) and of the Lie algebra su(2) 76

VI.3.2 Unitary representations of SU(2) 78

VI.3.3 Tensor products of irreducible representations of SU(2) 83

VI.4 Behavior of quantum-mechanical observables under rotations 85VI.4.1 Rotation of a quantum-mechanical system 85

VI.4.2 Irreducible tensor operators 87

VI.4.3 Wigner–Eckart theorem 89

VII Representations of GL(n) and its continuous subgroups • • • • • • • • • •91VII.1 Low-dimensional irreducible representations of GL(n,C) 91

VII.1.1 Representations of dimension 1 91

VII.1.2 Vector representation 91

VII.1.3 Complex conjugate representation 91

VII.1.4 Dual representation 91

VII.2 Tensor representations of GL(n,C) 92VII.2.1 Tensor products of the vector representation 92

VII.2.2 Operations on the indices of a tensor 93

VII.2.3 Irreducible representations built from the vector representation 94

VII.2.4 Tensor products of the dual and conjugate representations 97

VII.2.5 Representations of SL(n,C) and SL(n,R) 99

VII.2.6 Representations of SU(n) 100

v

VIII Lorentz and Poincaré groups • • • • • • • • • • • • • • • • • • • • • 102VIII.1 Definitions and first properties 102

VIII.1.1 Poincaré group 102

VIII.1.2 Lorentz group 103

VIII.2 The Lorentz and Poincaré groups as Lie groups 107VIII.2.1 Generators of the Lorentz group 107

VIII.2.2 Generators of the Poincaré group 108

VIII.3 Representations of the Lorentz group 110

Appendix to Chapter VIII • • • • • • • • • • • • • • • • • • • • • • • • • • 111VIII.A Proof of the linearity of Minkowski spacetime isometries 111

IX Symmetries in classical field theory • • • • • • • • • • • • • • • • • • • 113IX.1 Basics of classical field theory 113

IX.1.1 Definitions 113

IX.1.2 Euler–Lagrange equations 114

IX.2 Symmetries in field theories and conserved quantities 115IX.2.1 Symmetries in a field theory 115

IX.2.2 Noether theorem 115

IX.2.3 Examples 117

IX.3 Gauge field theories 119

A Elements of topology• • • • • • • • • • • • • • • • • • • • • • • • • • 120A.1 Topological space 120

A.2 Manifolds 120

Bibliography • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 121

CHAPTER I

Basics of group theory

An intro is missing

I.1 Group structureDefinition I.1. A group (G, · ) consists of a set G and a binary operation · , the group law , thatfulfills the following axioms:

• closure: for all g, g′ ∈ G, the result g · g′ is also in G; (G1)

• associativity: for all g, g′, g′′ ∈ G, (g · g′) · g′′ = g · (g′ · g′′); (G2)

• there exists a neutral element e such that for every g ∈ G, e · g = g · e = g; (G3)

• for every g ∈ G there is an inverse element g−1 ∈ G such that g−1 · g = g · g−1 = e. (G4)

Remarks:

∗ The closure axiom (G1) is often omitted, at the cost of calling the group law an “internal com-position law”.

∗ The associativity property (G2) means that parentheses are not needed — as long as one onlyconsiders the group law.

∗ The axioms (G3) and (G4) can be replaced by weaker, yet equivalent versions, namely the ex-istence of a left neutral element e such that e · g = g for every g ∈ G and that of a left inverseg−1 ∈ G to g such that g−1 · g = e. These are then automatically also right neutral element andright inverse, respectively.

∗ The postulated existence of a neutral element, also called unit or identity element , guaranteesthat the set G is not empty.

∗ In the following, the result g ·g′ will also be written as gg′, and referred to as the “product” or the“composition” of the elements g and g′. Accordingly, g · g will be written as g2 and more generallyg · gn with n ∈ N∗ as gn+1.

∗ Similarly, in keeping with a widespread habit I shall regularly refer to “a group G”, withoutspecifying the group law. This assumes either that the latter is obvious, or that it is the genericgroup law of an unspecified group.

Theorem I.2. Every group G contains a single neutral element, and each element g ∈ G has a uniqueinverse element g−1.

2 Basics of group theory

Proof: If e and e′ are two identity elements, then e = e · e′ = e′, where the first resp. secondequality expresses the right- resp. left-neutrality of e′ resp. e.

In turn, if g−1 and g′ are two inverse elements to a given g ∈ G, then g−1 = g−1gg′ = g′, wherethe first resp. second equality uses the fact that g′ is a right inverse resp. g−1 a left inverse to g.

A straightforward consequence of the existence of an inverse element is the

Theorem I.3. For every triplet g, g′, g′′ of elements of a group (G, · ), the equality g · g′ = g · g′′(or g′ · g = g′′ · g) implies g′ = g′′.

Definition I.4. A group (G, · ) is called Abelian(a) if the group law is commutative, i.e. if for allg, g′ ∈ G, g · g′ = g′ · g.

Remark: The group law of an Abelian group is often denoted with +.

Definition I.5. The number of elements in the set G of a group (G, · ) is called the order of thegroup and will be hereafter denoted by |G|.

Depending on the finiteness of the group order one distinguishes between finite groups (|G| ∈ N∗)and infinite groups. In turn, infinite groups can either be countable or not countable, where thelatter can further be split into compact and non-compact groups.(1)

In the case of a finite group (G, · ), one can (in principle) write down all possible compositionsg · g′ = gg′ in a Cayley(b) table, i.e. as

g1 g2 · · · gj · · · gn

g1 (g1)2 g1g2 · · · g1gj · · · g1gng2 g2g1 (g2)2 · · · g2gj · · · g2gn...

......

......

......

gi gig1 gig2 · · · gigj · · · g2gn...

......

......

......

gn gng1 gng2 · · · gngj · · · (gn)2

(I.6)

with G = g1, g2, . . . , gn, where the ordering of the group elements is irrelevant — although the“first” element in the examples listed hereafter (§ I.2.1 a) will always be the identity element of thegroup. Note that the entry in the i-th line and j-th column is by convention the product gigj .

One easily proves that each element g ∈ G appears only once in every line and every column ofthe group (Cayley) table. This is a particular case of the following result, valid for both finite andinfinite groups:

Theorem I.7 (Group rearrangement theorem). Let G be a group and g ∈ G one of its element.Defining a set gG = gg′ | g′ ∈ G, one has gG = G.

Proof: Let us show that both inclusions gG ⊆ G and gG ⊇ G are fulfilled, starting with the firstone. Consider an arbitrary g′ ∈ G. Then gg′ ∈ G and therefore gG ⊆ G.Conversely, take an arbitrary g′ ∈ G. The element g−1g′ is again in G and so is gg−1g′ = g′,which shows gG ⊇ G.

(1)These terms will be defined and discussed in further detail in Chapter V.

(a)N. H. Abel, 1802–1829 (b)A. Cayley, 1821–1895

I.2 Examples of groups 3

I.2 Examples of groupsWe now list a few examples of groups, starting with finite ones (Sec. I.2.1) and then going on toinfinite ones (Sec. I.2.2). Further examples will be given later in this chapter as well as in the nextchapters.

I.2.1 Finite groups

introductory blabla

::::::I.2.1 a

::::::::::::::::::::Small finite groups

Order 1One may define a group structure on any set S consisting of a single element a, i.e. S = a,

by defining an internal composition law · on S by a · a = a. (S, · ) trivially fulfills the 4 groupaxioms (G).

As “symmetry group” of the transformations that leave a physical or mathematical object invari-ant, a group of order 1 is obviously quite boring and corresponds to an object having “no symmetry”,since the only transformation leaving it unchanged is the identity transformation.

Order 2A first example of a group of order 2 consists of the set 0, 1, where 0 and 1 denote the “usual”

natural numbers, while the group law is the addition modulo 2, i.e. such that 1 + 1 = 0. Thecorresponding Cayley table is

+ 0 1

0 0 1

1 1 0

(I.8)

and the group is traditionally called (Z2,+).A second example is that of the set 1,−1 with the usual multiplication of integers, yielding

the Cayley table× 1 −1

1 1 −1

−1 −1 1

(I.9)

A further example can be built as follows. Let F (R3) denote a set of functions defined on R3 —the target set of the functions is irrelevant. One defines two operators I and P acting on this set,i.e. associating to a function f ∈ F (R3) another function g ∈ F (R3), by

I : f(~r) 7→ g(~r) = f(~r) and P : f(~r) 7→ g(~r) = f(−~r) (I.10)

respectively. That is, I is the identity operator, while P reverses the argument of the function itacts upon. As internal composition law on the set I,P one considers the successive operation,i.e. the composition (of operators) often denoted by . One then at once sees that applying P twiceto a function f gives back the same function, i.e. P 2 = P P = I. More generally, one finds that(I,P , ) is a group of order 2 with the Cayley table

I P

I I P

P P I

(I.11)

Inspecting the Cayley tables (I.8), (I.9), and (I.11) reveals that they are all similar, namely ofthe form

· e a

e e a

a a e

(I.12)


with e the neutral element of the group. This table describes an abstract group e, a with aninternal composition law · such that a2 = e, which is Abelian.

In turn, the groups (Z2,+), (1,−1,×), and (I,P , ) are various representations of thisabstract group law — which happens to be the only possible one for a set with two elements, as iseasily checked.

Order 3Considering now groups of order 3, we shall start with a set e, a, b and show that there is only

one possibility to define an internal composition law that turns it into a group, actually an Abeliangroup, with e as the identity element.

Writing down the Cayley table, the line and column associated with the identity element e followautomatically from axiom (G3):

· e a b

e e a b

a a

b b

Turning now to the second line, associated to a, we know from the result preceding Theorem (I.7)that it should still contain e and b, and that b may not appear in the third column, which alreadycontains b, so that necessarily a2 = b. A similar reasoning gives for the third line b2 = a — whichis quite normal, a and b play symmetric roles —, and thus

· e a b

e e a b

a a b

b b a

Eventually, the remaining entries can be filled in by realizing that e is still missing from the secondand third lines, leading eventually to ab = ba = e and therefore to the table

· e a b

e e a b

a a b e

b b e a

(I.13)

In particular, both a and b obey a3 = b3 = e.

A possible representation of the abstract group (I.13) consists of the three rotations throughangles 2kπ/3 with k ∈ 0, 1, 2 — that with k = 0 being obviously the identity transformation —about an arbitrary point, e.g. the origin O, of a (two-dimensional!) plane, with the composition ofgeometrical transformations as group law. Altogether, these three rotations constitute the symmetrygroup of an equilateral triangle centered on the origin with oriented sides, as in Fig. I.1.

Figure I.1

As alternative groups of order 3, one may consider the set of complex numbers 1, e2iπ/3, e4iπ/3with the usual product of complex numbers as group law, or the set Z3 ≡ 0, 1, 2 of integers withthe addition modulo 3.


Order 4Consider now a set of four elements e, a, b, c. Proceeding as in the previous paragraph, one can

investigate internal composition laws on this set that would turn it into a group of order 4 withidentity element e, by systematically building the Cayley table of the group. It turns out that thereare now two different possible group laws, up to renaming of the elements.

A first possibility is that of a group with a2 = b and a3 = c or equivalently c2 = b and c3 = a, i.e.in which two of the elements (a and c) play symmetric roles, while the third non-identity element(b) behaves differently since b2 = e and thus b3 = b:

· e a b c

e e a b c

a a b c e

b b c e a

c c e a b

(I.14)

Note that this is again an Abelian group.A possible representation of the abstract group (I.14) are the four rotations through angles kπ/2

with k ∈ 0, 1, 2, 3 about an arbitrary point O of a plane, with the composition of geometricaltransformations as group law: this is the symmetry group of a square with oriented sides centeredon O (Fig. I.2). Alternatively, one may consider the set of complex numbers 1, i,−1,−i with theusual product of complex numbers as group law, or the set Z4 ≡ 0, 1, 2, 3 of integers with theaddition modulo 4.

Figure I.2

There is a second possible group structure on a set with four elements, in which the three non-identity elements play symmetric roles: a · b = b · a = c, a · c = c · a = b, b · c = c · b = b, anda2 = b2 = c2 = e:

· e a b c

e e a b c

a a e c b

b b c e a

c c b a e

(I.15)

This is again an Abelian group, known as the Klein(c) group or Vierergruppe (“four-group”) andoften denoted by V4.

A possible representation consists of geometrical transformations in a plane: the identity trans-formation, a (“two-dimensional”) rotation through π about a point O, and the reflections acrosstwo perpendicular axes going through O. This is the symmetry group of a non-equilateral rect-angle (Fig. I.3. Alternatively, the reflections can be viewed as rotations through π about the twoaxes in three-dimensional space, while the “two-dimensional” rotation is now seen as a third “three-

(c)F. Klein, 1849–1925


Figure I.3

dimensional” rotation about an axis perpendicular to the other two axes, restoring the symmetrybetween the three transformations.

::::::I.2.1 b

::::::::::::::::::::::::::::::Four families of finite groups

More generally, one can identify several families of finite groups, four of which we now list.

:::::::::::::::::Cyclic groups Cn

The groups with respective Cayley tables (I.12), (I.13), (I.14) obviously have an importantfeature in common, namely the existence of a (non-identity) element a such that the n = 2, 3 or4 elements of the group can be expressed as a, a2. . . , an−1, and an = e.

More generally, one defines such a group with n ∈ N∗ elements, Cn = a, a2, . . . , an−1, an = e,(2)

called the cyclic group of order n, with the Cayley table

· e a a2 . . . an−1

e e a a2 . . . an−1

a a a2 a3 . . . e

a2 a2 a3 a4 . . . a...

......

......

...an−1 an−1 e a . . . an−2

(I.16)

These groups are clearly Abelian for any n.

One can generalize the representations given in § I.2.1 a in the cases n = 2, 3 or 4 to representa-tions of Cn. For instance, the rotations through angles 2kπ/n with k ∈ 0, 1, . . . , n−1 about a fixedpoint constitute, with the composition of rotations, such a representation, which is the symmetrygroup of a regular polygon with n oriented sides.

Alternatively, the abstract group Cn can be represented by the set Zn ≡ 0, 1, 2, . . . , n − 1 ofintegers with the addition modulo n, or by the set of complex numbers 1, e2iπ/n, e4iπ/n, . . . , e2ikπ/nwith the usual multiplication of complex numbers.

::::::::::::::::::::Dihedral groups Dn

Consider an regular polygon with n ≥ 3 sides(3) centered on a point O. The geometric transfor-mations leaving the polygon invariant are on the one hand the rotations rk through 2kπ/n aboutthe center O, with k ∈ 0, 1, . . . , n − 1, and on the other hand n reflections sk across lines goingthrough O and through either a vertex of the polygon or the midpoint of a side.

If n is odd, the n symmetry axes are indeed joining a vertex of the n-gon to the midpoint ofthe opposite side. If n is even, there are n/2 diagonals joining opposite vertices, and n/2 linesjoining the midpoints of opposite sides.

The composition of two rotations or two reflections gives a rotation, while the composition of areflection and a rotation yield a reflection, and all these transformations together form a group.Denoting s one of the reflections, any one of them, then all other reflections are of the form srk.(2)The group is also denoted Zn.(3)In contrast to the triangle and square pictures in Figs. I.1–I.2, the sides are not oriented.


In addition, r0 is obviously the identity transformation e and denoting r ≡ r1 one has rk = rk fork ≥ 1, so that the group eventually is

e, r, r2, . . . , rn−1, s, sr, sr2 . . . , srn−1 with rn = s2 = e. (I.17)

This is a finite group of order 2n, called dihedral group of order 2n and often denoted Dn.

Remark: Unfortunately, the dihedral group of order 2n is equally frequently denoted D2n in theliterature. If the subscript is an odd integer there is no ambiguity; otherwise, let the reader beware!

One generalizes the construction to the cases n = 1 and n = 2, for which the geometric repre-sentation is meaningless. Thus D1 = e, s with s2 = e is actually the same as the cyclic group C2

of order 2. In turn, D2 = e, r, s, sr has the same structure as the Klein group V4. Both D1 andD2 are thus Abelian, which is not true of the higher-order dihedral groups Dn with n ≥ 3.

::::::::::::::::::::::Symmetric groups Sn

Consider a set X with n ∈ N∗ elements, for instance X = 1, 2, . . . , n. The bijectionsX → X are also called permutations of X . Since the composition of two such permutations isagain a permutation, one easily checks the following theorem:

Theorem & Definition I.18. The permutations on a set of n elements, together with the compositionof functions, form a group Sn, called the symmetric group of degree n.

Property I.19. The symmetric group Sn is of order n!.

A usual notation for a permutation σ ∈ Sn consists of two lines: the elements of X are listedin the first row, and for each element its image under σ below it in the second row:

σ =

(1 2 · · · n

σ(1) σ(2) · · · σ(n)

). (I.20)

For instance, the identity element of Sn takes the form

e =

(1 2 · · · n1 2 · · · n

). (I.21)

The order of the elements of X in the first row is irrelevant, so that the permutation σ of Eq. (I.20)may equivalently be written

σ =

(i1 i2 · · · in

σ(i1) σ(i2) · · · σ(in)

), (I.22)

where (i1, . . . , in) is a permutation(!) of (1, . . . , n). Using this property, the inverse of σ reads

σ−1 =

(σ(1) σ(2) · · · σ(n)

1 2 · · · n

)(I.23)

since obviously 1 = σ−1(σ(1)

), 2 = σ−1

(σ(2)

)and so on. The trick is also useful to construct the

“product” of two permutations, i.e. their composition. Writing

τ =

(1 2 · · · n

τ(1) τ(2) · · · τ(n)

)=

(σ(1) σ(2) · · · σ(n)

τ(σ(1)) τ(σ(2)) · · · τ(σ(n))

),

one has

τσ ≡ τ σ =

(σ(1) σ(2) · · · σ(n)

τ(σ(1)) τ(σ(2)) · · · τ(σ(n))

)(1 2 · · · n

σ(1) σ(2) · · · σ(n)

)(I.24)

=

(1 2 · · · n

τ(σ(1)) τ(σ(2)) · · · τ(σ(n))

)=

(1 2 · · · n

τσ(1) τσ(2) · · · τσ(n)

). (I.25)

In general, τσ 6= στ , i.e. the group Sn is non Abelian — this is the case for n ≥ 3.


Decomposition of a permutation into independent cyclesConsider an arbitrary permutation σ of Sn, for instance

σ =

(1 2 3 4 5 6 7 8 94 3 9 6 1 7 5 8 2

)∈ S9. (I.26)

One remarks that the three sets of elements 1, 4, 5, 6, 7, 2, 3, 9, and 8 “evolve” independentlyof each other under the application of σ:

• 1σ→ 4

σ→ 6σ→ 7

σ→ 5σ→ 1;

• 2σ→ 3

σ→ 9σ→ 2;

• 8σ→ 8.

These three sets form respective cycles or cyclic permutations, for which one introduces a shorter,one-line notation as e.g.(

1 4 6 7 5)≡(

1 4 6 7 54 6 7 5 1

)≡(

1 4 6 7 5 2 3 8 94 6 7 5 1 2 3 8 9

),

which only affects the elements 1, 4, 5, 6, 7, leaving the other ones unchanged. Using this notation,the permutation of Eq. (I.26) can be written as

σ =

(1 2 3 4 5 6 7 8 94 3 9 6 1 7 5 8 2

)=(1 4 6 7 5

) (2 3 9

) (8).

As a last simplification, one drops the cycles of length one — here (8) —, with the convention thatnon-written elements belong to such cycles:(4)

σ =

(1 2 3 4 5 6 7 8 94 3 9 6 1 7 5 8 2

)=(1 4 6 7 5

) (2 3 9

).

The permutation σ has now be decomposed into the product of independent (or “disjoint”) cycles.Since the latter affect different elements, their product actually commute:

σ =(1 4 6 7 5

) (2 3 9

)=(2 3 9

) (1 4 6 7 5

).

The above procedure can be applied to any permutation, which proves the following theorem:

Theorem I.27. Every permutation of n elements can be expressed as the (commutative) product ofindependent cycles.

Definition I.28. A cycle with two elements is called a transposition of those elements.

For any k ∈ N∗ with k ≥ 2, and any k-tuple of numbers (i1, i2, . . . , ik), one checks the identity(i1 i2 i3 · · · ik

)=(i1 ik

)· · ·(i1 i3

)(i1 i2

), (I.29)

which proves the

Theorem I.30. Any cycle can be written as a product of transpositions.Note that the transpositions on the right-hand side of Eq. (I.29) are not independent of each other.

By combining the theorems (I.27) and (I.30), one finds that any permutation can be written asa product of transpositions. The number of such transpositions is not fixed, yet one can show thatfor a given permutation σ ∈ Sn, the number of transpositions in any decomposition always has thesame parity, i.e. is either always even or always odd. This leads to the consistency of the followingdefinition:(4)The notation becomes singular in the case of the identity permutation, which only consists of cycles of length 1!

I.3 Subgroups 9

Definition I.31. The parity or signature of a permutation σ ∈ Sn is the parity, characterized by anumber ε(σ) ∈ −1,+1, of the number of transpositions in all decompositions of σ.

Example: Using for instance the decomposition (I.29), one finds that a cycle with k elements hasparity (−1)k−1.

Let us eventually introduce a last definition:

Definition I.32. A permutation that leaves no element unchanged is called a derangement .

The decomposition into disjoint cycles of a derangement contains no cycle of length 1.

:::::::::::::::::::::Alternate groups An

Eventually, one easily checks the following results:

Theorem & Definition I.33. The permutations of n ∈ N∗ elements with even parity form a group,called the alternating group An.

Property I.34. The alternating group An is of order n!/2.

One checks that only A1, A2 and A3 are Abelian — they consist of only 1, 2 or 3 elements,respectively. For n ≥ 4, An is non Abelian.

I.2.2 Infinite groups

Let us now give a few examples of infinite groups.An important one is the set of integers with the addition, (Z,+). This group is countably

infinite, as is Z, and Abelian since n+m = m+ n for all m,n ∈ Z.

Turning now to non-countably infinite groups, one may quote the set of real numbers with theaddition, (R,+), or the set of non-zero real numbers with the usual multiplication (R∗,×) — whereleaving aside the 0 is necessary to ensure the existence of the inverse for every element of the group.Both these groups are clearly Abelian.

Another example of non-countably infinite, Abelian group is the set of complex numbers withmodulus 1 with the multiplication as group law, (eiϕ |ϕ ∈ [0, 2π],×).

Let now X be an arbitrary infinite set and Bij(X ) denote the set of bijections of X on itself.Using the composition of functions as internal law, (Bij(X ), ) is always a group— which generalizesto the infinite case the symmetric groups Sn. In general, (Bij(X ), ) is non-Abelian.

Taking as set X an n-dimensional real or complex vector space, say Rn or Cn, and consideringthe bijective linear applications on that vector space, the latter can be represented (after choosinga basis) by regular n × n matrices with real resp. complex entries. The set of such matrices, withthe usual matrix product as internal composition law, forms a group GL(n,R) resp. GL(n,C).

I.3 Subgroups

I.3.1 Definition

Definition I.35. Let (G, · ) be a group. A non-empty set H ⊆ G is called a subgroup of G if it isitself a group under the same composition law · as that of G.

For any group G with identity element e, both G itself and e are (trivial) subgroups of G.

Definition I.36. A proper subgroup of a group G is a subgroup H which differs from both G and e,i.e. e ( H ( G.


Examples:

∗ The Klein group V4 has three proper subgroups e, a, e, b, and e, c, as can be read off itsCayley table (I.15).

∗ The set 2Z of even numbers is a proper subgroup of the group (Z,+) of integers, which is itselfa subgroup of (R,+). More generally, the set pZ of the multiples of a number p ∈ N∗ is a propersubgroup of (Z,+).

Theorem I.37. Let (G, · ) be a group. A non-empty subset H ⊂ G is a subgroup of G if and only iffor all g, h ∈ H, g ·h−1 ∈ H. If G is finite, then the condition g ·h ∈ H for all g, h ∈ H is sufficient.

I.3.2 Generating set

Theorem I.38. Let (G, · ) be a group. If H and K are two subgroups of G, then their intersectionH ∩K is also a subgroup of G.

Theorem & Definition I.39. Let S be a subset of a group (G, · ). The intersection of the subgroupsH of G that include S is again a subgroup including S, denoted by 〈S〉.

Property I.40. 〈S〉 is the “smallest” subgroup including S, in that it is included in every subgroupH of G including S. Note, however, that two arbitrary subgroups H,K of G may not be comparable,i.e. neither H ⊆ K nor K ⊆ H holds.(5)

Definition I.41. A group G is called finitely generated if there exists a finite set S = a1, . . . , an ofelements of G such that G = 〈S〉, which is also denoted G = 〈a1, . . . , an〉.

Examples:

∗ Every finite group is obviously finitely generated. For instance, the Klein group is generated byany two of its non-identity elements: V4 = a, b = a, c = b, c.

∗ The group (Z,+) or integers is finitely generated: Z = 〈1〉.

Definition I.42. A group is called cyclic if it is generated by a single element.

Property I.43. A cyclic group is always Abelian.

Definition I.44. The order of an element g of a group G is the order of the subgroup 〈g〉 generatedby g.

Property I.45. The order of an element g is either infinite, or it is the smaller positive integer ssuch that gs = e.

I.3.3 Cosets of a subgroup

Definition I.46. Let H denote a subgroup of a group (G, · ) and g ∈ G. The set gH = g ·h |h ∈ His called left coset of H in G with respect to g.Similarly, one defines the right coset Hg = h · g |h ∈ H.

In the remainder of this section, we give results for the left cosets of a subgroup in a group only.The reader can state and prove equivalent results for the right cosets.

(5)Stated differently, inclusion is not a total order on the set of subgroups (or more generally subsets) of G.

I.4 Conjugacy and conjugacy classes 11

Theorem I.47. The various left cosets of a subgroup H in a group G with respects to all elementsg ∈ G are either identical or disjoint, and they provide a partition(6) of G.

Proof: let g1, g2 ∈ G be such that the left cosets g1H and g2H have (at least) a commonelement. Then there exist h1, h2 ∈ H such that g1h1 = g2h2. Multiplying this identity leftby g−12 and right by h−11 yields g−12 g1 = h2h

−11 , where the latter product is an element of H.

Multiplying both members of this identity right with any h ∈ H yields g−12 g1h = h2h−11 h ∈ H,

i.e. g−12 g1H ⊂ H, which leads at once to g1H ⊂ g2H.Coming back to g1h1 = g2h2 and multiplying now left by g−11 and right by h−12 amounts toexchanging the roles of g1 and g2, i.e. leads in turn to g2H ⊂ g1H, so that g1H = g2H: bothcosets are identical as soon as they have one common element.

Since the left cosets are disjoint, we only have to show that their union⋃g gH, which is neces-

sarily included in G, actually equals G to prove that they form a partition of the whole group.Now, from e ∈ H follows g ∈ gH for all g ∈ G, leading to G ⊂

⋃g gH. 2

Definition I.48. Let H denote a subgroup of a group G. The number of distinct left cosets of H inG is called the index of H in G and denoted |G : H|.

Theorem I.49. Let H denote a subgroup of a group G. If two of the numbers |G|, |H| and |G : H|are finite, then so is the third and they are then related by

|G| = |G : H| |H|. (I.50)

Proof: all left cosets have the same order |gH| = |H| for every g ∈ G, which leads to the wantedresult.

In the case of finite groups, the theorem (I.49) leads to several straightforward consequences:

Corollary I.51 (Lagrange’s(d) theorem). The order |H| of every subgroup H of a finite group Gdivides the order |G| of the group.

Corollary I.52. The order |〈g 〉| of every element g of a finite group G divides the order |G| of thegroup.

Corollary I.53. For every element g of a finite group (G, · ), g|G| = e.

This follows from the property (I.45) of the order of an element and from the previous corollary.

Corollary I.54. If G is a finite group, whose order |G| is a prime number, then G is cyclic.

I.4 Conjugacy and conjugacy classes

I.4.1 Equivalence relation

Definition I.55. An equivalence relation ∼ on a set S is a binary relation on S which is reflexive,symmetric, and antisymmetric, that is:

• reflexivity: for all a ∈ S, a ∼ a (E1)

• symmetry: for all a, b ∈ S, a ∼ b if and only if b ∼ a (E2)

• transitivity: for all a, b, c ∈ S, if a ∼ b and b ∼ c then a ∼ c (E3)(6)A partition of a set S is a family Aii∈I of subsets Ai ⊂ S such that the intersection of any two different Ai is

empty (Ai ∩Aj = ∅ for i 6= j), while the union of all Ai is S,⋃iAi = S.

(d)J.-L. Lagrange, 1738–1813


Definition I.56. Let ∼ be an equivalence relation on a set S. The subset of S consisting of theelements b such that a ∼ b is called the equivalence class of a in S with respect to ∼.

Theorem I.57. Let ∼ be an equivalence relation on a set S. The various equivalence classes withrespect to ∼ provide a partition(6) of S.

I.4.2 Conjugacy

Definition I.58. Let (G, · ) be a group. An element b ∈ G is said to be conjugate to an elementa ∈ G if there exists g ∈ G such that b = g · a · g−1. g is then called conjugating element .

Remark: The conjugating element is not necessarily unique.

Theorem & Definition I.59. Let G be a group. The conjugacy relation “element a is conjugate toelement b in G” is an equivalence relation on G, whose equivalence classes are called conjugacyclasses.Accordingly, the conjugacy classes of a group G provide a partition of G; the elements of a givenconjugacy class share many properties, as will now be illustrated on a few examples.

Remark I.60. One sees at once that the identity element e of a group will always form a conjugacyclass by itself.

I.4.3 Examples of conjugacy classes

::::::I.4.3 a

::::::::::::::::Abelian groups

Coming back to definition (I.58), one sees that every element a of an Abelian group G is onlyconjugate to itself, since for all g ∈ G, g · a · g−1 = a. That is, the conjugacy classes of an Abeliangroup are singletons, i.e. sets with a single element.

In particular, since the cyclic group Cn is Abelian for any n ∈ N∗, each of its element forms aconjugacy class by itself.

::::::I.4.3 b

:::::::::::::::::::::Symmetric group Sn

Let σ ∈ Sn be a permutation of n objects. The conjugacy class of σ consists of the permutationsof Sn which have the same cycle structure as σ.

To prove this result, let us first write the decomposition of σ into r disjoint cycles of respectivesizes `1, . . . , `r with 1 ≤ r ≤ n, including the cycles of length 1:

σ =(i1 i2 . . . i`1

)(j1 . . . j`2

)(k1 . . . k`3

)· · · (I.61)

with `1 ≥ `2 ≥ · · · ≥ `r ≥ 1 and `1 + `2 + · · · + `r = n. In this decomposition, the n numbersi1, . . . , i`1 , j1, . . . , j`2 , · · · ∈ 1, . . . , n are all different from each other.

Any permutation τ ∈ Sn can be written as

τ =

(1 2 · · · n

τ(1) τ(2) · · · τ(n)

)=

(i1 i2 . . . i`1 j1 . . . j`2 . . .

τ(i1) τ(i2) . . . τ(i`1) τ(j1) . . . τ(j`2) . . .

)with the inverse permutation τ−1 [cf. Eq. (I.23)]

τ−1 =

(τ(i1) τ(i2) . . . τ(i`1) τ(j1) . . . τ(j`2) . . .i1 i2 . . . i`1 j1 . . . j`2 . . .

).

A straightforward calculation then gives

τστ−1 =(τ(i1) τ(i2) . . . τ(i`1)

)(τ(j1) . . . τ(j`2)

)(τ(k1) . . . τ(k`3)

)· · · , (I.62)

I.4 Conjugacy and conjugacy classes 13

i.e. a permutation with a decomposition into r cycles with respective sizes `1, . . . , `r, similar to thatof σ.

Conversely, one easily checks that two permutations of Sn having the same cycle structure areconjugate to each other — Equations (I.61) and (I.62) show how one can find a conjugating elementrelating them.

All in all, the various conjugacy classes of the symmetric group Sn are in one-to-one correspon-dence with the different integer partitions(7) of n, i.e. of the possible ways of writing n as a sum ofpositive integers (up to the order of the summands), namely as

n = `1 + `2 + · · ·+ `r with `1 ≥ `2 ≥ · · · ≥ `r ≥ 1 and `1, `2, . . . , `r ∈ N∗. (I.63)

This will now be illustrated with the cases of S2, S3 and S4.

The two permutations of S2, namely the identity — with cycle decomposition (1)(2) — and thetransposition of the two elements (1 2) have different cycle structures and thus belong to differentconjugacy classes. This is actually consistent with the fact that S2 is Abelian, so that each of itselements builds a conjugacy class by itself.

The corresponding partitions of 2 are 2 = 1 + 1 and 2 = 2.

The six permutations of S3 are distributed into three conjugacy classes. A first class consists ofthe identity permutation (1)(2)(3), in agreement with remark I.60. Then come the three permuta-tions with a single transposition of two elements, namely (1 2)(3), (1 3)(2), and (2 3)(1). Eventuallythere remain the two circular permutations (1 2 3) and (1 3 2).

The corresponding partitions of 3 are 3 = 1 + 1 + 1 and 3 = 2 + 1, and 3 = 3, respectively.

Coming now to S4, one finds five conjugacy classes, which respectively correspond to the par-titions 4 = 1 + 1 + 1 + 1, 4 = 2 + 1 + 1, 4 = 2 + 2, 4 = 3 + 1, and 4 = 4. First, the identitypermutation (1)(2)(3)(4) forms its own class. There are then 6 permutations consisting of a singletransposition: (1 2)(3)(4), (1 3)(2)(4), (1 4)(2)(3), (2 3)(1)(4), (2 4)(1)(3), and (3 4)(1)(2). Threepermutations can be decomposed into two disjoint transpositions, namely (1 2)(3 4), (1 3)(2 4), and(1 4)(2 3). With a cycle of length 3, one finds the 8 permutations (1 2 3)(4), (1 3 2)(4), (1 2 4)(3),(1 4 2)(3), (1 3 4)(2), (1 4 3)(2), (2 3 4)(1), and (2 4 3)(1). Eventually, there are 6 distinct circularpermutations of all 4 elements: (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2).

:::::::::::::::::Young diagrams

A convenient way to depict a partition of the integer n consists of a Young(e) diagram, in whichn boxes ( ) are arranged in r rows, with `s boxes on the s-th row.

Thus, the two partitions of 2 are respectively represented by

and , (I.64)

the three partitions of 3 by

, and , (I.65)

and the five partitions of 4 by

, , , , . (I.66)

(7)Note that a partition of an integer n is not the same notion as the partition of a set as defined in footnote (6).(e)A. Young, 1873–1940


Remark: The number of permutations of n elements in the conjugacy class corresponding to adecomposition with r1 cycles of length 1, r2 cycles of length 2, . . . , rn cycles of length n — wherethe ri can now take their values in 0, . . . , n — is given by

n!

r1!r2! · · · rn!1r12r2 · · ·nrn,

with the usual convention 0! = 1. Setting r1 = n and all other ri = 0, one again finds that theidentity permutation is alone in its class. In turn, with rn = 1 and all other ri = 0, one seesthat there are (n− 1)! different circular permutations of all n elements — as was indeed found forn = 2, 3, 4.

::::::I.4.3 c

:::::::::::::::::::::::::::::Three-dimensional rotations

Considering now rotations about axes going through a fixed point O of three-dimensional space,let us denote by R~n(α) the rotation through an angle α about the axis with direction ~n.

If R′ is any arbitrary rotation about an axis going through O, one checks that the compositionof transformations R′R~n(α)R′−1 is again a rotation through α, now about the axis with direction~n′ = R′~n. Conversely, one finds that two rotations through the same angle α are conjugate to eachother.

::::::I.4.3 d

::::::::::::::::::::Invertible matrices

In the group of regular real-valued resp. complex-valued n×n matrices GL(n,R) resp. GL(n,C),the conjugacy relation is the so-called matrix similarity : two similar matrices actually represent thesame linear application in two different bases.

I.5 Normal subgroupsIn the previous two sections, we have introduced two different partitions of a group G: first, byconsidering a subgroup H of G and the collection of its left cosets gH (Sec. I.3.3); then, by lookingat the conjugacy classes of G (Sec. I.4). Since the conjugacy class of the identity is always asingleton, which is not the case of the other classes if the group is non-Abelian, while all cosets ofa subgroup have the same number of elements, the conjugacy classes are in general not the cosetsof a subgroup — the exception being the case of the conjugacy classes of an Abelian group, whichare the cosets of the trivial subgroup e.

Nevertheless, one can identify specific subgroups with a well-defined behavior under conjugacy,whose cosets can be provided with a group structure.

I.5.1 Definition

Definition I.67. A subgroup N of a group G is called normal (or invariant) in G if the conjugate ofevery element of N by any element g of G is still in N , i.e. gN g−1 ⊆ N or, in less compact notation,for all n1 ∈ N and for all g ∈ G, there exists n2 ∈ N such that n2 = gn1g

−1.

Remarks:

∗ One actually even has the equality gN g−1 = N for all g ∈ G.

∗ The trivial subgroups e and G of a group G, where e is the identity element of G, are clearlyalways normal in G.

Notation: If N is a proper normal subgroup of G, one writes N G.

Property I.68. If G is an Abelian group, then all its subgroups are normal.

I.5 Normal subgroups 15

I.5.2 Quotient group

From definition I.67 follows at once the following result.

Property I.69. If N is a normal subgroup of a group G, then the left cosets of N in G equal theright cosets, i.e. gN = N g for all g ∈ G.

This property is actually the key to the proper definition an internal composition law · on theset gN | g ∈ G of the (left) cosets of N in G.

Theorem & Definition I.70. Let N be a normal subgroup of a group G. The set G/N ≡ gN | g ∈ Gof the cosets of N in G, together with the internal composition law · defined by

(gN ) · (g′N ) = (gg′)N , (I.71)

forms a group, called the quotient group (or factor group) of G by N .

Property I.72. The identity element of G/N is eN = N and the inverse element of gN in G/N isg−1N .

To prove the theorem I.70, one must first show that Eq. (I.71) indeed defines a composition lawon the set of cosets. This already requires two logical steps.

Assume first that g, g′ ∈ G are given. Then for all n1, n2 ∈ N , gn1 resp. g′n2 is in the left cosetgN resp. g′N , and the associativity of the group law on G gives (gn1)(g′n2) = g(n1g

′)n2. In thelatter right member, n1g

′ is an element of the right coset N g′, which according to property I.69equals the left coset g′N ; that is, there exists n3 ∈ N such that n1g

′ = g′n3. One may thus write(gn1)(g′n2) = g(n1g

′)n2 = g(g′n3)n2 = (gg′)(n3n2), where associativity in G was again used: n3n2

is an element of N , so that (gg′)(n3n2) ∈ (gg′)N : for all n1, n2 ∈ N , (gn1)(g′n2) ∈ (gg′)N , so thatEq. (I.71) makes sense.

To conclude on the consistency of Eq. (I.71) as a well-defined internal law, one still needs tocheck that the product is independent of the group element g chosen to label a given coset gN .That is, if g1N = g2N and g′1N = g′2N with g2 6= g1, g′2 6= g′1, then (g1N ) · (g′1N ) = (g2N ) · (g′2N ).some more needed!

Remark I.73. The order of the quotient group of G by N equals the index of N in G:

|G/N| = |G : N|.

I.5.3 Examples

As a first example, consider the group G = 0, 1, 2, 3, 4, 5 with the addition modulo 6 as grouplaw — which is in fact a representation of the cyclic group C6. Since the group is Abelian, all itssubgroups are automatically normal (property I.68), as for instance N = 0, 3. The correspondingcosets are the sets g + N with g ∈ G, namely N = 3 + N , 1 + N = 4 + N = 1, 4, and2 +N = 5 +N = 2, 5. The quotient group of G by N is thus

G/N =0, 3, 1, 4, 2, 5

and examples of group additions on G/N are 0, 3+1, 4 = 1, 4 (since 0, 3 = N is the identityelement) or 1, 4+ 2, 5 = 0, 3, meaning that 1, 4 is the inverse of 2, 5.

Next, one shows using the decomposition of permutation into transpositions that the alternatinggroup An is normal in the symmetric group Sn — for every σ ∈ Sn, σ and its inverse σ−1 have thesame parity.

Taking G = Z, with the addition of integer numbers as group law, and N = 2Z, i.e. the setof even numbers, one finds that there are two cosets 2Z itself and the set 1 + 2Z of odd numbers.


The quotient group G/N = Z/2Z thus contains two elements only — it may thus be identifiedZ2 = 0, 1.

I.5.4 Further definitions and results

Definition I.74. A group G is called a simple group if it contains no proper normal subgroup.

Remark: The classification of the finite simple groups has been achieved in 2008. For example, allcyclic groups Cp with a prime order p are simple, since they include no proper subgroup, as are thealternating groups An with n ≥ 5 (as well as those with n ≤ 3, which are Abelian).

::::::::::::::::::::Composition seriesDefinition I.75. A composition series of a group G is a finite series of the type

e = H0 H1 · · ·Hr = G (I.76)

such that each quotient group Hi+1/Hi with i ∈ 0, . . . , r − 1 is simple. The quotient groupsHi+1/Hi are called composition factors.

Remark: A series of the type (I.76), and more generally a similar sequence of subgroups of infinitelength, is called a subnormal series.

Example: The symmetric group S3 admits the alternating group A3 as proper normal subgroup,and A3 itself is simple:

eA3 S3,

where e denotes the identity permutation of three elements. Now, A3/e is (isomorphic(8) to) A3

itself, and is thus simple. In turn, S3/A3 is of order 2, i.e. will be (isomorphic to) C2 and thus againsimple.

More generally, every finite group has a composition series. However, this is not necessarily truefor infinite groups, as e.g. for (Z,+): for any proper (normal)(9) subgroup N of Z, one easily findsa proper normal subgroup N ′ of N , for instance 2N , such that N ′/N is simple, so that one cannotfind a subnormal series of finite length.

If a group has a composition series, then it is basically unique:

Theorem I.77 (Jordan(f)–Hölder(g) theorem). The composition factors Hi+1/Hi of two compositionseries (I.76) of a given group G are the same, up to a permutation of the factors.

I.6 Direct product

I.6.1 Cartesian product of groups

Theorem & Definition I.78. Given two groups G1 and G2, their direct Cartesian product G1 × G2 isthe set of ordered pairs (g1, g2) with g1 ∈ G1, g2 ∈ G2:

G1 × G2 = (g1, g2) | g1 ∈ G1, g2 ∈ G2. (I.78a)

The internal composition law · defined by

(g1, g2) · (g′1, g′2) = (g1g

′1, g2g

′2) for all g1, g

′1 ∈ G1, g2, g

′2 ∈ G2, (I.78b)

provides G1 × G2 with a group structure.(8)The notion of being ”isomorphic” will be defined in Sec. I.7.1.(9)(Z,+) is Abelian, so that all its subgroups are normal.

(f)C. Jordan, 1838–1922 (g)O. Hölder, 1859–1937

I.6 Direct product 17

Remark I.79. The identity element of G1×G2 is (e1, e2), where e1 and e2 are the respective identityelements of G1 and G2. In turn, the inverse of (g1, g2) ∈ G1 × G2 is (g−1

1 , g−12 ).

Property I.80. The subsets G1 = G1 × e2 and G2 = e1 × G2, where e1 and e2 are the respectiveidentity elements of G1 and G2, obey the following properties:

• G1 and G2 are normal subgroups of the direct product G1 × G2;

• the intersection of G1 and G2 contains a single element, namely the identity element of thedirect product, G1 ∩ G2 = e;

• every element of G1 commutes with every element of G2.

Remark: Note that the latter property holds even if the groups G1 and G2 are not Abelian.

Given s ≥ 2 groups G1,G2, . . . ,Gs, their direct Cartesian product G1×G2× · · ·×Gs follows froma straightforward generalization of definition I.78. The order of this direct product is then

|G1 × G2 × · · · × Gs| =s∏i=1

|Gi|. (I.81)

One easily generalizes the property I.80 to the Cartesian direct product of more than two groups.

I.6.2 Inner direct productDefinition I.82. A group G is called (inner) direct product group of normal subgroups N1, . . . , Nsif the following properties hold:• G = N1 · · · Ns; (I.82a)

• Ni ∩ (N1 · · · Ni−1Ni+1 · · · Ns) = e for all i ∈ 1, . . . , s, (I.82b)with e the identity element of G.

Equation (I.82b) means that every element g ∈ G can be written asg = n1n2 · · ·ns (I.83)

with ni ∈ Ni for all i ∈ 1, . . . , s.

Property I.84. Up to a permutation of the factors ni (see property I.85), the decomposition (I.83)is unique.

Property I.85. The elements of Ni commute with those of Nj for i 6= j.

Example I.86. Letm and n be coprime integers. The group (Zmn,+) contains two proper subgroupsconsisting of the multiples modulo mn of m and of those of n: N1 = 0,m, 2m, . . . , (n−1)m andN2 = 0, n, 2n, . . . , (m−1)n. As Zmn is Abelian, both N1 and N2 are normal (property I.68). Sincem and n are coprime, each number k ∈ Zmn can be written as the sum modulo mn of an elementof N1 and an element of N2.(10) The intersection of these subgroups reduces to the singleton 0.All in all, Zmn is thus the direct product of N1 and N2, which are respectively isomorphic(8) to Znand Zm.

Property I.87. If G is the direct product of normal subgroups N1 and N2, G = N1 ×N2, then N1

is isomorphic(8) to the quotient group G/N2.The converse does not hold! For instance(11) the dihedral group D3 contains a normal subgroup,consisting of the rotations through 2kπ/3 with k = 0, 1, 2, which is isomorphic to C3. The quotientgroup D3/C3, consisting of two elements, is isomorphic to C2. Yet the direct product group C2×C3,which is Abelian, cannot be isomorphic to the non-Abelian D3.(10)The proof follows from Bézout’s(h)lemma applied to the integers mk and nk, whose greatest common divisor is k.(11)If she wishes, the reader may replace D3 by the symmetric group S3 and C3 by the alternating group A3 in this

example.(h)E. Bézout, 1730–1783


I.7 Group homomorphisms

I.7.1 Definitions

Definition I.88. Let (G, · ) and (G′, ∗) be two groups. A mapping f : G → G′ is called a (group)homomorphism if it is compatible with the group structure, i.e. if

f(g1 · g2) = f(g1) ∗ f(g2) for all g1, g2 ∈ G. (I.89)

Definition I.90. A bijective homomorphism is called an isomorphism.

Definition I.91. A homomorphism from a given group G into itself is called an endomorphism of G.

Definition I.92. A bijective endomorphism of G is called an automorphism of G.

Theorem & Definition I.93. If there exists an isomorphism f : G → G′, then the groups G and G′ aresaid to be isomorphic. This defines an equivalence relation between groups, denoted G ∼= G′.

Examples:

∗ Considering on the one hand the group (Zn,+) and on the other hand the group of the n-throots of the unity (e2ikπ/n, k ∈ 0, 1, . . . , n−1,×), there is the obvious isomorphism k 7→ e2ikπ/n.

∗ For any a ∈ R, the linear mapping x 7→ ax is an endomorphism of (R,+). If a 6= 0, it is anisomorphism.

∗ The well-known property ln(xy) = lnx+ln y expresses that the logarithm ln is a homomorphismfrom the group (R∗+,×) of positive real numbers (with the multiplication as internal law) into thegroup (R,+) of real numbers with the addition as group law.Conversely, the exponential is a homomorphism from (R,+) into (R∗+,×), since ex+y = exey.

∗ In turn, the identity det(AB) = (detA)(detB) means that the determinant is a homomorphismfrom the group GL(n,C) of regular n × n matrices with complex entries on the group (C\0,×)of non-zero complex numbers.

∗ The signature ε (definition I.31) is a homomorphism from the symmetric group Sn (with n ∈ N∗)into the group (1,−1,×).

Theorem I.94 (Cayley theorem). Every (finite) group of order n ∈ N∗ is isomorphic to a subgroupof the symmetric group Sn.

To prove the theorem, one only needs to realize that each line of the Cayley table of the groupis precisely a permutation of the group elements.

I.7.2 Properties of homomorphisms

Let G and G′ be two groups with respective identity elements e and e′, and let f : G → G′be a homomorphism. One easily checks that the following results hold for the examples of grouphomomorphisms given above.

Theorem I.95. The homomorphism f maps the identity element e ∈ G to the identity elemente′ ∈ G′:

f(e) = e′. (I.95)

Theorem I.96. For every element g ∈ G, the homomorphism f maps the inverse of g to the inverseof the image f(g):

f(g−1) =(f(g)

)−1. (I.96)

I.7 Group homomorphisms 19

Theorem & Definition I.97. Denoting im f the image by the homomorphism f of the group G,

im f = f(g) | g ∈ G, (I.97)

im f is a subgroup of G′.

For instance, one may consider that the homomorphism k 7→ e2ikπ/n maps Zn into a subgroupof the group C\0.

Theorem & Definition I.98. The kernel ker f of the homomorphism f , i.e. the set of elements g ∈ Gwhich are mapped to the identity element e′,

ker f = g ∈ G, f(g) = e′, (I.98)

is a normal subgroup of G.

Example: The kernel of the signature ε : Sn → 1,−1 is the alternating group An.

Remark: Conversely, every normal subgroup of a group G can be used as kernel of a homomorphism,as exemplified in the proof of the isomorphism theorem I.101.

Theorem I.99. A homomorphism f is injective if and only if ker f = e.

Theorem I.100. More generally, the image by the homomorphism f of a subgroup H ⊂ G is asubgroup of G′, while the “preimage” by f of a subgroup H′ ⊂ G′ is a subgroup of G.

Since ker f is a normal subgroup of G, one can define a quotient group G/ ker f .

Theorem I.101 (Isomorphism theorem). If f : G → G′ is a group homomorphism, then its image isisomorphic to the quotient group G/ ker f :

im f ∼= G/ ker f. (I.101)

To prove this theorem, one has to check that the mapping F : G/K → im f , which maps thecoset gK to F (gK) = f(g) with g ∈ G is an isomorphism, where for the sake of brevity we introducethe shorter notation K = ker f .(more details later)

The results listed in this section strongly constrain the form of homomorphisms, as we nowillustrate on a simple example. Consider the homomorphisms from the symmetric group S3 (orequivalently the dihedral group D3 since they are isomorphic) into another group G. Theorem I.98already restricts the kernel of such a homomorphism f , which has to be a normal subgroup of S3,leaving only three possibilities: ker f = e (the identity permutation), the alternating group A3,or S3 itself.

If ker f = e, then f is injective, so that its image is isomorphic to S3, i.e. f is (equivalent to)an isomorphism of S3. In turn, if ker f = S3, then f is a constant mapping since its image reducesto the identity element of the target group G.

Eventually, if ker f = A3, the only proper normal subgroup of S3, then the isomorphism theo-rem I.101 tells us that the image of f is isomorphic to the quotient group S3/A3, which consists oftwo elements only. More precisely, f is necessarily (equivalent to) the signature ε.

I.7.3 Inner automorphisms

Definition I.102. Let G be a group and a ∈ G one of its elements. The automorphism φa defined by

φa :

G → G

g 7→ φa(g) = aga−1(I.103)

is called an inner automorphism of G.


Theorem I.104. The set of inner automorphisms of G is a subgroup Inn(G) of the group of auto-morphisms of G.

Proof: For all a ∈ G one has (φa)−1 = φa−1 and thus for all a, b, g ∈ G

φb (φa)−1(g) = b(a−1ga)b−1 = ba−1gab−1 = (ba−1)g(ba−1)−1 = φba−1(g),

i.e. φb (φa)−1 = φba−1 ∈ Inn(G).

Remark: Coming back to definition I.67, one sees that a normal subgroup of a group G is a subgroupwhich is invariant under Inn(G), which justifies the alternative denomination “invariant subgroup”.

CHAPTER II

Representation theory

An intro is missing

II.1 Group actionDefinition II.1. An action of a group G on a set X is a correspondence that associates to eachelement g ∈ G a mapping φg : X →X in such a way that

• for all g1, g2 ∈ G, φg1g2 = φg1 φg2 (II.1a)

• φe is the identity mapping on X . (II.1b)

Instead of φg(x) with x ∈X , one often writes gx.

Theorem & Definition II.2. The action of a group G on a set X induces an equivalence relation onthe latter: two elements x1, x2 ∈ X are equivalent if there exists g ∈ G such that x2 = gx1. Theequivalence class of an element x ∈X is called the orbit of x under the action of G.

II.2 Group representation: first definitions and resultsDefinition II.3. Let V be a vector space over a base field K. A representation of a group G on Vis a homomorphism D of G into the group of the regular operators of V to itself.

Coming back to the definition I.88 of a homomorphism, the operators D(g) with g ∈ G obey thefollowing properties:

• for all g1, g2 ∈ G, D(g1g2) = D(g1)D(g2), (II.4a)

where the composition of operators is denoted as a product;

• D(e) = 1V , (II.4b)

where e is the identity element of G and 1V the identity operator on V ; and

• for all g ∈ G, D(g−1) = D(g)−1, (II.4c)

which explains why the operators D(g) have to be regular.

Remark: Comparing with definition II.1, one sees that a group representation is an instance of groupaction on the vector space V .

Definition II.5. The vector space V is the representation space and its dimension dim V is calledthe dimension or degree if the representation.

Accordingly, one talks of an s-dimensional representation, often abbreviated “s-rep”.

Definition II.6. A representation is said to be real resp. complex when the base field of the vectorspace is K = R resp. K = C.

22 Representation theory

Definition II.7. If the mapping G → D(G) is an isomorphism, then the representation D is calledfaithful

Definition II.8. A linear representation is a representation whose operators D(g) with g ∈ G areall linear, i.e. D(g) ∈ GL(V ) for all g ∈ G, where GL(V ) precisely denotes the (“General Linear”)group of regular linear operators on V .

Definition II.9. A representation is called unitary resp. orthogonal if• K = C resp. K = R;

• V is a Hilbert(i) space, i.e. in particular a vector space with a scalar product;

• the operators D(g) with g ∈ G belong to the group U(V ) of unitary operators resp. to thegroup O(V ) of orthogonal operators on V , i.e. im D ⊂ U(V ) resp. im D ⊂ O(V ).(12)

In the case of a finite-dimensional linear representation, one can choose a basis on V to representthe operators D(g) as non-singular square matrices — which we shall denote D(g) —, resulting ina matrix representation of the group. Denoting dim V = s, D(g) ∈ GL(s,K) for all g ∈ G, and thegroup law is mapped to the product of matrices.

Eventually, let us quote two results:

Theorem II.10. If H is a subgroup of a group G, then every representation of G is also a represen-tation of H.

The proof follows at once from the existence of a natural homormorphism (the “canonical in-jection”) H → G which maps g on itself, and from the fact that the composition of two grouphomomorphisms is again a homomorphism.

Theorem II.11. For every representation D of an Abelian group G, D(g1)D(g2) = D(g2)D(g1) forall g1, g2 ∈ G.

II.3 Examples of group representations

II.3.1 Trivial representation

Theorem & Definition II.12. For any group G, the mapping which maps every element g ∈ G to theidentity operator 1C of C is a representation of G, called the trivial representation.

Since C is a one-dimensional complex vector space, the trivial representation is one-dimensional.For any scalar product ( · , · ) on C, the identity

(1C(z), 1C(z′)

)= (z, z′) holds for all z, z′ ∈ C —

by definition, 1C(z) = z —, so that the trivial representation is also unitary.In any basis of the representation space C, the operator 1C is represented by the (unitary!) 1×1-

matrix (1), which explains why one often says that the trivial representation “maps every groupelement to 1”.

When the group G contains more than one element, the trivial representation is not faithful.

II.3.2 Representations of the dihedral group D3

Let us now consider the non-Abelian dihedral group D3, i.e. the group of the geometrical trans-formations that leave an equilateral triangle invariant, which is isomorphic to the symmetric groupS3. We call A, B, C the corners of the triangle and introduce for further convenience a set of(12)A unitary operator U resp. orthogonal operator O is an operator that preserves the scalar product ( , ) on V ,

i.e. ∀x ∈ V ,(U(x), U(x)

)= (x, x) resp.

(O(x), O(x)

)= (x, x).

(i)D. Hilbert, 1862–1943

II.3 Examples of group representations 23

orthonormal basis vectors in the plane of the triangle, taking the origin of coordinates at the centerof the triangle and A along the x-direction (Fig. II.1).

A

B

C

O ~ex

~ey

Figure II.1

The elements of the group are

• the identity transformation, which we denote by Id;

• the three reflections with respect to the medians of the triangle, we which denote (A B),(A C), (B C) according to which pair of points they exchange;

• the rotations about O through 2π/3 and 4π/3, respectively denoted (A B C) and (A C B).

Note that the notations reflect the isomorphism between the geometrical transformations and thepermutations of the three corners.

We now introduce three different matrix representations of D3.

::::::II.3.2 a

:::::::::::::::::::::::Trivial representation

As discussed in Sec. II.3.1, one may represent all six group elements by the same matrix

D (1)(g) = (1) ∀g ∈ D3, (II.13)

which provides a non-faithful, unitary one-dimensional representation of D3.

::::::II.3.2 b

:::::::::::::::::::::Sign representation

Another one-dimensional unitary representation of D3 consist of mapping the identity transfor-mation and the two rotations to the matrix (1):

D (ε)(Id) = D (ε)((A B C)

)= D (ε)

((A C B)

)= (1) (II.14a)

and the three reflections to the matrix (−1):

D (ε)((A B)

)= D (ε)

((A C)

)= D (ε)

((B C)

)= (−1). (II.14b)

By noticing that this is the same thing as considering the signature ε of the associated permutations,which is a homomorphism from S3

∼= D3 to the group 1,−1, one checks that this is indeed arepresentation, called the sign, signature or alternating representation.

Remark: This representation can be introduced for any of the symmetric groups Sn. For n ≥ 2 it“differs from” — that is, anticipating on a notion introduced in Sec. II.4, it is not equivalent to —the trivial representation.

::::::II.3.2 c

::::::::::::::::::::::::::Standard representation

Introducing Cartesian coordinates in the plane of the triangle, one can associate to each cornerA, B, C its coordinates (xA, yA), (xB, yB), (xC , yC), and then represent each element of D3 by the2 × 2-matrix that performs the proper transformation of these coordinates. Accordingly we definesix matrices

D (s)(Id) =

(1 0

0 1

)(II.15a)


D (s)((A B)

)=

(−1

2

√3

2√3

212

), D (s)

((A C)

)=

(−1

2 −√

32

−√

32

12

), D (s)

((B C)

)=

(1 0

0 −1

)(II.15b)

and

D (s)((A B C)

)=

(−1

2 −√

32√

32 −1

2

), D (s)

((A C B)

)=

(−1

2

√3

2

−√

32 −1

2

). (II.15c)

One can check that these unitary matrices indeed form a group, which is isomorphic to D3, andthus constitutes a faithful, two-dimensional unitary representation of D3.

II.3.3 Three-dimensional rotations

As last example (for the moment), let us consider the group of three-dimensional rotations,where a typical rotation will be denoted R. The example will also help us illustrate why group rep-resentations are of paramount importance in physics and how/where they play a role in a quantum-mechanical problem (§ II.3.3 b).

::::::II.3.3 a

::::::::::::::::::::::::::Standard representation

An obvious matrix representation of the group of three-dimensional rotations is that involving“rotation matrices”, i.e. real 3× 3-matrices R such that RRT = RTR = 13 and with determinant 1.These matrices constitute a group called SO(3).

::::::II.3.3 b

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Representations of the group of rotations in a quantum-mechanical problem

Consider a physical system Σ, whose position with respect to a fixed reference frame is describedby a vector ~r. In a quantum-mechanical context, and using the language of wave mechanics, thestate of Σ is represented by a wave function ψ(~r), where for the sake of simplicity we omit any timedependence.

Let us rotate the system Σ, so that its new position is now described by a vector ~r ′ = R~r.The state of the system is now represented by a new wave function ψ′(~r), such that the probabilitydensity |ψ′(~r)|2 has been rotated with respect to |ψ(~r)|2.

Remark: The “physical” rotation R of the system, say through an angle α about direction ~n, whichis considered here is a so-called active transformation. An alternative point of view consists inleaving Σ at the same place, while using a new reference frame, rotated about the same direction ~nthrough the opposite angle −α, i.e. corresponding to R−1: this is a passive transformation. Bothpoints of view lead to the same mathematical equation (II.16).

Now, intuitively the new wave function ψ′ is such that it has at the “rotated” position ~r ′ thesame value as the old wave function ψ at the original position ~r, i.e. (13)

ψ′(~r ′) = ψ(~r) with ~r ′ = R~r.

Inverting the transformation between ~r and ~r ′, we have ~r = R−1~r ′ and thus

ψ′(~r ′) = ψ(R−1~r ′),

valid for all ~r ′. In this equation, ~r ′ is only a dummy variable appearing on both sides, which canalso be written ~r, leading to

ψ′(~r) = ψ(R−1~r). (II.16)

Summarizing, the rotation ~r → ~r ′ = R~r of the system induces a transformation

ψ → ψ′ = D(R)ψ (II.17)(13)In truth, the only statement which can be made is that the probability density at the new position |ψ′(~r ′)|2 should

be equal to that at the old position |ψ(~r)|2.

II.3 Examples of group representations 25

of the wave function, where ψ′ obeys Eq. (II.16). The operator D(R) acts on the functional spaceof wave functions, and D is a representation of the group of rotations on that functional space.

The latter is in general infinite-dimensional, yet by introducing a properly chosen (Hilbert)basis, one can in practical cases work in a finite-dimensional subspace, i.e. with a finite-dimensionalrepresentation of the group of rotations.

As an illustration, consider a quantum-mechanical problem involving a single particle, and thusonly one position-vector ~r, in a central potential V (~r) = V (r) with r ≡ |~r|. A stationary solution ofthe problem is an energy-eigenfunction, which may be written after introducing spherical coordinatesas ψn`m(~r) = Rn`(r)Y`m(θ, ϕ) with Rn` a problem-specific radial function and Y`m a sphericalharmonic.

If the system is rotated, it still remains in a stationary state, described by a new wave functionψn′`′m′(~r). Let us discuss the relation of the “new” quantum numbers n′, `′,m′ to the original onesn, `,m.

• The principal quantum number n and the orbital quantum number ` ∈ N are related tothe energy and the squared angular-momentum operator ~L 2 of the system, which are bothindependent of the spatial orientation. Accordingly, a rotation will leave n and ` unchanged,i.e. n′ = n, `′ = `.

• In turn, the magnetic quantum number m ∈ −`,−`+ 1, . . . , `− 1, ` quantifies the amountof angular momentum Lz along a given direction, i.e. bears some relation to the orientationof the system: under a rotation, m will be modified.

Generally, an energy eigenfunction ψ′n`m′ for the rotated system with well-defined magneticquantum number m′ will be a linear combination of the energy eigenfunctions ψn`m of the non-rotated system with the same values of n and `, yet with different m′:

ψ′n`m′ =∑m=−`

D(`)m′m(R)ψn`m.

The coefficients D(`)m′m(R) in this linear combination depend on the rotation R, and they are actually

the matrix elements of a representation of the rotation group on the 2`+1-dimensional space spannedby the ψn`m with fixed n and `.

We shall come back to these representations of the rotation group in further detail later

II.3.4 Direct sum and tensor product of representations

Let D (α), D (β) be two linear representations of the same group G on respective vector spacesV (α), V (β).

::::::II.3.4 a

::::::::::::::::::::::::::::::::Direct sum of representations

Theorem & Definition II.18. The mapping D (α⊕β) from G into the group of linear operators of thedirect-sum space V (α) ⊕ V (β) to itself, such that for all g ∈ G, xα ∈ V (α), xβ ∈ V (β)

D (α⊕β)(g)(xα + xβ) = D (α)(g)(xα)⊕ D (β)(g)(xβ) (II.18)

is a (linear) representation of the group G, called the direct sum representation.

The dimension of the direct-space representation is the sum of the dimensions of the individualrepresentations, namely dim V (α) + dim V (β).

If the representations D (α), D (β) are finite-dimensional, by choosing a basis of V (α) ⊕ V (β)

consisting of the union of a basis of V (α) and a basis of V (β), one obtains a matrix representationof the operators D (α⊕β)(g) of the block form


D (α⊕β)(g) =

D (α)(g) 0

0 D (β)(g)

, (II.19)

with D (α)(g), D (β)(g) matrix representations.

One generalizes in a straightforward way the definition to that of the direct sum of r represen-tations D (1), D (2), . . . , D (r), for which one introduces the somewhat more compact notation

r⊕i=1

D (i) = D (1) ⊕ D (2) ⊕ · · · ⊕ D (r). (II.20)

In case some of the representations appear several times in the direct sum, one even writess⊕i=1

aiD(i) = D (1)⊕ · · · ⊕ D (1)︸︷︷︸

a1 copies

⊕ · · · ⊕ D (s)⊕ · · · ⊕ D (s)︸︷︷︸as copies

. (II.21)

::::::II.3.4 b

::::::::::::::::::::::::::::::::::::Tensor product of representations

Theorem & Definition II.22. The mapping D (α⊗β) from G into the group of linear operators of thetensor-product space V (α) ⊗ V (β) to itself, such that for all g ∈ G, xα ∈ V (α), xβ ∈ V (β)

D (α⊗β)(g)(xα + xβ) = D (α)(g)(xα)⊗ D (β)(g)(xβ) (II.22)

is a (linear) representation of the group G, called the tensor product representation.

The dimension of the tensor-product representation is the product of the dimensions of theindividual representations, namely (dim V (α))(dim V (β)).

In the case of finite-dimensional representations D (α), D (β) with respective matrix representa-tions D (α), D (β) in bases e(α)

i , e(β)j , the tensor-product representation admits a matrix repre-

sentation D (α⊗β) on the basis e(α)i ⊗ e

(β)j of the tensor-product space such that for all g ∈ G(

D (α⊗β)(g))

(i,j),(k,l)=(D (α)(g)

)i,k

(D (β)(g)

)j,l

for all i, j, k, l. (II.23)

That is, D (α⊗β)(g) is the Kronecker(j) product of the matrices D (α)(g) and D (β)(g).

II.4 Classifying representationsThe two operations introduced in Sec. II.3.4 mean that the set of possible representations of agiven group G is infinitely large. To bring some order into that set, we now introduce two notions,diminishing the number of “independent” representations (Sec. II.4.1) and highlighting the role of“elementary building blocks” of specific representations (Sec. II.4.2).

II.4.1 Equivalent representations

Definition II.24. Two matrix representations D (α), D (β) of the same dimension of a given group Gare called equivalent if there exists an invertible matrix P such that for every element g ∈ G,

D (α)(g) = PD (β)(g)P−1. (II.24)

(j)L. Kronecker, 1823–1891

II.4 Classifying representations 27

That is, the matrices D (α)(g) and D (β)(g) are similar for all g ∈ G, and the similarity matrix Pis the same for all pairs

(D (α)(g),D (β)(g)

). Accordingly, for every g ∈ G the linear operators D (α)(g)

and D (β)(g) are the same, yet represented in different bases of the representation space, with thesame basis transformation for all g ∈ G.

Example: For instance, the two-dimensional rotation about a given point through an angle θ maybe equivalently represented by the 2× 2-matrices(

cos θ sin θ

− sin θ cos θ

)and

(cos θ − sin θ

sin θ cos θ

).

Both representations are equivalent, since for all θ ∈ [0, 2π] one has(cos θ sin θ

− sin θ cos θ

)=

(1 0

0 −1

)(cos θ − sin θ

sin θ cos θ

)(1 0

0 −1

)−1

.

“Being equivalent” obviously defines an equivalence relation on the set of representations of agiven group, which is compatible with the group structure: for all g, g′ ∈ G, one has

D (α)(gg′) = PD (β)(gg′)P−1 = PD (β)(g)D (β)(g′)P−1,

where we used the group property of the matrices D (β)(g). Introducing an identity matrix writtenas 1 = P−1P , one then finds

D (α)(gg′) = PD (β)(g)P−1PD (β)(g′)P−1 = D (α)(g)D (α)(g′),

as it should be.Instead of the representations, one investigates their equivalence classes, which means that

two equivalent (matrix) representations are identified — which reduces the number of “different”representations.

Remark: Two one-dimensional representations of a group are equivalent if and only if they are indeedequal, since Eq. (II.24) is then an equality between numbers, which commute with each other.

II.4.2 Reducible and irreducible representations

Definition II.25. A representation D of a group G is called reducible if there exists at least oneproper subspace of the representation space which is invariant(14) under all operators D(g) withg ∈ G. A representation which is not reducible is said to be irreducible.

Remark: In the literature, one often finds “irrep” instead of “irreducible representation”.

Property II.26. A representation of dimension 1 is automatically irreducible, since the representa-tion space admits no proper subspace.

When the reducible representation has a finite dimension n1 + n2, with n1 the dimension of theinvariant subspace, the corresponding matrices take the form

D(g) =

D (1)(g) A(g)

0n2×n1 D (2)(g)

(II.26)

(14)A subspace W of a vector space V is said to be invariant under a linear operator O on V if O(W ) ⊂ W , i.e. if forall x ∈ W , O(x) ∈ W .


in a basis whose first n1 vectors are a basis of the invariant subspace. D (1)(g) resp. D (2)(g) is an1 × n1- resp. n2 × n2-matrix, A(g) a n1 × n2-matrix, and 0n2×n1 denotes the zero n2 × n1-matrix.

The block product of two matrices D(g) and D(g′) of the form (II.26), with g, g′ ∈ G, gives

D(g)D(g′) =

D (1)(g)D (1)(g′) D (1)(g)A(g′) +A(g)D (2)(g′)

0n2×n1 D (2)(g)D (2)(g′)

,

i.e. is again of the same form. On the other hand, the group law gives D(g)D(g′) = D(gg′) with

D(gg′) =

D (1)(gg′) A(gg′)

0n2×n1 D (2)(gg′)

.

Comparing both expressions, one finds

D (1)(g)D (1)(g′) = D (1)(gg′) , D (2)(g)D (2)(g′) = D (2)(gg′),

which expresses the fact that D (1) and D (2) are smaller-dimensional (matrix) representations of G.

If D (1) and/or D (2) are themselves reducible, one can pursue the reduction in every invariant sub-space, until one is only left with irreducible representations. The matrix representation eventuallytakes the “block triangular” form

D(g) =

D (1)(g) A(1)(g)

0 D (2)(g) A(2)(g)

0 0 D (3)(g) A(3)(g)

0 0 0. . .

...

0 0 0 0 D (r)(g)

for all g ∈ G, (II.27)

with D (1), . . . , D (r) irreducible matrix representations.

An even more favorable situation is when the latter are in fact fully decomposable, which wenow discuss.

Definition II.28. A representation is called fully reducible or decomposable if to each invariantsubspace there corresponds an invariant complementary subspace.

In that case, by choosing on the representation space a basis consisting in the union of basesof complementary subspaces, one can bring the matrices of the representation — if it is finite-dimensional — in a block-diagonal form:

II.4 Classifying representations 29

D(g) =

D (1)(g) 0

0 D (2)(g) 0

0 0 D (3)(g) 0

0 0 0. . . 0

0 0 0 0 D (r)(g)

for all g ∈ G, (II.29)

with D (1), . . . , D (r) irreducible matrix representations.Comparing with Eq. (II.19), one sees that a decomposable representation is thus the direct sum

of irreducible representations.

Accordingly, an important topic we shall deal with in the following chapters is the search for theirreducible representations of a given group. In addition, we shall learn how to decompose a fullyreducible representation into irreducible representations.

Remark II.30. As illustrated in the following example, the reducibility of a representation maydepend on the choice of the base field K of the representation space.

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Example: Representations of the group of two-dimensional rotations

Consider the rotations about a fixed origin O in the (y, z)-plane. Let Rθ denote the rotationthrough a given angle θ ∈ [0, 2π].

Viewing Rθ as a three-dimensional rotation about the x-axis, one may represent it with the helpof the 3× 3-matrix

D (3)(Rθ) =

1 0 0

0 cos θ sin θ

0 − sin θ cos θ

, (II.31)

i.e. with an element of the matrix group SO(3).Physically, it is clear that both the (x, y)-plane and the z-axis, i.e. complementary subspaces,

are invariant under all rotations Rθ, so that the three-dimensional representation D (3) should befully reducible. This is also clear when inspecting the matrix (II.31), which may be recast as

D (3)(Rθ) =

1 0 0

0 cos θ sin θ

0 − sin θ cos θ

,

i.e. in the generic form (II.29) with D (1)(Rθ) a 1× 1-matrix and D (2)(Rθ) a 2× 2-matrix.Since D (1)(Rθ) = (1) for every rotation Rθ, D (1) is actually the trivial representation. In turn,

the matrices

D (2)(Rθ) =

(cos θ sin θ

− sin θ cos θ

), (II.32)

of the matrix group SO(2) provide a faithful representation of the group of two-dimensional ro-


tations. This representation is irreducible on R, yet considered on C it is reducible since thematrix (II.32) can then be diagonalized as

D (2′)(Rθ) =

(eiθ 0

0 e−iθ

), (II.33)

i.e. with complementary invariant subspaces, on which we have two inequivalent one-dimensionalrepresentations D (1′)(Rθ) = (eiθ) and D (1′)∗(Rθ) = (e−iθ).

II.5 Schur’s lemmaAn important theorem, which is used to derive a lot of results on the irreducible representations offinite and compact groups, is that known as Schur’s(k) lemma:

Theorem II.34 (Schur’s lemma).(1) Let D , D ′ be two irreducible representations of a group G on respective vector spaces V , V ′. LetT : V → V ′ be a linear operator such that(15)

T D(g) = D ′(g) T for all g ∈ G. (II.34a)

Then either T = 0, or T is an isomorphism.(2) Let D be a complex irreducible representation of a group G on a finite-dimensional vector spaceV . If T : V → V is a linear operator such that

T D(g) = D(g) T for all g ∈ G, (II.34b)

then T is a homothety, i.e. there exists λ ∈ C such that T = λ1V .

In this lemma, 0 denotes the zero operator, which maps every vector x ∈ V to the zero vectorof the target space, while 1V is the identity operator on V .

To prove the first part of Schur’s lemma, let us note that Eq. (II.34a) implies two properties.First, the kernel of T is invariant by all operators D(g) with g ∈ G:

Let x ∈ ker T and g ∈ G. Then D ′(g) T (x) = D ′(g)(T (x)

)= D ′(g)(0V ′) = 0V ′ , with 0V ′ the

zero vector of V ′. Eq. (II.34a) then gives T D(g)(x) = 0V ′ = T(D(g)(x)

), i.e. D(g)(x) ∈ ker T .

In turn the image of T is invariant by all operators D ′(g):

Let g ∈ G and x′ ∈ im T : there exists x ∈ V such that x′ = T (x). Applying Eq. (II.34a) to thevector x gives

T D(g)(x) = D ′(g) T (x) i.e. T(D(g)(x)

)= D ′(g)

(T (x)

)= D ′(g)(x′),

which shows that D ′(g)(x′) is indeed an element of im T .

Now, since the representations D resp. D ′ is by assumption irreducible, ker T resp. im T has tobe an improper subspace of V resp. V ′. If ker T is the whole vector space V , then T is the zerooperator (and im T reduces to the zero vector of V ′). Otherwise, it means that ker T reduces tothe zero vector of V , i.e. T is injective. Since im T is an improper subspace, and differs from 0V ′,it has to be V ′ itself, i.e. T is also surjective, and thereby bijective, i.e. it is an isomorphism. 2

Remark: If V and V ′ have the same dimension, then the existence of a bijective linear operator Tobeying Eq. (II.34a) means that D and D ′ are equivalent.

(15)T is called an intertwining operator .(k)I. Schur, 1875–1941

II.5 Schur’s lemma 31

Let us now prove the second part of Schur’s lemma II.34. Since it now deals with operators ona complex finite-dimensional vector space, the characteristic polynomial det(T − λ1V ) admits atleast one root λ ∈ C, i.e. an eigenvalue of T to which corresponds at least a non-zero eigenvector.Subtracting −λD(g) = −λ1V D(g) = D(g)

(−λ1V

)from Eq. (II.34b) yields(

T − λ1V

) D(g) = D(g)

(T − λ1V

)for all g ∈ G,

from where follows that the kernel of T − λ1V is invariant under all operators D(g) with g ∈ G.Since the representation D is irreducible, ker(T −λ1V ) is either reduced to the zero vector, which isexcluded by the fact that λ is by construction eigenvalue to T , or the whole vector space V , whichprecisely means T = λ1V . 2

A first, important consequence of Schur’s lemma is the following result.

Corollary II.35. The only irreducible finite-dimensional complex representations of an Abelian groupare its representations of dimension one.

The operators of any representation D of an Abelian group G commute with each other: fixingg′ ∈ G, D(g)D(g′) = D(g′)D(g) for all g ∈ G. If D is a finite-dimensional complex irrep., theoperator D(g′) obeys Eq. (II.34b) and the assumptions of the second part of Schur’s lemma arefulfilled, so that D(g′) is an homothety — and this holds for all g′ ∈ G. Since every subspace ofa vector space is invariant by an homothety, D can only be irreducible if it is of dimension one,so that the representation space has no proper invariant space. 2

To finish this Section, let us give without proof a partial converse to Schur’s lemma:

Theorem II.36. Let D be a fully reducible finite-dimensional representation of a group G on a vectorspace V . If the only linear operators on V that commute with all operators D(g) with g ∈ G are thehomotheties of V , then D is irreducible.

CHAPTER III

Representations of finite groups

In this chapter, we focus on the case of the representations of finite groups. We first argue in Sec. III.1that such a representation is always decomposable, which displaces the focus towards irreduciblerepresentations. For the latter, we first derive in Sec. III.2 a set of relations. These are thenexploited in Sec. III.3 to constrain a number of properties of the unitary irreducible representations,in particular their characters. In Sec. III.4, we discuss how one can determine which irreduciblerepresentations, and with which multiplicity, enter the decomposition of a given representation. Inparticular, we introduce the regular representation of a finite group as well as the notion of thegroup algebra. Eventually, Section III.5 deals with the special case of the representations of thesymmetric groups Sn.

III.1 Full reducibility of the representationsWe first state a theorem which is valid not only for finite groups, but also for infinite ones.

Theorem III.1. Every reducible finite-dimensional unitary representation of an arbitrary group G isdecomposable.

Let D be a unitary representation on a finite-dimensional vector space V and ( · , · ) denote thescalar product on V for which D is unitary. If D is reducible, then there exists a proper subspaceW of V which is invariant under all operators D(g) with g ∈ G. Let W ⊥ denote the orthogonalcomplement of W ; since V is finite-dimensional, W ⊥ is a complementary subspace of W in V . Weshall show that W ⊥ is invariant under all operators D(g) with g ∈ G, thereby proving the theorem.

For all g ∈ G, x ∈ W ⊥ and y ∈ W , consider the scalar product(D(g)(x), y

). Invoking the

unitarity property of D(g−1) = D(g)−1, one has(D(g)(x), y

)=(x, D(g)−1(y)

)=(x, D(g−1)(y)

).

From the invariance of W under D , i.e. in particular under D(g−1), follows D(g−1)(y) ∈ W , so thatthe rightmost scalar product in the above equation is zero. Thus

(D(g)(x), y

)= 0 for all y ∈ W ,

i.e. D(g)(x) ∈ W ⊥, which shows the invariance of W ⊥ under D(g). 2

Theorem III.2. Any representation of a finite group is equivalent to a unitary representation.

Let D be a representation of a finite group G on the vector space V , on which a scalar product( · , · ) has been defined. We shall first introduce a second scalar product 〈 · , · 〉 on V , for whichevery operator D(g) with g ∈ G will be unitary. Then we shall find a representation D ′ equivalentto D and such that the operators D(g) are actually also unitary for the scalar product ( · , · ).

For x, y ∈ V , the formula

〈x, y〉 ≡ 1

|G|∑g∈G

(D(g)(x), D(g)(y)

)(III.3)

defines a mapping which is bilinear and symmetric in x and y, and which is positive definite sincethis is the case of the scalar product ( · , · );it is thus a scalar product (or inner product) on V . For

III.1 Full reducibility of the representations 33

every g ∈ G and x, y ∈ V , one has⟨D(g)(x), D(g)(y)

⟩=

1

|G|∑g′∈G

(D(g′)(D(g)(x)), D(g′)(D(g)(y))

)i.e. ⟨

D(g)(x), D(g)(y)⟩

=1

|G|∑g′∈G

(D(g′) D(g)(x), D(g′) D(g)(y)

).

The group property of the operators D(g) gives D(g′) D(g) = D(g′g). In addition, at fixed g ∈ G,a sum over all g′ ∈ G is equivalent to a sum over all g′′ = g′g ∈ G (group rearrangement theorem I.7),leading eventually to⟨

D(g)(x), D(g)(y)⟩

=1

|G|∑g′′∈G

(D(g′′)(x), D(g′′)(y)

)= 〈x, y〉.

This identity, valid for all g ∈ G and x, y ∈ V , means that the operator D(g) is unitary for thescalar product 〈 · , · 〉.

Consider now two bases ei and e′j of the space V , which are respectively orthonormal forthe scalar products ( · , · ) resp. 〈 · , · 〉. There exist a non-singular linear operator M such thatM(ei) = e′i for all i — this is precisely the linear operator corresponding to the transformationmatrix of the change of basis. Now, using the definitions of M and of the orthonormal bases, onecan write for all i, j

〈M(ei), M(ej)〉 = 〈e′i, e′j〉 = δij = (ei, ej)

with δij the usual Kronecker symbol. The leftmost and rightmost terms in this equation are bilinear,which leads for every x, y ∈ V to ⟨

M(x), M(y)⟩

= (x, y). (III.4)

Let us define a new representation D ′ of G by D ′(g) = M−1 D M for all g ∈ G: by constructionD ′ is equivalent to D . For all g ∈ G and x, y ∈ V one has(

D ′(g)(x), D ′(g)(y))

=(M−1 D(g) M(x), M−1 D(g) M(y)

)=⟨D(g) M(x), D(g) M(y)

⟩where we used Eq. (III.4). Using the unitarity of D(g) for the scalar product 〈 · , · 〉 then yields(

D ′(g)(x), D ′(g)(y))

=⟨M(x), M(y)

⟩.

Eventually, invoking again Eq. (III.4), one obtains(D ′(g)(x), D ′(g)(y)

)= (x, y),

i.e. D ′(g) is unitary for the scalar product ( · , · ). 2

Remark: It is interesting to note where the finiteness of the group was used in the proof. The factor1/|G| in the definition (III.3) is actually irrelevant for the reasoning, as one easily checks. What isreally crucial is the possibility to sum over the elements of the group: when we turn to compactinfinite groups, this sum will be replaced by an integral, yet this is not possible for non-compactinfinite groups.

Combining the theorems III.1 and III.2 leads to Maschke’s(l) theorem:

Theorem III.5 (Maschke’s theorem). All finite-dimensional reducible representations of a finitegroup are decomposable.

Remark III.6. One can also show that every irreducible representation of a finite group is in factfinite-dimensional.

All in all, we see that to find all representations of a finite group G, it is sufficient to find itsunitary irreducible representations, which is the topic of the next two Sections.(l)H. Maschke, 1853–1909

34 Representations of finite groups

III.2 Orthogonality relationsTheorem III.7. Let D , D ′ be two irreducible representations of a finite group G on respective vectorspaces V , V ′. Given a linear operator φ : V → V ′, one defines an operator V → V ′ by

TG(φ) ≡ 1

|G|∑g∈G

D ′(g) φ D(g−1) (III.7)

• If D and D ′ are not equivalent, then TG(φ) = 0.

• If D = D ′, then TG(φ) is a homothety: TG(φ) =Tr φ

dim V1V .

In this theorem, 0 denotes the zero operator, which maps every vector x ∈ V to the zero vectorof the target space V ′, while 1V is the identity operator on V .

Proof: One first checks that TG(φ) D(g) = D ′(g) TG(φ) for all g ∈ G:

TG(φ) D(g) =1

|G|∑g′∈G

D ′(g′) φ D(g′−1) D(g) =1

|G|∑g′∈G

D ′(g′) φ D(g′−1g),

where we used the group law of the operators D(g). Now g′−1g = (g−1g′)−1 and we writeD ′(g′) = D ′(gg−1g′) = D ′(g)D ′(g−1g′):

TG(φ) D(g) =1

|G|∑g′∈G

D ′(g)D ′(g−1g′) φ D((g−1g′)−1)

= D ′(g)1

|G|∑g′∈G

D ′(g−1g′) φ D((g−1g′)−1).

Letting g′′ ≡ g−1g′, g′′ runs over the whole group G when g′ does at fixed g, which gives theannounced result TG(φ) D(g) = D ′(g) TG(φ).

Invoking Schur’s lemma II.34 gives that either TG(φ) = 0, namely precisely when the tworepresentations are inequivalent, or TG(φ) = λ1V with some λ ∈ C, which we can computewhen D = D ′. Writing that the right hand side of Eq. (III.7) with D ′ = D equals λ1V andcomputing the trace gives, with the linearity of the trace and Tr 1V = dim V ,

λ dim V =1

|G|∑g∈G

Tr[D(g) φ D(g−1)

]=

1

|G|∑g∈G

Tr[D(g−1) D(g) φ

],

where we used the cyclicity of the trace. The sum on the right hand side of this equation is just|G| times the trace Tr φ, which gives λ = Tr φ/|dim V |. 2

Corollary III.8. Let D , D ′ be two irreducible matrix representations of a finite group G.

• If D and D ′ are not equivalent, then1

|G|∑g∈G

[D ′(g)]ij [D(g−1)]kl = 0 ∀i, j, k, l. (III.8a)

• If D = D ′, then1

|G|∑g∈G

[D(g)]ij [D(g−1)]kl =δilδjkdim V

∀i, j, k, l. (III.8b)

Proof: In matrix representation, the Eq. (III.7) reads

[TG(φ)]il =1

|G|∑g∈G

∑j′,k′

[D ′(g)]ij′ [φ]j′k′ [D(g−1)]k′l ∀i, l (III.9)

where the indices i, j′, k′, l takes values between 1 and the relevant vector-space dimension.Choosing for φ the operator whose matrix elements are [φ]j′k′ = δjj′δkk′ , where j and k arearbitrary, and applying the theorem III.7 yields the results (III.8a)–(III.8b). 2

III.3 Irreducible representations of a finite group 35

Let us from now on label the irreducible representations of a given finite group G — or moreaccurately, the various equivalence classes of irreducible representations of G — with a superscriptbracketed Greek letter: D (α), D (β), . . . . The corresponding representation spaces will be similarlydenoted V (α), V (β), and so on.

Using this notation, both equations (III.8a)–(III.8b) can be gathered into a single one:

1

|G|∑g∈G

[D (α)(g)

]ij

[D (β)(g−1)

]kl

=δαβδilδjk

dim V (α)∀i, j, k, l (III.10a)

known as (Schur’s) orthogonality relations, where the Kronecker symbol δαβ equals 1 if the repre-sentations are the same, and 0 if they are inequivalent.

Invoking theorem III.2, we may assume that the irreducible matrix representations we are dealingwith are unitary. In that case, using D (β)(g−1) = D (β)(g)−1, which follows from the group law, onehas

D (β)(g−1) = D (β)(g)†,

where † denotes the Hermitian conjugate matrix, i.e. for the matrix elements[D (β)(g−1)

]kl

=[D (β)(g)

]∗lk∀k, l.

The orthogonality relations can thus be recast as

1

|G|∑g∈G

[D (α)(g)

]ij

[D (β)(g)

]∗lk

=δαβδilδjk

dim V (α)∀i, j, k, l. (III.10b)

In the following Section we shall exploit these relations to derive constraints obeyed by theirreducible representations of a finite group.

III.3 Irreducible representations of a finite group

III.3.1 First consequences of the orthogonality relations

Theorem & Definition III.11. Let G be a finite group. The expression(ψ,ϕ

)G ≡

1

|G|∑g∈G

ϕ(g)ψ(g)∗ (III.11)

with ϕ,ψ : G → C defines a scalar product on the space of complex-valued functions on G.

Proof: One easily checks that the expression on the right-hand side is sesquilinear, positive, anddefinite.

Remark: We used the physicists’ convention for sesquilinearity, namely linearity resp. antilinearityin the second resp. first argument.

At fixed i and j resp. k and l, the matrix elements[D (α)(g)

]ijand

[D (β)(g)

]∗lk

are the values ing ∈ G of complex-valued functions

[D (α)

]ij,[D (β)

]lk

on G, and the term on the left hand side of theorthogonality relation (III.10b) is the scalar product of these functions as defined in Eq. (III.11):([

D (β)]lk,[D (α)

]ij

)G =

δαβδilδjk

dim V (α)∀i, j, k, l. (III.12)

Assume first α = β, i.e. identical representations, so that δαβ = 1 and the four indices i, j, k, lcan each take dim V (α) different values. Unless i = l and j = k, the scalar product of

[D (α)

]ij

and


[D (α)

]lk

vanishes, i.e. the functions are orthogonal to each other. As i, j ∈ 1, . . . ,dim V (α) vary,there are (dim V (α))2 such mutually orthogonal functions.

Considering now another irreducible representation D (β) with β 6= α, the (dim V (β))2 corre-sponding functions

[D (β)

]ijare likewise mutually orthogonal, but relation (III.12) with δαβ shows,

that they are also orthogonal to the functions[D (β)

]ij. Repeating the argument for all (classes

of) inequivalent irreducible representations of the group G, one finds that their respective matrixelements provide a set of ∑

irreps α

(dim V (α))2

pairwise orthogonal complex-valued functions on G. Now, such a function is defined by the |G|values it can take, so that the space of these functions is a vector space on C of dimension |G|. Ona |G|-dimensional vector space, there can be at most |G| mutually orthogonal vectors. All in all, thedimensions of the various inequivalent irreducible representations of a finite group G must obey theinequality ∑

irreps α

(dim V (α))2 ≤ |G|. (III.13)

An important consequence of this inequality is that the number of inequivalent irreduciblerepresentations of a finite group is finite, and their dimensions are bounded from above.

Example: In Sec. II.3.2 we introduced three representations of the dihedral group D3: the trivialrepresentation D (1), the sign representation D (ε), and the standard representation D (s). The readercan check that the latter is actually irreducible (also in C), as are the first two ones since they areone-dimensional. We have thus found three representations with respective dimensions 1, 1 and 2,for a group with 6 elements, thereby saturating the inequality (III.13).

Using the regular representation of G, we shall show in Sec. III.4.2 that in fact the equalityalways holds in Eq. (III.13):

Theorem III.14. The sum of the squares of the dimensions of a complete set of inequivalent irre-ducible representations of a finite group |G| equals the order of the group:

∑irreps α

(dim V (α))2 = |G|. (III.14)

III.3.2 Character theory

:::::::III.3.2 a

:::::::::::::::::::::::::::::::Definition and first properties

Definition III.15. Let D be a finite-dimensional complex representation of a group G. The characterof D is the complex-valued function χD on G defined by

χD(g) = Tr[D(g)

]for all g ∈ G. (III.15)

Property III.16. For all g, h ∈ G, the cyclicity of the trace gives χD(hgh−1) = χD(g), i.e. thecharacter of a representation D is constant on a conjugacy class of the group G. Accordingly, χD isreferred to as a class function.

Definition III.17. A character χD is called simple or irreducible if the corresponding representationD is itself irreducible. In contrast, the character of a reducible representation is called a compoundcharacter .


Going back to the relevant definitions, one easily proves the following properties of characters:

• Equivalent representations D , D ′ of a group have the same character χD = χD ′ .

• The character of a direct-sum representation D ⊕ D ′ is the sum of the respective charactersof D and D ′:

χD⊕D ′ = χD + χD ′ . (III.18)

• More generally, if D =⊕α

aαD (α), then

χD =∑α

aαχD(α) . (III.19)

In particular, if the representations D (α) are irreducible, this identity shows that a compoundcharacter is a linear combination (with positive integer coefficients) of simple characters.

• The character of a tensor-product representation D ⊗ D ′ is the product of the respectivecharacters of D and D ′:

χD⊗D ′ = χD · χD ′ , (III.20)

where the dot stands for the multiplication in C of the values taken by the characters.

• The characters of a representation D and its conjugate D∗ are conjugate to each other:

χD∗ =(χD

)∗. (III.21)

• If D is a unitary representation, then

χD(g−1) = χD(g)∗ for all g ∈ G. (III.22)

Property III.23. If D is a one-dimensional representation of a group G, then the complex numbersχD(g) should obey the group law, i.e. χD(gg′) = χD(g)χD(g′) for all g, g′ ∈ G.

:::::::III.3.2 b

:::::::::::::::::::::::::::::Orthogonality of characters

Let us consider again the orthogonality relation (III.10b). If we multiply the term on the lefthand side with δij and sum over all possible values of i and j, we are actually computing the traceof D (α)(g). According to definition III.15, this trace is precisely the value in g of the character ofD (α), which we shall denote χ(α). Similarly, one can also trace the relation (III.10b) over the indicesk and l, which yields

1

|G|∑g∈G

χ(α)(g)χ(β)(g)∗ =δαβ

dim V (α)

∑i,j,k,l

δilδjkδijδkl.

The term on the left hand side is precisely the scalar product (III.11) of the characters χ(α) andχ(β). In turn, the sum appearing on the right hand side is readily computed and equals dim V (α),canceling the similar factor present in the denominator. All in all, one thus finds

(χ(β), χ(α)

)G =

1

|G|∑g∈G

χ(α)(g)χ(β)(g)∗ = δαβ, (III.24)

which means that the characters of inequivalent irreducible representations of a given group G areorthogonal — and in fact orthonormal — for the scalar product (III.11).

For every element g ∈ G, let us denote [g] the conjugacy class of g in G, and∣∣[g]∣∣ the number of

elements in this conjugacy class. Writing a sum over all elements of G as a sum over all conjugacyclasses of the sums over all elements in a class, and taking into account the constancy of the


characters over a conjugacy class, the orthogonality relation of characters (III.24) can be recast inthe equivalent form

1

|G|∑[g]

∣∣[g]∣∣χ(α)([g])χ(β)([g])∗ = δαβ, (III.25)

where we somewhat loosely use the same notations χ(α), χ(β) for functions whose argument is nowa conjugacy class [g].

Let r be the number of conjugacy classes in G. Viewing∣∣[g]∣∣1/2χ(α)([g]) and

∣∣[g]∣∣1/2χ(β)([g]) as

r-dimensional vectors, Eq. (III.25) states that these are orthogonal if α 6= β, i.e. for inequivalentirreducible representations. Repeating a similar reasoning as in Sec. III.3.1 where we now use thefact that the pairwise orthogonal vectors

∣∣[g]∣∣1/2χ(α)([g]) belong to an r-dimensional vector space,

we can conclude that there exists at most r inequivalent irreducible representations of G. In fact,one can even show the following:

Theorem III.26. The number of inequivalent irreducible representations of a finite group G is equalto the number of conjugacy classes in G.

Examples:∗ Since the dihedral group D3 contains 3 conjugacy classes, the three irreducible representationsintroduced in Sec. II.3.2 constitute a whole set of inequivalent irreducible representations of thegroup.

∗ In the case of an Abelian group G, there are as many conjugacy classes as elements in the group.According to theorem (III.26), there are thus |G| inequivalent irreducible representations of G. Tofulfill the inequality (III.13), these irreps. must all be one-dimensional, which provides a furtherdemonstration of the corollary II.35.

:::::::III.3.2 c

::::::::::::::::Character table

To characterize the irreducible representations of a finite group G, one uses a character table,which lists the characters of the various (classes of) irreducible representations. More precisely, thecolumns of the table correspond to the different conjugacy classes of the group, while on each rowstands a different irreducible representation:

conjugacy classes[g1] [g2] . . . [gr]

χ(1) χ(1)(g1) χ(1)(g2) . . . χ(1)(gr)

χ(2) χ(2)(g1) χ(2)(g2) . . . χ(2)(gr)

......

......

...

χ(r) χ(r)(g1) χ(r)(g2) . . . χ(r)(gr)

Table III.1 – Character table of a group.

For instance, Table III.2 shows a character table for the group D3, using the irreducible represen-tations D (1) D (ε), D (s) introduced in Sec. II.3.2. In the first row, [(A B)] stands for the conjugacyclass of the reflexions, consisting of the three elements (A B), (A C), and (B C); in turn, [(A B C)]is the conjugacy class of the rotations through 2π/3 or 4π/3, i.e. [(A B C)] = (A B C), (A C B.

When constructing the table III.2, we already knew irreducible representations of D3 and thuswe only had to compute the respective traces to find the table entries. Even without explicitlyknowing irreducible representations, one can often still determine the character table of a finitegroup |G|, by exhausting the available information:


[Id] [(A B)] [(A B C)]

χ(1) 1 1 1

χ(ε) 1 −1 1

χ(s) 2 0 −1

Table III.2 – Character table of the dihedral group D3.

i. The number of rows and columns of the table is given by theorem (III.26), i.e. it is the numberof conjugacy classes of the group.

ii. The (squared) dimensions of inequivalent irreducible representations D (α) of the group shouldobey Eq. (III.14).(16)

iii. Assuming that the first column of the table corresponds to the conjugacy class of the identityelement e of G, then the entries in that column are precisely the representation dimensionsdim V (α) since χ(α)(e) = Tr 1V (α) = dim V (α).

iv. There is always at least a one-dimensional irreducible representation, namely the trivial rep-resentation. All entries in the corresponding line — which it is convenient to list first — aresimply 1.

v. If there are further one-dimensional irreducible representations, their respective charactersobey property III.23.

vi. The orthogonality relation (III.25) should be fulfilled for every pair of representations, i.e.every pair of rows of the table.

Let us rediscover the character table of D3 by applying this set of recipes. We assume thata preliminary study of the group has concluded to the existence of r = 3 conjugacy classes [Id],[(A B)], and [(A B C)], with respectively 1, 3, and 2 elements. According to point i., the charactertable should have 3 rows and 3 columns.

Denoting dα the representation dimensions, these should according to point ii. be a solution ofthe equation (d1)2 + (d2)2 + (d3)2 = |D3| = 6: up to a relabeling of the irreducible representations,the only possible solution is d = 1 = d2 = 1, d3 = 2.

Invoking then points iii. and iv., we can already construct the following table, in which fourentries are still missing.

[Id] [(A B)] [(A B C)]

χ(1) 1 1 1

χ(2) 1 · ·

χ(3) 2 · ·

The second line can be filled with the help of point v.: noting that two successive applications ofa symmetry gives the identity, e.g. (A B)2 = Id, property III.23 gives χ(2)((A B))2 = χ(2)(Id) = 1,i.e. χ(2)((A B)) = ±1. In turn, one checks that the composition of a reflexion and a rotation givesa reflexion with respect to an other axis, e.g. (A B)(A B C) = (B C), which gives

(16)Such an equation∑α(dα)

2 = n where all dα and n are (positive) integers is called a Diophantine equation.


χ(2)((A B))χ(2)((A B C)) = χ(2)((B C)).

From χ(2)((A B)) = χ(2)((B C)), both reflexions being in the same conjugacy class, and the factthat this character is either 1 or −1, i.e. non-zero, one deduces χ(2)((A B C)) = 1. This at onceleaves only the possibility χ(2)((A B)) = −1, otherwise the first and second lines of the table wouldbe identical:

[Id] [(A B)] [(A B C)]

χ(1) 1 1 1

χ(2) 1 −1 1

χ(3) 2 · ·

The third line can be dealt with entirely by using the orthogonality relation (III.25), as indicatedin point vi. More precisely, one must have∣∣[Id]

∣∣χ(3)([Id])χ(β)([Id])∗ +∣∣[(A B)]

∣∣χ(3)([(A B)])χ(β)([(A B)])∗

+∣∣[(A B C)]

∣∣χ(3)([(A B C)])χ(β)([(A B C)])∗ = 0

for β = 1 or 2, i.e. using∣∣[Id]

∣∣ = 1,∣∣[(A B)]

∣∣ = 3,∣∣[(A B C)]

∣∣ = 2 and the already known entries ofthe first column

2 + 3χ(3)([(A B)])χ(β)([(A B)])∗ + 2χ(3)([(A B C)])χ(β)([(A B C)])∗ = 0.

Denoting x = χ(3)((A B)), y = χ(3)((A B C)) the two missing characters, one obtains a system oftwo linear equations with two unknowns:

2 + 3x+ 2y = 0

2− 3x+ 2y = 0.

The obvious solution is x = 0, y = −1, leading eventually to the table III.2.

III.4 Reduction of a representation

III.4.1 Decomposition of a representation into irreducible representations

Let D (α) be a set of inequivalent unitary irreducible representations of a finite group G andD a finite-dimensional representation of G. Following Maschke’s theorem III.5, D is fully reducible,and the equivalence class of each irreducible representations in its decomposition contains one ofthe D (α) (theorem III.2). Accordingly, one writes

D =⊕

irreps. α

aαD (α) with aα ∈ N, (III.27)

where the integer coefficients aα signal the possible multiple appearance of the irreducible represen-tation D (α) in the direct sum, as in Eq. (II.21).

Remark: The equal sign in Eq. (III.27) in fact stands for an equivalence. Similarly, this equivalenceis used to (re)order the irreducible representations according to the (here unspecified) index α.

Let χ(α) denote the character of the irreducible representation D (α) and χD that of the repre-sentation D . Applying both sides of Eq. (III.27) to an element g of G, which yields an equalitybetween operators, and building the trace of the latter, one obtains thanks to property (III.19)

III.4 Reduction of a representation 41

χD(g) =∑

irreps. α

aαχ(α)(g) for all g ∈ G. (III.28)

That is, a compound character is a linear combination with positive integer coefficients of simplecharacters.

This identity is the key to finding the coefficients aα of the decomposition. Invoking theorthogonality relation (III.24) of the irreducible characters, one has

aα =(χ(α), χD

)G =

1

G∑g∈G

χD(g)χ(α)(g)∗. (III.29)

Proof: denoting β the dummy summation index for irreps. in Eq. (III.28) and using the linearityof the scalar product (III.11) in its second argument, one has(

χ(α), χD

)G =

∑irreps. β

aβ(χ(α), χ(β)

)G =

∑irreps. β

aβδαβ = aα. 2

Theorem III.30 (Criterion for irreducibility). Let χD denote the character of a representation D ofa finite group G. Then D is irreducible if and only if the scalar product

(χD , χD

)G equals 1.

Proof: Let D (α) be a set of inequivalent unitary irreducible representations of G and

D =⊕

irreps. α

aαD (α)

with positive integer coefficients aα be the decomposition of D into irreps. Using Eq. (III.28),the sesquilinearity of the scalar product (III.11), and the orthonormality relation (III.24), onehas (

χD , χD

)G =

∑irreps. α,β

a∗αaβ(χ(α), χ(β)

)G =

∑irreps. α,β

a∗αaβδαβ =

∑irreps. α

|aα|2.

Since every aα is a positive integer, this also holds for |aα|2, and the sum on the right hand sideequals 1 if and only if all coefficients aα vanish but a single one, which has to be equal one —i.e. precisely if D is an irreducible representation. 2

III.4.2 Regular representation

:::::::III.4.2 a

:::::::::::::::Group algebra

::::::::Algebra

Let us first define a new algebraic structure, the algebra, which we shall need both in the presentsection and more generally in the study of continuous groups in Chapter V.

Definition III.31. An algebra is a vector space A on which a bilinear operation ? mapping A×Ainto A is defined with the following properties:

• for all x, y, z ∈ A, (x+ y) ? z = x ? z + y ? z; (A1)

• for all x, y, z ∈ A, x ? (y + z) = x ? y + x ? z; (A2)

• for all x, y ∈ A, λ ∈ K, (λx) ? y = λ(x ? y) = x ? (λy), (A3)

where K is the base field of the vector space. One often specifies “algebra over the field K”.

Remarks:

∗ The internal law ? is often referred to as the “vector product” or the “multiplication” on thealgebra.


∗ Thanks to the bilinearity, it is sufficient to know the products ei ? ej of the vectors of a basis eiof the algebra to fully determine the multiplication:

if x =∑i

xiei, y =∑j

yjej then x ? y =∑i,j

xiyjei ? ej . (III.32)

∗ Denoting 0A the zero vector of the algebra A, one deduces from Eq. (A3) that 0A is an absorbingelement for ?, i.e.

for all x ∈ A , x ? 0A = 0A ? x = 0A. (III.33)

Definition III.34. An algebra A is called associative if for all x, y, z ∈ A, x ? (y ? z) = (x ? y) ? z.

Definition III.35. An algebra A is called commutative if for all x, y ∈ A, x ? y = y ? x.

Definition III.36. An algebra is called unital (or unitary) if it has an identity element with respectto the operation ?.

Examples:

∗ The set M (n,K) of n × n matrices with entries in K, with the usual matrix product as multi-plication, is a unital associative algebra.

∗ The set R3 of three-dimensional vectors, with the usual cross product × as vector product, is analgebra over R, which neither associative nor commutative (~a ×~b = −~b ×~a!), nor has an identityelement.

∗ The linear operators on a Hilbert space, with the commutator as multiplication, form an algebra.

Definition III.37. The group algebra of a finite(17) group G over a field K is the algebra, denotedKG, consisting of elements of the form

x =∑g∈G

xgg , with xg ∈ K, (III.38a)

with the addition and multiplication respectively defined for x =∑g∈G

xgg and y =∑

g∈G ygg by

x+ y =∑g∈G

(xg + yg)g and x ? y =∑h∈G

∑g,g′∈Ggg′=h

(xgyg′)h. (III.38b)

Remarks:

∗ Equation (III.38a) is to be understood as meaning that the group elements form a basis of thegroup algebra. Thus if G is finite, then KG is of dimension |G|.

∗ In turn, the vector product ? defined in Eq. (III.38b) is induced by the group law: if the latterreads gg′ = h for g, g′, h ∈ G, then in KG one has g ? g′ = h. Accordingly, one quickly drops thesymbol ? when denoting the (vector) product of elements of the group algebra.

Property III.39. The group algebra KG is associative.

Property III.40. The identity element e of the group G is also an identity element of the groupalgebra KG.

Property III.41. If the group G is Abelian, then the group algebra KG is commutative.

These results follow at once from the properties of the group law.

(17)If the group is infinite, the sums in Eqs. (III.38a)–(III.38b) should run over only a finite number of terms.


:::::::III.4.2 b

::::::::::::::::::::::::::::::::::::::::::Regular representation and its reduction

Theorem & Definition III.42. Let G be a finite group and K a field. The mapping D (r) : G → Aut(KG)which to an element g ∈ G associates the linear operator D (r)(g) on the group algebra KG such thatfor all h ∈ G ⊂ KG

D (r)(g)(h) = gh (III.42)

is a representation of G on KG, called the (left) regular representation.

Denoting g1, . . . , gn the elements of G, where n = |G|, the matrix of the operator D (r)(g) in thebasis gi1≤i≤n of the group algebra is such that D (r)(g)

g1

...gn

=

gg1...ggn

. (III.43)

Since g is one of the basis vectors, the matrix D (r)(g) contains a single 1 and n− 1 times 0 on eachline and on each column. More precisely, the matrix representing the group identity element e is the|G|-dimensional identity matrix, D (r)(e) = 1|G|, as is always the case; while for g 6= e, the mappingh→ gh is a derangement of the group elements, leaving none of them invariant, so that the matrixD (r)(g) has only 0 on its diagonal.

Accordingly, denoting χ(r) the character of the regular representation, the latter obeys

χ(r)(e) = |G| (III.44a)

χ(r)(g) = 0 for g 6= e. (III.44b)

Let D (α) be a set of inequivalent unitary irreducible representations of G on respective vectorspaces V (α), with characters χ(α). The reduction (III.27) of the regular representation reads

D (r) =⊕

irreps. α

a(r)α D (α)

with integer coefficients a(r)α given by Eq. (III.29), namely

a(r)α =

(χ(α), χ(r)

)G =

1

G∑g∈G

χ(r)(g)χ(α)(g)∗.

Using Eqs. (III.44), the sum on the right hand side of the previous equation actually only involvesa single non-zero summand for g = e, namely χ(r)(e)χ(α)(e)∗ = |G|χ(α)(e)∗ = |G|

(dim V (α)

)∗, i.e.a(r)α = dim V (α).

We thus have proven the following theorem:

Theorem III.45. The regular representation D (r) of a finite group G contains each irreducible rep-resentation D (α) of the group(18) with a multiplicity equal to the dimension of D (α):

D (r) =⊕

irreps. α

(dim V (α)

)D (α). (III.45)

Taking the trace of the reduction (III.45) yields

χ(r) =∑

irreps. α

(dim V (α)

)χ(α) (III.46)

for the characters. Applying this equality between functions to the group identity element e, using(18)More accurately, it contains an element of each equivalence class of irreducible representations.


the fact that the character of e is the dimension of the representation space, gives

|G| =∑

irreps. α

(dim V (α)

)2,

which proves theorem III.14.

Let us now sketch how one can find subspaces of the group algebra KG which are invariant underthe regular representation, thus reducing D (r). More precisely, we search for “minimal” invariantsubspaces Ii, i.e. such that they admit no proper subspace which is itself invariant. The ideasintroduced here will be illustrated in Sec. III.5.

Let g ∈ G and y ∈ Ii. Decomposing y on the basis of KG consisting of the group elements, onefinds D (r)(g)(y) = g ?y. The invariance of Ii under D (r) thus means that for all g ∈ G and all y ∈ Ii,g ? y ∈ Ii. Since this holds for every group element g and given that Ii is a vector space, any linearcombination of vectors g ? y with g ∈ G is again in Ii, i.e. x ? y ∈ Ii, which one also expresses asx ? Ii ⊂ Ii for all x ∈ KG.(19)

Assume that one has found elements ei of the group algebra KG fulfilling the following set ofrequirements:

i. for all i, ei ? ei = ei, which defines an idempotent element;

ii. for all i 6= j, ei ? ej = 0 where 0 denotes the zero vector of KG;

iii. each ei is primitive, i.e. cannot be expressed as the sum e′1 + e′2 of two idempotents obeyingrequirement ii.;

iv. denoting Ii = KG ? ei = x ? ei, x ∈ KG, the various Ii are proper subspaces of KG whosedirect sum

⊕i Ii equals KG;

Then one easily checks that the subspaces Ii thus defined are invariant under the regular represen-tation and actually provide a reduction in irreducible representations.

The key idea is to recognize that the identity element e of the group can be written in the forme = e1 + e2 + · · · and to use x ? e = x for all x ∈ KG.

The reduction of the regular representation can thus be brought down to the search for primitiveidempotents ei of the group algebra, which will “generate” the invariant subspaces Ii.Remarks:∗ A single primitive idempotent can generate a subspace Ii = KG ? ei of dimension greater thanone. That is, there will in general be less than |G| primitive idempotents, so that they do notconstitute a basis of KG.

∗ The notation ei is traditional for the primitive idempotents defined here, as is the notation eifor basis vectors of a vector space! In these lecture notes the distinction is made by denoting theprimitive idempotents in italic and the basis vectors in roman font, but the author can guaranteethat this is not a universal convention.

∗ An equivalent formulation to that adopted here is to talk of projectors pi on mutually orthog-onal subspaces of the group algebra: a projector is indeed idempotent (pi pi = pi) and orthogonalprojectors obey pi pj = pj pi = 0, where 0 now denotes the zero operator, which is clearlyreminiscent of requirements i. and ii.

∗ In practical cases, it might be less time-consuming to search for primitive idempotents “up to afactor”, i.e. such that ei ? ei = λiei with λi ∈ K, λi 6= 0K, since they generate the same invariantsubspaces.

(19)A subset I of an algebra A obeying x ? I ⊂ I for all x ∈ A is called a left ideal of A, hence the notation Ii.


III.4.3 Reduction of a tensor-product representation

Let D and D ′ be two representations of a finite group G with respective characters χ and χ′.The tensor-product representation D ⊗ D ′ was introduced in § II.3.4 b.

Theorem III.47. The tensor-product representation D ⊗ D ′ is a representation of the direct-productgroup G × G. If D and D ′ are irreducible, so is D ⊗ D ′.

Since G is (isomorphic to) a subgroup of G × G, the tensor-product representation D ⊗ D ′

is also a representation of G. Even if D and D ′ are irreducible, yet D ⊗ D ′ can in general bereducible (although theorem III.50 below provides a generic exception). Introducing a set D (α)of inequivalent unitary irreducible representations of G, one may thus write [cf. Eq. (III.27)]

D ⊗ D ′ =⊕

irreps. α

aαD (α) with aα ∈ N. (III.48a)

This decomposition is called the Clebsch(m)–Gordan(n) series of the tensor-product representations.Following Eq. (III.29), the integer coefficients aα are given by

aα =(χ(α), χD⊗D ′

)G =

1

G∑g∈G

χD⊗D ′(g)χ(α)(g)∗,

where χD⊗D ′ denotes the character of D ⊗D ′. Under consideration of Eq. (III.20) comes

aα =(χ(α), χDχD ′

)G =

1

G∑g∈G

χD(g)χD ′(g)χ(α)(g)∗. (III.48b)

Example III.49. Consider once again the dihedral group D3. Let D (s⊗ s) ≡ D (s) ⊗ D (s) denote thetensor product of the two-dimensional, irreducible standard representation with itself, and χ(s⊗ s)

the corresponding character. Since χ(s⊗ s) = χ(s)χ(s) [Eq. (III.20)], one finds the following characters:

[Id] [(A B)] [(A B C)]

χ(s⊗ s) 4 0 1

Comparing with the character table III.2 of the group, one finds χ(s⊗ s) = χ(1) + χ(ε) + χ(s), i.e.D (s⊗ s) = D (1) ⊕ D (ε) ⊕ D (s). The other possible tensor products of irreducible representations ofD3, involving either D (1) or D (ε), are taken care of by the following theorem.

Theorem III.50. The tensor product of an irreducible representation of a group G with a represen-tation of dimension 1 is irreducible.

::::::::::::::::::::::::::::::Clebsch–Gordan coefficients

Let D (γ) be a set of inequivalent irreducible representations of a finite group G. For each γ,we denote d(γ) the dimension of D (γ).

Given two irreducible representations D (α) and D (β) on respective vector spaces V (α) and V (β)

with basesψ

(α)i

i=1,...,d(α)

andφ

(β)j

j=1,...,d(β)

, let

D (α) ⊗ D (β) =⊕

irreps. γ

aαβγ D (γ) with aαβγ ∈ N (III.51)

be the decomposition (III.48a) of the tensor-product representation in irreducible representations.The vectors

ψ

(α)i ⊗ φ

(β)j

form a basis of the representation space V (α)⊗ V (β) of D (α) ⊗ D (β).

(m)A. Clebsch, 1833–1872 (n)P. Gordan, 1837–1912


Since the latter is decomposable, one can find an alternative basis Ψ(γ),nγk that is better “adapted”

to the decomposition (III.51) in the following sense. The set of new basis vectors can be partitionedinto disjoint subsets, labeled by γ and nγ ∈ 1, . . . , aαβγ — if the corresponding integer aαβγ in theClebsch–Gordan series is at least equal to 2, otherwise nγ is irrelevant. For given γ and nγ , there ared(γ) basis vectors Ψ

(γ),nγk with k ∈ 1, . . . , d(γ), which span of a subspace of V (α)⊗ V (β) invariant

under (one of) the representation D (γ) in the decomposition (III.51).The basis change between

ψ

(α)i ⊗ φ

(β)j

and Ψ(γ),nγ

k is of the form

Ψ(γ),nγk =

∑i,j

Cγ,nγ ;kα,β;i,j ψ

(α)i ⊗ φ

(β)j , (III.52)

where the sum over i resp. j runs from 1 to d(α) resp. d(β). The (complex) coefficients Cγ,nγ ;kα,β;i,j are

called Clebsch–Gordan coefficients. These are usually normalized to obey the relation∑i,j

∣∣Cγ,nγ ;kα,β;i,j

∣∣2 = 1 (III.53)

for all γ, nγ and k ∈ 1, . . . , d(γ). This choice ensures that the basis Ψ(γ),nγk is orthonormal for

the “natural” scalar product on V (α)⊗ V (β) if this is the case of the basesψ

(α)i

and

φ

(β)j

.

Definition III.54. A group is called simply reducible if the integer coefficient aαβγ in the Clebsch–Gordan series (III.51) is either 0 or 1 for all α, β, γ, that is when the tensor product of any twoirreducible representations decomposes into different irreducible representations.

An example is provided by the dihedral group D3, as can be seen from example III.49. In thecase of a simply reducible group, the label nγ is unnecessary in Eqs. (III.52)–(III.53).

III.5 Representations of the symmetric group Sn

Let us now investigate the representations of the group Sn of the permutations of n ∈ N∗ objects,using the ideas sketched in § III.4.2 b. For definiteness, we shall assume that the group algebra, onwhich the regular representation operates, is a complex vector space, thus to be denoted CSn. Themultiplication on CSn is (built from) the composition of permutations: for σ, τ ∈ Sn, τ ? σ = τ σ,which will be more shortly written τσ.

Gathering first a few results obtained in previous sections, we know that the group Sn is of order|Sn| = n! (property I.19) and that its conjugacy classes are in one-to-one correspondence with theinteger partitions of n, which are conveniently depicted with the help of Young diagrams (§ I.4.3 b).

III.5.1 Irreducible representations of SnAccording to § III.4.2 b, the reduction of the regular representation of Sn can be done using

primitive idempotents, i.e. adequate linear combinations of the group elements σ ∈ Sn that gener-ate invariant subspaces of CSn. In this specific case, the primitive idempotents are called Youngelements. Two of them (for n ≥ 2(20)) are easily found, namely

the symmetrizer : S ≡ 1

n!

∑σ∈Sn

σ (III.55a)

the antisymmetrizer : A ≡ 1

n!

∑σ∈Sn

ε(σ)σ, (III.55b)

with ε(σ) the signature of the permutation σ. Using the identities τS = S and τA = ε(τ)A forall τ ∈ Sn, one indeed checks that S and A are idempotent and that their product is 0 (the zero(20)The reduction of S1 is straightforward: it has a single irrep., the trivial representation.

III.5 Representations of the symmetric group Sn 47

vector of the group algebra). Additionally, one sees that both elements generate one-dimensionalinvariant subspaces, corresponding to the trivial and the signature (or alternating) representations,respectively. For S2, these are the only two irreducible representations.

For n ≥ 3, there are further irreducible representations, corresponding to other Young elements.We shall just state a few results regarding those, without proof, and illustrate them on the examplesof S3 or S4.(21) For that purpose, let us first introduce two definitions.

:::::::III.5.1 a

::::::::::::::::Young tableaux

Definition III.56. A Young tableau of order n is a Young diagram with n boxes in which the numbers1, . . . , n have been written in any arbitrary order.

Definition III.57. A Young tableau is called standard if the numbers occur in increasing order ineach row and in each column.

For instance, the following are four Young tableaux of order 7 corresponding to the same Youngdiagram; only the third and fourth tableaux are standard:

3 1 8 74 26

,1 3 6 74 52

,1 3 6 72 45

,1 2 3 45 67

.

As further examples, we list in Fig. III.1 resp. III.2 all standard Young tableaux with 3 resp. 4boxes, arranging them according to the Young diagrams.

• 1 2 3 • 1 23

, 1 32

•123.

Figure III.1 – Standard Young tableaux for S3.

• 1 2 3 4 • 1 2 34

, 1 2 43

, 1 3 42

• 1 23 4

, 1 32 4

•1 234

,1 324

,1 423

•1234

.

Figure III.2 – Standard Young tableaux for S4.

To the Young diagrams (of any order n) with a single row or a single column, there correspondsonly one standard Young tableau, respectively. In contrast, when there are at least two rows orcolumns in the diagram, there are several standard tableaux. To know the number of the latterwithout enumerating them, let us introduce the hook length of a given box in a Young diagram,which is the number of boxes (including itself) either directly to its right or directly below. Weillustrate this definition on an example in Fig. III.3, where the number in each box of the rightdiagram is precisely the hook length of the box.

Hook lengths for the diagram :6 4 2 13 11

Figure III.3 – Example of hook lengths (right) for the Young diagram depicted left.(21)Note that the irreducible representations of S3, which is isomorphic to D3, were actually already introduced in

Sec. II.3.2.


Using the hook lengths, the number of standard Young tableaux for a given Young diagram oforder n is given by the hook length formula:

dα =n!∏

(hook lengths), (III.58)

where the product in the denominator runs over all boxes of the diagram, while the subscript αmerely labels that diagram. For the example of Fig. III.3, one would find 7!/(6 ·4 ·2 ·1 ·3 ·1 ·1) = 35,different standard Young tableaux, which the reader probably does not want to enumerate. Onequickly checks that the formula (III.58) gives the right results for the diagrams with only one rowor only one column, as well as for those presented in Figs. III.1–III.2.

:::::::III.5.1 b

::::::::::::::::::::::::::::::::::::::::::::::::::::Young elements; irreducible representations of Sn

For each standard Young tableau of order n, corresponding to a Young diagram (α), one definesan element of the group algebra CSn as follows. Let P resp. Q denote the subset of Sn consistingof the permutations that leave every row resp. every line of the standard Young tableau invariant.Denoting

P ≡∑σ∈P

σ and Q ≡∑σ∈Q

ε(σ)σ,

both elements of CSn, their productY = QP (III.59)

is again an element of CSn which is a primitive idempotent up to a factor: Y is a Young element.For instance, the standard Young diagram built from the Young diagram with only a single rowresp. a single column leads to the symmetrizer (III.55a) resp. to the antisymmetrizer (III.55b).

Remark: P resp. Q is sometimes called the row resp. column stabilizer of the standard Youngtableau under consideration. Both are actually subgroups of Sn.

To each Young diagram (α) correspond dα standard tableaux, with dα given by Eq. (III.58),which lead to as many Young elements. In turn, each of these Young elements generates (by leftmultiplication with the group algebra) a subspace of dimension dα that is invariant under the regularrepresentation. More precisely, each Young diagram (α) leads to dα copies of the same irreduciblerepresentation, namely of a representation of dimension dα — which we shall naturally denote D (α).

Going to the standard Young diagrams with 3 boxes as given in Fig. III.1, this means weshould find 4 Young elements S , Y , Y ′, A , respectively. S and A generate two one-dimensionalinvariant spaces with different irreducible representations, which we already know. In turn, Y andY ′ generate two invariant spaces of dimension 2, corresponding to two copies of the same irreduciblerepresentation — which is the standard representation of § II.3.2 c.

Consider for example the standard Young diagram 1 23

. The corresponding row stabilizer isP = Id, (1 2) and the column stabilizer Q = Id, (1 3), leading to

P = Id + (1 2) and Q = Id− (1 3),

leading to the Young element Y = [Id − (1 3)][Id + (1 2)], that is, after factorizing out andperforming the products

Y = Id + (1 2)− (1 3)− (1 2 3).

The left multiplication of Y with each element of S3 gives (somewhat tediously)

Id Y = Y (III.60a)(1 2 3) Y = (1 2 3) + (1 3)− (2 3)− (1 3 2) (III.60b)

(1 3 2) Y = (1 3 2) + (2 3)− (1 2)− Id = −Y − (1 2 3) Y (III.60c)(1 2) Y = (1 2) + Id− (1 3 2)− (2 3) = Y + (1 2 3) Y (III.60d)

(2 3) Y = (2 3) + (1 3 2)− (1 2 3)− (1 3) = −(1 2 3) Y (III.60e)(1 3) Y = (1 3) + (1 2 3)− Id− (1 2) = −Y . (III.60f)


All 6 products can thus be expressed as linear combinations of Y and (1 2 3) Y , which showsthat the subspace generated by multiplying Y left with CS3 is of dimension 2.

For every element σ of S3, one can also compute the product σ(1 2 3) Y : using Eqs. (III.60),one can then construct a 2× 2-matrix representing σ in the basis consisting of Y and (1 2 3) Y ,i.e. precisely a two-dimensional representation of S3.

Eventually, note that Eqs. (III.60) allow one to compute Y 2 = 3Y , which shows that Y isindeed idempotent up to a factor, as well as S Y = A Y = 0.

Regarding S4, the standard Young diagrams depicted in Fig. III.2 lead to two representationsof dimension 1 — the trivial and alternating representations —, two inequivalent irreducible repre-sentations of dimension 3 — which are both contained three times in the regular representation —,and one irreducible representation of dimension 2, contained twice in the regular representation.

:::::::III.5.1 c

::::::::::::::::::::::::::::Conjugate Young diagrams

Definition III.61. Given a Young diagram of order n, the conjugate diagram (or transposed diagram)is that obtained by exchanging the roles of the rows and columns. A self-conjugate diagram is thena Young diagram which coincides with its conjugate diagram.

Examples:∗ Considering the Young diagrams of order 4, the diagrams

and resp. and

form two pairs of conjugate diagrams, while the diagram is self-conjugate.

∗ More generally, for any n the diagrams consisting of a single row and that consisting of a singlecolumn, corresponding respectively to the Young symmetrizer and antisymmetrizer, are conjugateto each other.

Coming to the irreducible representations associated to the standard Young tableaux corre-sponding to conjugate diagrams — which according to the hook length formula will have the samedimension —, there exist a handful of results, two of which we now quote.

Theorem III.62. Let D (α) be the irreducible representation of Sn associated with a given Youngdiagram of order n ≥ 2. The irreducible representation associated with the conjugate Young diagramis obtained by multiplying D (α) with the alternating representation.

The fact that the Young symmetrizer generates the trivial representation and the Young anti-symmetrizer the alternating representation is a special case of that theorem.

Corollary III.63. In a self-conjugate irreducible representation, the characters of the odd permuta-tions are zero.

III.5.2 Product of irreducible representations of the symmetric group

Consider two irreducible representations D(α)(n) of Sn and D

(β)(p) of Sp, where n and p are not

necessarily equal. The corresponding Young diagrams will be denoted (α) and (β). One can definean “outer product” of these representations, or equivalently of the Young diagrams, which leads toa representation of Sn+p that will in general be reducible:

D(α)(n) ⊗ D

(β)(p) =

⊕irreps. γ

cαβγ D(γ)(n+p), (III.64)


where the D(γ)(n+p) are now irreducible representation of Sn+p while the integer coefficients cαβγ are

their respective multiplicities. In this section, we introduce the recipe for this outer product, withoutproviding any proof.(22)

Remark III.65. A somewhat “cleaner” setup to motivate the outer product and Eq. (III.64) is tochoose n elements i1, . . . , in among 1, . . . , n + p, for which choice there are

(n+pn

)possibilities,

denoting in+1, . . . , in+p the remaining p elements. The permutations that only affect the subseti1, . . . , in, leaving in+1, . . . , in+p invariant, form a subgroup H1 of Sn+p which is isomorphicto Sn,. Conversely, the permutations that only affect in+1, . . . , in+p while leaving i1, . . . , inunchanged form a subgroup H2 isomorphic to Sp. The outer product (III.64) is then the productof irreducible representations of different subgroups of Sn+p.

One easily checks that every permutation of Sn+p can be written as the (commutative) compo-sition of an element of H1 and an element of H2, and that the intersection H1 ∩H2 reduces tothe singleton consisting of the identity. H1 and H2 are thus so-called complementary subgroupsof Sn+p. Since H1 and H2 are in general not normal, Sn+p is not their direct product.

Let us illustrate the idea, as well as the recipe to find the decomposition on the right hand sideof Eq. (III.64), on an example:

⊗

(α) (β)

Here the Young diagram (α) resp. (β) corresponds to an irreducible representation of S4 resp. S3.To perform their outer product, one proceeds as follows.

• Take one of the diagrams — usually the smallest one, thus here (β) — and place an a in eachbox of its first row, a b in each box of the second row and so on:

a ab

.

• Add the boxes a to the diagram (α), to the right (or at the bottom of the first column) andin such a way that (i.) two of them do not appear in the same column and (ii.) the resultingdiagrams are valid ones, i.e. the number of boxes decreases from top to bottom:

a a , aa

,a

a,

a a, a

a.

• Add the boxes b to each of the diagrams obtained at the previous step, respecting the rules(i.) and (ii.) as well as an extra restriction, namely that (iii.) when reading the new diagramfrom top to bottom and from right to left, the number of boxes b always remains smallerthan or equal to the number of boxes a :

a ab

,a a

b, a

a b,

aa

b

ab

a,

a

ab

, a ab

, aa b

, aab

.

If diagram (β) contains further rows, add the corresponding boxes c , d , . . . using thesame rules (i.), (ii.), and (iii.).

(22)The outer product of Young diagrams will prove useful again when we consider the representations of the generallinear groups.


• Eventually, remove the letters from the boxes: some diagrams may appear several times,which means that the corresponding irreducible representation is contained with a multiplicitycαβγ ≥ 2 in the direct sum on the right hand side of Eq. (III.64).

⊗ = ⊕ ⊕ ⊕ 2

⊕ ⊕ ⊕ ⊕ .

Remarks:

∗ One can compute the dimension of each of the irreducible representations involved in the outerproduct. For the previous example, one finds for the representations of S7 on the right hand side14, 15, 14, (2×)35, 20, 21, 21, 35 — where one recognizes that conjugate representations have thesame dimension. Their sum, under consideration of the twofold appearance of one of them, thusequals 210. On the other hand, the irreducible representations of S4 and S3 on the left hand sideare of dimension 3 and 2, respectively: multiplying these dimensions, and multiplying the result (6)by(

74

)=(

73

)= 35 — corresponding to the number of ways to choose 4 (or 3) elements among 7,

see remark III.65 — also yields 210.

∗ The outer product of two Young diagrams (or of the corresponding irreducible representations)is commutative, i.e. one can exchange the roles of (α) and (β) and still obtain the same result.

CHAPTER IV

Physical applications

IV.1 Electric dipole of a moleculecoming soon

IV.2 Molecular vibrationscoming soon


IV.3 Crystal field splittingThe energy levels of electrons orbiting around a given atomic nucleus depend on the environmentof the atom. If the latter is “isolated”, as e.g. in a gas, one can assume that the potential felt by anelectron has spherical symmetry, i.e. is preserved by any three-dimensional rotation about an axisgoing through the nucleus. As the reader learned in quantum mechanics, this symmetry leads todegeneracy in the energy spectrum: all levels with a given azimuthal quantum number ` yet withdifferent magnetic quantum numbers m` ∈ −`, . . . , `− 1, ` have the same energy. However, if theatom is part of a crystal, the spherical symmetry is broken, i.e. the symmetric group is smaller,which leads to (partial) lifting of some of the degeneracy. In this Section, we illustrate this loss ofdegeneracy, known as “crystal field splitting”,(23) on an example.

Figure IV.1

Consider a cubic lattice of identical ions forming a crystal, as shown in Fig. IV.1. Focusingon the nearest neighbors, a given “tagged” ion in this crystal is surrounded by 6 other ions at thesame distance a — the lattice spacing — (Fig. IV.2, left), which constitute the vertices of a regularoctahedron with the tagged ion at its center (Fig. IV.2, right).

aa

a

a

a

a

Figure IV.2(23)Or “ligand field splitting” in chemistry.

54 Physical applications

Accordingly, the rotations that leave the geometry of the system invariant are those that preservea regular octahedron — or equivalently a cube —, which we now enumerate. These rotationsconstitute a group O, whose conjugacy classes we also give without proof.

• The identity transformation Id, which as always is in a conjugacy class on its own, which willhereafter will be denoted [Id].

• The octahedron possesses four axes with threefold symmetry, namely the straight lines goingthrough the centers of opposite faces (see Fig. IV.3). For each of these axes, there are two

Figure IV.3 – Example of threefold-symmetry axis.

proper rotations, namely through 120o and 240o. This gives all in all 8 such rotations, whichconstitute a conjugacy class C3.

• The three diagonals of the octahedron, i.e. the straight lines joining vertices that do not belongto a same face (Fig. IV.4), are axes with fourfold symmetry. All in all, there are thus 9 proper

Figure IV.4 – The three axes with fourfold symmetry of a regular octahedron.


rotations, through 90, 180 or 270o, around these axes. The 3 rotations through 180o constitutea conjugacy class 3C2, while the 6 other rotations are in a class C4.

• Eventually, the 12 edges of the octahedron form 3 squares. Consider the straight line joiningthe middle points of opposite sides of such a square, as shown in Fig. IV.5: a rotation through180o about such an axis leaves the octahedron invariant. There are 6 such axes with twofoldsymmetry, yielding 6 new rotations through 180o, which constitute a conjugacy class 6C2.

Figure IV.5

All in all, O thus consists of 24 rotations (including the identity), which are partitioned into 5conjugacy classes.

Remarks:

∗ Under each of the rotations of O, two opposite faces of the octahedron are “locked” together, sothat only four pairs of faces are permuted. Accordingly, the 24 rotations of O are in one-to-onecorrespondence with the 24 permutations of S4.

∗ The symmetry group of a regular octahedron actually also includes reflections and their productwith the rotations considered here. For simplicity we leave those transformations aside in thefollowing.

Since the group O has five conjugacy classes, it admits 5 (inequivalent) irreducible representa-tions. The corresponding character table is given (without proof) in Table IV.1.

[Id] C3 3C2 6C2 C4

D (1) 1 1 1 1 1

D (1′) 1 1 1 −1 −1

D (2) 2 −1 2 0 0

D (3) 3 0 −1 −1 1

D (3′) 3 0 −1 1 −1

Table IV.1 – Character table of the group O of octahedron-preserving rotations.

56 Physical applications

The group O of octahedron-preserving rotations is a subgroup of the group of three-dimensionalrotations. Accordingly, every representation of the latter will also be a representation of the former.Anticipating on the results of a later chapter, the group of three-dimensional rotations admits forevery ` ∈ N an irreducible (2`+ 1)-dimensional representation D` with character

χ`(θ) =∑

m`=−`eim`θ =

2`+ 1 if θ = 0

sin[(`+ 12)θ]

sin θ2

for θ 6= 0(IV.1)

for a rotation through the angle θ [see Eq. (VI.48)]. From what was just said, D` is also a represen-tation of O, yet not necessarily irreducible. We shall hereafter find the reduction of D` in irreduciblerepresentations of O.

In the context of the physical problem of an electron in a central potential, D` is the repre-sentation that operates on the energy eigenfunctions with azimuthal quantum number `, which allcorrespond to the same energy. If D` becomes reducible when the symmetry group is broken to O,this means that the degeneracy of the energy eigenstates is possibly lifted.

Now, using Eq. (IV.1) one can compute the character χ` for each conjugacy class of O. Thus,since the identity transformation is a rotation through 0o one finds

χ`([Id]) = 2`+ 1, (IV.2)

which simply means that D` is (2` + 1)-dimensional. The conjugacy classes 3C2 and 6C2 bothcorrespond to rotations through 180o — which are all in the same conjugacy class in the group ofthree-dimensional rotations. Using χ`(π), one thus finds

χ`(3C2) = χ`(6C2) = (−1)`. (IV.3)

The character (IV.1) is an even function of θ, which in particular gives χ`(2π/3) = χ`(−2π/3)and χ`(π/2) = χ`(−π/2). One thus obtains on the one hand

χ`(C3) =

1 for ` = 0 mod 3

0 for ` = 1 mod 3

−1 for ` = 2 mod 3,

(IV.4)

and on the other hand

χ`(C4) =

1 for ` = 0 or 1 mod 4

−1 for ` = 2 or 3 mod 4.(IV.5)

The characters of the representations D` with ` = 0, 1, 2, 3, 4 for the five conjugacy classes of O aresummarized in table IV.2.

[Id] C3 3C2 6C2 C4

D`=0 1 1 1 1 1

D`=1 3 0 −1 −1 1

D`=2 5 −1 1 1 −1

D`=3 7 1 −1 −1 −1

D`=4 9 0 1 1 1

Table IV.2


Comparing this table with the character table IV.1 of O, one sees

χ`=0 = χ(1) i.e. D`=0 = D (1)

andχ`=1 = χ(3) i.e. D`=1 = D (3).

The representations D`=0 and D`=1 of the group of three-dimensional rotations are thus irreduciblerepresentations of O. The corresponding physical states of an electron in a central potential — inthe usual denomination of atomic physics, s-states with l = 0 and p-states with ` = 1 — are thusstill in irreducible representations of O, so that their respective degeneracies persist.

Using now the results of Sec. III.4.1, as e.g. Eq. (III.29) for the integer coefficients of the reductionof a representation in irreducible representations, one easily finds

χ`=2 = χ(2) + χ(3′) i.e. D`=2 = D (2) ⊕D (3′),

which shows that the eigenstates with azimuthal quantum number ` = 2, the d-states, will belongto two different irreducible representations of the symmetry group O. Accordingly, the 5 orthogonalenergy eigenstates with ` = 2 will split into a doublet (corresponding to the two-dimensional irrep.D (2)) and a triplet (corresponding to D (3′)), with possibly different energies.

Similarly, one finds

χ`=3 = χ(1′) + χ(3) + χ(3′) i.e. D`=3 = D (1′) ⊕D (3) ⊕D (3′)

andχ`=4 = χ(1) + χ(2) + χ(3) + χ(3′) i.e. D`=4 = D (1) ⊕D (2) ⊕D (3) ⊕D (3′),

which illustrates the reduction of higher-dimensional representations D`, which will also lead to aloss of degeneracy.

CHAPTER V

Lie groups and Lie algebras

In physics, one encounters not only finite or discrete symmetry groups, but also “continuous symme-tries”, corresponding to transformations described by one or several real parameters that take theirvalues in an interval of R or in a continuous subset of some Rs. Obvious examples are translationsor rotations in two- or three-dimensional space, or the multiplication of the state vector |ψ〉 of aquantum-mechanical system by a phase factor eiδ where δ ∈ R is arbitrary.

All in all, we are thus led to consider groups consisting of elements g(θ1, . . . , θs) labeled by anumber s ∈ N∗ of real parameters θ1, . . . , θs, such that every parameter is relevant — i.e. one cannotmake s smaller — and that each θa takes values in an interval of R. Such a group will be referredto as an s-parameter continuous group, and its parameters will sometimes be collectively denotedeither θa or θ.

Remark: When counting the parameters of a continuous group, only those parameters which cantake continuous values are to be taken into account.

In addition, we shall require that the correspondence θ 7→ g(θ) be continuous — we shall infact even require analyticity —, which necessitates the introduction of topological notions on theset underlying the group (Sec. V.1 and Appendix A) .

V.1 Lie groupsThe continuous groups of interest in physics are actually more than merely continuous: their depen-dence on the parameters is actually (real) analytic, constituting the so-called Lie(o) groups, which wenow define (Sec. V.1.1) and of which we give the most currently encountered examples (Sec. V.1.2).

V.1.1 Definition and first results

Definition V.1. A real resp. complex Lie group G is a real resp. complex differentiable manifold onwhich two analytic functions ϕ : G × G → G and f : G → G are defined, which provide G with agroup structure: the group law is such that g1 · g2 = ϕ(g1, g2) and the inverse element given byg−1 = f(g).

Stated differently, a Lie group is a group, as defined by axioms G1–G4, whose operations —multiplication and inversion — are analytic (or at least smooth).

Remarks:

∗ The dimension of G as a manifold(24) is the number of parameters needed to label a groupelement.

(24)The definition of a manifold and of closely related mathematical notions is recalled in Appendix A.2.

(o)S. Lie, 1842–1899

V.1 Lie groups 59

∗ That the functions ϕ and f provide a group structure constrains their possible forms. Forinstance, the associativity property G2 becomes

ϕ(ϕ(g, g′), g′′

)= ϕ

(g, ϕ(g′, g′′)

)∀g, g′, g′′ ∈ G,

while the existence of an identity element e (axiom G3)reads

ϕ(e, g) = ϕ(g, e) = g ∀g ∈ G.

In turn, the characteristic property of the inverse requires

ϕ(f(g), g

)= ϕ

(g, f(g)

)= e ∀g ∈ G.

One can show that these requirements, considered for elements g ∈ G in the neighborhood of theidentity element and using a chart on that neighborhood, can be used to deduce the existence ofthe structure constants introduced below.

∗ Instead of requiring the analyticity of ϕ and f , one can use weaker assumptions which lead tothe same result, especially in the finite-dimensional case we are mostly interested in.

In the following we shall only consider Lie groups consisting of matrices — the group law beingthe usual matrix multiplication. These groups obviously depend on only a finite number s ofreal parameters, and the identity element of the group will always be an identity matrix (whosedimension naturally depends on the specific group under consideration), which will generically bedenoted 1. More precisely, one has the following definition:

Definition V.2. A matrix Lie group is a closed subgroup of the group GL(n,C) of regular n × nmatrices with complex entries.

V.1.2 Examples

We now list a few families of groups which are often encountered in physics, in the role ofsymmetry groups either for “physical problems” or for toy models. In the following paragraphs(except in § V.1.2 d), n denotes a natural number.

::::::V.1.2 a

:::::::::::::::::::::::General linear groups

Theorem & Definition V.3. The n × n invertible matrices with real resp. complex entries form agroup, the general linear group GL(n,R) resp. GL(n,C).

GL(n,R) is a Lie group with n2 continuous parameters — the condition that the matrix de-terminant be non-zero does not constitute a strong enough condition to decrease the number ofparameters.(25) On the other hand, since the image of GL(n,R) by the determinant, i.e. a continu-ous mapping, is the disconnected set R∗ proves that GL(n,R) cannot be connected. More precisely,GL(n,R) consists of two connected components (two sheets) corresponding to the two possible signsof the determinant. Eventually, GL(n,R) is noncompact.

Remark: The connected component of the identity matrix 1n, i.e. the n × n invertible matriceswith positive determinant, form a (connected!) subgroup — this is a generic property —, denotedGL+(n,R), which is also of dimension n2.

GL(n,C) is a complex Lie group with 2n2 real parameters, which unlike GL(n,R) is connected,yet also noncompact.(25)A trick to quickly check that a general formula for the number of parameters is not wrong (if not correct) is to

check the case n = 1. Thus, GL(1,R) is R∗, i.e. a one-parameter group; GL(1,C) is C \ 0, i.e. a two-parametergroup; SL(1,R), SL(1,C), SO(1), SU(1) are reduced to the singleton 1, i.e. are zero-parameter groups; O(1)contains only the elements 1 and −1, i.e. is also a zero-parameter group; U(1) is (isomorphic to) the unit circle inthe complex plane, i.e. it is a one-parameter group.

60 Lie groups and Lie algebras

::::::V.1.2 b

:::::::::::::::::::::::Special linear groups

Theorem & Definition V.4. The n × n invertible real resp. complex matrices with determinant oneconstitute the special linear group SL(n,R) resp. SL(n,C).

The condition detM = 1 is strong enough to “cost” one real or complex parameter: a matrixof GL(n,R) resp. GL(n,C) can always be written as the product of its real- or complex-valueddeterminant with a matrix of SL(n,R) resp. SL(n,C). Accordingly SL(n,R) is of dimension n2− 1and SL(n,C) is of dimension 2(n2 − 1).(25) Both groups are connected and noncompact.

Remark: Geometrically, SL(n,R) corresponds to the geometrical transformations that preserve the(n-dimensional) volume and the orientation.

::::::V.1.2 c

:::::::::::::::::::::Orthogonal groups

Theorem & Definition V.5. The n × n real matrices O such that OTO = OOT = 1n form a group,the orthogonal group, denoted On.

Property V.6. The orthogonal matrices represent the isometries of the n-dimensional Euclideanspace that leave the origin invariant.

Stated differently, the orthogonal matrices O represent the transformations X 7→ X ′ = OX thatpreserve the Euclidean scalar product

XTY =n∑i=1

xiyi, (V.7)

where X resp. Y is a vector with n components (x1, . . . , xn) resp. (y1, . . . , yn).

Property V.8. The Lie group O(n) is of dimensionn(n− 1)

2.

Proof: A matrix O ∈ O(n) transforms an orthonormal basis for the scalar product (V.7) intoanother orthonormal basis, so that its columns (or equivalently its lines) are of norm 1 andpairwise orthogonal. Accordingly, one can construct O by choosing n − 1 independent realparameters for the first column — the n entries have to obey the normalization condition, whichremoves one degree of freedom —, n−2 parameters for the second column — the normalizationfixes one parameter, the orthogonality with the first column a second one —, and more generallyn− k independent parameters for the k-th column. All in all there are thus

n∑k=1

(n− k) = n2 − n(n+ 1)

2=n(n− 1)

2

independent real parameters. 2

The defining property OTO = OOT = 1n leads at once to | detO| = 1 and to the normalizationof each column to 1 for the Euclidean scalar product (V.7). Accordingly, the entries of an orthogonalmatrix are bounded, so that the group O(n) is compact. On the other hand, since both detO = +1and detO = −1 are possible, O(n) is disconnected, consisting of two connected components.

The connected component of the identity matrix is a Lie subgroup of O(n):

Theorem & Definition V.9. The n× n real matrices O such that OTO = OOT = 1n and detO = 1form a group, the special orthogonal group SO(n).

Property V.10. The special orthogonal matrices represent the isometries of the n-dimensional Eu-clidean space that leave the origin invariant and preserve the orientation, i.e. precisely the rotationsin n-dimensional space.

SO(n) is of dimensionn(n− 1)

2, connected(!), and compact.

V.1 Lie groups 61

::::::V.1.2 d

::::::::::::::::::::::::::::::Indefinite orthogonal groups

Let p and q be two natural numbers and η be the diagonal (p+q)× (p+q) matrix defined as

η = diag(1, 1, . . . , 1︸︷︷︸p

,−1, . . . ,−1︸︷︷︸q

). (V.11)

Theorem & Definition V.12. The (p+q)× (p+q) real matrices Λ such that ΛTηΛ = η form a group,the indefinite orthogonal group O(p, q).

One at once sees that the matrices Λ obey |det Λ| = 1, i.e. are invertible, and the definingproperty gives Λ−1 = η−1ΛTη, where actually η−1 = η.

Defining a (non-positive definite) scalar product by

XTηY =

p∑i=1

xiyi −p+q∑i=p+1

xiyi, (V.13)

where X resp. Y is a vector with p + q components (x1, . . . , xp+q) resp. (y1, . . . , yp+q), a matrix Λof O(p, q) represents a transformation that preserve that scalar product.

Remarks:

∗ Replacing η by −η (with p minus signs and q plus signs in the scalar product) leaves O(p, q)unchanged.

∗ The special case p = 3, q = 1 (or equivalently p = 1, q = 3, see previous remark) gives theLorentz (p) group of relativistic physics, which will be studied in further detail in Chap. VIII.

Denoting n = p+ q, O(p, q) is a noncompact group with n(n− 1)/2 parameters, and with fourconnected components, two of which correspond to matrices with determinant det Λ = +1, theother two to matrices with determinant −1.

Theorem & Definition V.14. The (p+q)× (p+q) real matrices Λ such that ΛTηΛ = η and det Λ = 1form a group, the indefinite special orthogonal group SO(p, q).

SO(p, q) is again noncompact and still consists of two connected components. The connectedcomponent of the identity matrix is a Lie group, denoted SO+(p, q).

Remarks:

∗ The group SO(3, 1) is that of proper Lorentz transformations.

∗ One can also find a Lie subgroup O+(p, q) of O(p, q), which consists of two of the four sheets. Inthe case of the Lorentz group O(3,1), this subgroup O+(3, 1) will be that of orthochronous Lorentztransformations.

::::::V.1.2 e

::::::::::::::::Unitary groups

Theorem & Definition V.15. The n × n complex matrices U such that U †U = UU † = 1n form agroup, the unitary group, denoted U(n).

Property V.16. The unitary matrices represent the isometries of the n-dimensional Hermitian spacethat leave the origin invariant.

Stated differently, the unitary matrices U represent the transformations X 7→ X ′ = UX thatpreserve the Hermitian product

X†Y =

n∑i=1

x∗i yi, (V.17)

where X resp. Y is a vector with n complex components (x1, . . . , xn) resp. (y1, . . . , yn).(p)H. Lorentz, 1853–1928


Property V.18. The Lie group U(n) is of dimension n2.

Proof: A matrix U ∈ U(n) transforms an orthonormal basis for the scalar product (V.17)into another orthonormal basis, so that its columns (or equivalently its lines) are of norm 1and pairwise orthogonal. Accordingly, one can construct U by choosing 2n − 1 independentreal parameters for the first column — the n complex entries have to obey the normalizationcondition, which removes one degree of freedom—, 2n−3 real parameters for the second column,and more generally 2n− (2k − 1) independent parameters for the k-th column. All in all thereare thus n∑

k=1

[2n− (2k − 1)] = 2n2 − 2n(n+ 1)

2+ n = n2

independent real parameters. 2

The defining property U †U = UU † = 1n leads at once to | detU | = 1 and to the normalizationof each column of U to 1 for the Hermitian product (V.17). The entries of a unitary matrix arethus bounded, so that the group U(n) is compact. Unlike the orthogonal group O(n), it is alsoconnected. It contains an important subgroup:

Theorem & Definition V.19. The n × n complex matrices U such that U †U = UU † = 1n anddetU = 1 form a group, the special unitary group SU(n).

Requiring that the determinant be equal to 1 suppresses a real parameter with respect to ageneral matrix of U(n) — instead of detU = eiδ with an arbitrary δ ∈ [0, 2π], the latter is now fixedto 0. Accordingly, SU(n) is of dimension n2 − 1. It is still connected and compact.

::::::V.1.2 f

::::::::::::::::::::Symplectic groups

Let 1n denote the n × n identity matrix with n ∈ N∗. One defines a 2n × 2n antisymmetricmatrix J2n as

J2n ≡

(0 1n

−1n 0

). (V.20)

Theorem & Definition V.21. The 2n× 2n real matrices M such that MTJ2nM = J2n form a group,the symplectic group, denoted Sp(2n,R).

Remark V.22. The defining property gives at once detM = ±1, which ensures the regularity of Mand thus the existence of an inverse, which is indeed given by M−1 = J−1

2nMTJ2n.

Property V.23. The symplectic matricesM represent the group of transformations X 7→ X ′ = MXthat preserve the bilinear product

XTJ2nY =

n∑i=1

(xiyi+n − xi+nyi), (V.23)

where X resp. Y is a vector with 2n real components (x1, . . . , x2n) resp. (y1, . . . , y2n).

Property V.24. The Lie group Sp(2n,R) is of dimension n(2n+ 1).

A possible proof runs similar to that of property V.8. Note, however, that Sp(2n,R) has thesame dimension as O(2n+ 1).

Unlike the orthogonal and unitary groups, the symplectic group Sp(2n,R) is noncompact. It isconnected, and one can how that the determinant of a symplectic matrix actually always equals 1.(26)

Remark: The symplectic group Sp(2,R) coincides with the special linear group SL(2,R).

(26)An elementary proof can be found in Ref. [1].

V.2 Lie algebras 63

V.2 Lie algebras

V.2.1 General definitions

::::::V.2.1 a

::::::::::::::Lie algebras

We recall that algebras on a field were defined in definition III.31, with in particular the threeproperties A1–A3 of the vector multiplication.

Definition V.25. A Lie algebra A is an algebra such that its vector multiplication fulfills two extrarequirements:

• alternativity: for all x ∈ A, x ? x = 0A; (LA1)

• Jacobi(q)identity: for all x, y, z ∈ A, x ? (y ? z) + y ? (z ? x) + z ? (x ? y) = 0A, (LA2)

where 0A is the zero vector of the algebra.

Property V.26. Being bilinear and alternating, the vector multiplication is also anticommutative,that is

for all x, y ∈ A, x ? y = −y ? x, (LA1’)

which is often used in lieu of Eq. (LA1) in the definition.(27)

This follows at once from expanding the identity (x+ y) ? (x+ y) = 0A.

Examples:

∗ The vector space R3 with the cross product × is a Lie algebra.

∗ The set M (n,R) resp. M (n,C) of n × n matrices with real resp. complex entries with thecommutator AB −BA as vector multiplication is a Lie algebra of dimension n2.

The 3 × 3 strictly upper triangular real matrices constitute a Lie subalgebra of M (3,R) calledthe Heisenberg(r) algebra, and which we shall denote h3. A basis of this Lie algebra is obviously

M1 =

0 1 00 0 00 0 0

, M2 =

0 0 00 0 10 0 0

, M3 =

0 0 10 0 00 0 0

.

One quickly checks M1 ? M2 ≡ M1M2 −M2M1 = M3, M1 ? M3 ≡ M1M3 −M3M1 = 0 andM2 ? M3 ≡ M2M3 −M3M2 = 0, and thus for all A,B,C ∈ h3, A ? (B ? C) = (A ? B) ? C = 0,where 0 stands for the zero 3 × 3 matrix. h3 is thus an example of associative Lie algebra, butone sees that requiring associativity, alternativity and the Jacobi identity leads to the strongconstraint A ? (B ? C) = 0 for all A,B,C ∈ h3.

Notation V.27. The vector multiplication x?y of a Lie algebra is traditionally denoted [x, y], calledthe Lie bracket , so that the Jacobi identity (LA2) reads

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0A for all x, y, z ∈ A. (LA2′)

In the following, we shall only consider matrix Lie algebras on which the vector multiplicationis the commutator AB − BA, so that the notation [A,B] should hopefully not confuse the readerused to employ it precisely for that purpose.

(27)There is indeed equivalence between Eq. (LA1) (with bilinearity) and Eq. (LA1’) in case the base field of thealgebra is R or C, yet not for a field of characteristic 2.

(q)C. G. Jacobi, 1804–1851 (r)W. Heisenberg, 1901–1976


::::::V.2.1 b

:::::::::::::::::::::Structure constants

Definition V.28. Let Xa1≤a≤s be a basis of a (matrix) Lie algebra A of dimension s. The productof two basis vectors — often called generators — Xa, Xb is a linear combination of the basis vectors

[Xa, Xb] =s∑c=1

cabcXc, (V.28)

which defines the structure constants cabc of the Lie algebra.

More generally, one may define the structure constants of an algebra which is not a Lie algebra,yet we do not need those in these lecture notes.

Notation V.29. The previous definition V.28 uses the mathematicians’ convention. Physicists ratherdenote the generators T a1≤a≤s and define “their” structure constants through the relation

[T a, T b] = is∑c=1

fabcTc. (V.29)

Setting T a = iXa for all a ∈ 1, . . . , s, one sees that Eqs. (V.28) and (V.29) define the samestructure constants fabc = cabc.

The coexistence of both conventions naturally means that both have advantages. Anticipatingon Sec. V.2.2, we shall see that the mathematicians’ choice gives “more natural” expressions for theform of the elements of a Lie group in the neighborhood of the identity matrix, see Eq. (V.32), whilethe physicists’ convention lead to Hermitian generators T a for the usual matrix groups (see below),instead of antihermitian ones.

Remarks:

∗ Multiplying each generator Xa or T a by a constant number λ in Eqs. (V.28) and (V.29) obviouslyleads to multiplying the structure constants by the same factor λ. However, once a normalizationcondition has been chosen for the generators, then one can show that the structure constants arein fact independent of the basis. Thus, they really encode an intrinsic property of the Lie algebra,irrespective of a basis choice.

∗ For compact Lie groups, the position of the indices labeling the structure constants has nomeaning. Thus the reader will often find in the (physics) literature fabc, fabc , fabc . . . used somewhatinterchangeably. In the case of a non-compact group, some more care is needed.

Property V.30. The structure constants fabc satisfy generic relations, which are inherited from theanticommutativity (LA1’) and the Jacobi identity (LA2), namely

fabc = −f bac ∀a, b, c, (V.30a)

i.e. the structure constants are antisymmetric in their two upper indices, ands∑c=1

(fabcf

cde + f bdcf

cae + fdacf

cbe

)∀a, b, d, e. (V.30b)

If you wish to memorize this equation, note its structure: e is fixed, c is summed over, and a, b,d are cyclically permuted.

We shall give examples of structure constants, on which the reader can check the above proper-ties, in the following sections.

V.2 Lie algebras 65

V.2.2 Lie algebra associated to a Lie group

::::::V.2.2 a

:::::::::::::::::::::::::::::::::::::::::::::Tangent space at the identity in a Lie group

Let G be a matrix Lie group of dimension s. Restricting oneself to a chart that encompasses theidentity matrix 1, one chooses a system of coordinates θa1≤a≤s on that chart such that 1 is thegroup element g(θ1, . . . , θs) corresponding to the origin of the coordinates, i.e.

g(0, . . . , 0) = 1. (V.31)

Invoking the differentiability of the Lie group and the smoothness of the charts, one may writea group element in the neighborhood of 1 — i.e. for small enough θ1, . . . , θs — in the form

g(θ1, . . . , θs) = g(0, . . . , 0) +∑a

θaXa +O(θ2

a) = 1+∑a

θaXa +O(θ2

a), (V.32a)

or equivalently, inserting an imaginary unit i in front of the small parameters

g(θ1, . . . , θs) = g(0, . . . , 0)− i∑a

θaTa +O(θ2

a) = 1− i∑a

θaTa +O(θ2

a). (V.32b)

Obviously, one has for each a ∈ 1, . . . , s

Xa = limθa→0

g(0, . . . , 0, θa, 0, . . . , 0)− 1θa

, T a = i limθa→0

g(0, . . . , 0, θa, 0, . . . , 0)− 1θa

. (V.33)

This shows that Xa resp. T a is a vector of the tangent space T1G at the identity of the Lie group.

Example V.34. Consider a so-called one-parameter subgroup of G, i.e. a continuous set of elementsg(t) of the Lie group labeled by a single real parameter t taking its values in a neighborhood of 0,with the following properties:

g(0) = 1 (V.35a)∀t1, t2, g(t1 + t2) = g(t1)g(t2), (V.35b)

where the latter holds at least in a neighborhood of t = 0. Stated differently, g(t) is a “continuouscurve” drawn on the Lie group G and going through the identity matrix, and which is at least locallya group homomorphism from (R,+) into G.

More precisely, we require that g(t) be a differentiable function of t. For a small enough δt onemay then write

g(δt) = g(0) +Xδt+O((δt)2) = 1+Xδt+O((δt)2) with X =dg(t)

dt

∣∣∣∣t=0

= g′(0).

X is a tangent vector at the identity to the Lie group G, X ∈ T1G.For any t, one may also write g(t + δt) = g(t) + g′(t)δt + O((δt)2), while the one-parameter

group property (V.35b) yields

g(t+ δt) = g(t)g(δt) = g(t)[1+Xδt+O((δt)2)

]= g(t) + g(t)Xδt+O((δt)2),

where we used the expression of g(δt) given above. Identifying both expressions of g(t + δt), onethus finds after dividing by δt and taking the limit δt→ 0

g′(t) = g(t)X.

One recognizes a linear differential equation of first order, whose solution must also obey the initialcondition (V.35a), yielding

g(t) = etX .

That is, the one-parameter subgroup g(t) follows from exponentiating a tangent vector — which, Gbeing a matrix Lie group, is a matrix — at the identity. Accordingly the definition and propertiesof the exponential of a matrix will be discussed in § V.2.2 c hereafter.


Similarly, we shall see that the relations (V.32) are actually approximations to first order in theparameters θa to the more general formulas

g(θ1, . . . , θs) = exp

(∑a

θaXa

)(V.36a)

g(θ1, . . . , θs) = exp

(−i∑a

θaTa

), (V.36b)

which we shall also encounter on specific examples.

Remark: The example V.34 shows why the tangent space T1G at the identity of the group seemsto play a special role in the study of Lie groups — which will be further discussed in the followingparagraph. Thanks to the group law, one can map the behavior of the Lie group in the vicinity ofany of its points g to that in the neighborhood of the identity — this is the step encoded in theequation g(t + δt) = g(t)g(δt) = g(t)[1 + · · · ] in the example —, so that the tangent spaces TgGat g is isomorphic to that at the identity. In other words, the behavior in a neighborhood of theidentity matrix, described by T1G, is actually representative of the behavior in a neighborhood ofany element of the Lie group.

::::::V.2.2 b

:::::::::::::::::::::::::::Lie algebra of a Lie group

By studying a Lie group G in the neighborhood of the identity matrix, one identifies a Liealgebra.

Theorem V.37. The tangent space at the identity of a matrix Lie group G is a Lie algebra, referredto as the Lie algebra of the group and traditionally denoted g,(28) with the matrix commutator asLie bracket.

Proof: Let us first show that for any tangent vector Y ∈ T1G and any element g ∈ G of the Liegroup, the (matrix) product gY g−1 defines a one-parameter subgroup of G, i.e. is also a tangentvector. For that purpose, the property (V.46) of the exponential map yields for every t suchthat etY ∈ G the identity

et(gY g−1) = g etY g−1.

As product of elements of the Lie group, et(gY g−1) is itself in G, which gives the desired result

gY g−1 ∈ T1G.Taking now g = eλX , where λ ∈ R and X ∈ T1G is another tangent vector, one has g−1 = e−λX

and thus eλXY e−λX ∈ T1G. The latter result hold for λ = 0 and for any λ in a small enoughneighborhood of 0: using the linearity of a vector space, the difference eλXY e−λX − Y is againin the tangent space, as well as its quotient by λ. This still holds in the limit λ → 0, i.e. thederivative of eλXY e−λX with respect to λ, and compute at λ = 0, is still in the tangent space:

d

dλ

(eλXY e−λX

)∣∣∣λ=0

=(XeλXY e−λX − eλXY e−λXX

)∣∣∣λ=0

= XY − Y X ∈ T1G.

Since the commutator of two matrices obeys the properties of a Lie bracket, the tangent spaceT1G is a Lie algebra. 2

One should note at once that while the Lie algebra associated to a given Lie group is unique,yet different Lie groups can have the same Lie algebra — the most obvious example being that ofa disconnected group and its subgroup consisting of the connected component of the identity, likeO(n) and SO(n).

Corollary V.38. The Lie algebra g associated to a matrix Lie group G is the set of complex matricesX such that etX ∈ G for all t ∈ R.

We shall use this result in Sec. V.2.3 below to write down the Lie algebras of the classical Liegroups of Sec. V.1.2.(28)The Lie algebra of a group G is also sometimes denoted Lie(G).

V.2 Lie algebras 67

Let X, Y be two tangent vectors at the identity of a Lie group G, i.e. X,Y ∈ g. At least fort, u close enough to 0, etX and euY belong to G. Let us compute their commutator ,(29) using theapproximation eαA = 1+ αA+ 1

2α2A2 +O(α3). To quadratic order, one finds

etXeuY e−tXe−uY = 1+ tu(XY − Y X) +O(t3, u3). (V.39)

One thus finds that the Lie bracket [X,Y ] = XY − Y X ∈ g measures the lack of commutativity ofthe elements of the one-parameter subgroups etX and euY .

::::::V.2.2 c

::::::::::::::::::Exponential map

Theorem & Definition V.40. Let A be an n×n matrix with real or complex entries. The exponentialof A is the matrix given by the power series

exp(A) ≡ eA ≡∞∑k=0

Ak

k!, (V.40)

with the convention that A0 is the identity n× n matrix.

One indeed checks that the series converges for any A.

Let us list without proof a few properties of the exponential map. Throughout, A or B are n × nmatrices with real or complex entries.

• e0n = 1n with 0n resp. 1n the zero resp. identity n× n matrix. (V.41)

• det eA = eTr A (V.42)

• If A and B are two commuting matrices, i.e. AB = BA, then eA+B = eA eB. (V.43)Note that the converse is not true: the matrices

A =

(iπ 00 −iπ

)and B =

(0 10 −2iπ

)yield eA = −12, eB = 12 and eA+B = −12 = eA eB , although they do not commute.

•(eA)−1

= e−A. (V.44)

• For all t, u ∈ C, e(t+u)A = etA euA. (V.45)

• If B is invertible, then B eAB−1 = eBAB−1. (V.46)

• d

dtetA = A etA = etAA, and in particular

d

dtetA∣∣∣∣t=0

= A (V.47)

According to corollary V.38, the exponential map provides a mapping between a Lie algebra gand a Lie group G. More precisely, one can show the following results:

Property V.48. Given a matrix Lie group G and its Lie algebra g, there exist neighborhoods U0 ofthe zero matrix in g and U1 of the identity matrix in G such that the exponential map X 7→ eX isa bijection from U0 onto U1.

Remark: Obviously, the mention of a neighborhood of the zero matrix in g presupposes the intro-duction of a topological structure on the Lie algebra. Since the latter is a finite-dimensional spaceof matrices, this represents no difficulty.

Property V.49. A sufficient condition to ensure that the exponential map X ∈ g 7→ eX ∈ G issurjective is if the Lie group G is connected and compact.(29)The commutator of two elements g, h of a group G is the element ghg−1h−1.


The condition is not necessary: for example, every matrix of the general linear group GL(n,C),which is non-compact, can be written as the exponential of a matrix of gl(n,C), i.e. of an n × nmatrix.Property V.50. A necessary condition for the exponential map to be injective is that the Lie groupG is simply connected.

The condition is not sufficient: the exponential maps both 2×2 traceless antihermitian matrices(±iπ 0

0 ∓iπ

)∈ su(2),

to the special unitary matrix −12 ∈ SU(2), i.e. the exponential map is not injective on su(2),although SU(2) is simply connected.

An example of non-simply connected Lie group is the abelian group U(1), whose Lie algebrais u(1) = iR, i.e. the set of imaginary complex numbers — which is obviously isomorphic to thespace of 1× 1 antihermitian matrices. Indeed, every element of u(1) of the form 2kπi with k ∈ Z ismapped by the exponential function to the same element 1 of U(1).

According to property V.48, if two matrices X,Y ∈ g are “close enough” to the zero matrix,then their respective exponentials are elements of a Lie group G of which g is the Lie algebra, botheX and eY being in the vicinity of the identity matrix 1. Accordingly, the product eXeY is also anelement of G, which should also lie close to 1, and thus be writable in the form eZ with Z ∈ g. Anatural question then arises, as to whether, and if yes how, Z is related to X and Y .

If the latter are “small enough” (for the infinite series involved in the formula to converge), Zindeed exists and is given by the Baker (s)-Campbell (t)-Hausdorff (u) formula

eXeY = eZ(X,Y ) with Z(X,Y ) = X+Y +1

2[X,Y ]+

1

12

([X, [X,Y ]

]+[Y, [Y,X]

])+ · · · , (V.51)

where all successive terms only depend on iterated commutators of X and Y .One can write down several closed forms for the series on the right hand side, especially involving

the adjoint mapping introduced below. Instead, let us give special cases of the formula, in whichthe series is truncated at a low order:

• If X and Y commute, one recovers property (V.43), namely eXeY = eX+Y .

• If the commutator [X,Y ] commutes with both X and Y , then

eXeY = eX+Y+ 12

[X,Y ] = eX+Y e12

[X,Y ]. (V.52)

Eventually, let us mention a related formula:

eXY e−X = Y + [X,Y ] +1

2!

[X, [X,Y ]

]+

1

3!

[X,[X, [X,Y ]

]]+ · · · , (V.53)

which is also useful in physical applications.

V.2.3 Lie algebras associated to the classical Lie groups

::::::V.2.3 a

::::::::::::::::::gl(n,R), gl(n,C)

n× n matrices with real resp. complex entries

::::::V.2.3 b

::::::::::::::::::sl(n,R), sl(n,C)

traceless n× n matrices with real resp. complex entries

::::::V.2.3 c

::::::::::::::o(n) = so(n)

traceless antisymmetric n× n matrices

::::::V.2.3 d

::::::::::::::u(n) = su(n)

traceless antihermitian n× n matrices(s)H. F. Baker, 1866–1956 (t)J. E. Campbell, 1862–1924 (u)F. Hausdorff, 1868–1942

CHAPTER VI

The groups SO(3) and SU(2)

and their representations

Two continuous groups of transformations that play an important role in physics are the specialorthogonal group of order 3, SO(3), and the special unitary group of order 2, SU(2), which are infact related to each other, and to which the present chapter is devoted. We first recall in Secs. VI.1and VI.2 their most useful properties, which the reader probably knows from previous lectures, andintroduce their respective Lie algebras. We then discuss their representations (Sec. VI.3).

VI.1 The group SO(3)In this section, we begin with a reminder on the rotations in three-dimensional Euclidean space(Sec. VI.1.1) — which as the reader knows models the spatial arena in which the physical objectsof non-relativistic physics evolve.

VI.1.1 Rotations in three-dimensional space

Consider first the rotations in three-dimensional Euclidean space that leave a given point Oinvariant. For the usual composition, they form a continuous group, which is non-Abelian sincerotations about different axes do not commute. By choosing a Cartesian coordinate system with Oat its origin, one maps these rotations one-to-one to those on R3.

To describe such a rotation, one mostly uses either a set of three angles, or the direction of therotation axis and the rotation angle, as we now recall.

:::::::VI.1.1 a

::::::::::::::Euler angles

A given rotation R is fully characterized by its action on three pairwise orthogonal directionsin three-dimensional Euclidean space — or equivalently, on an orthonormal basis of R3. Thus, Rtransforms axes Ox, Oy, Oz into new axes Ou, Ov, Ow, as illustrated in Fig. VI.1. Note thatit is in fact sufficient to know the transformations from Ox to Ou and from Oz to Ow, since theremaining axes Oy and Ov are entirely determined by the first two directions of a given triplet.

As was already mentioned before, in these lecture notes we consider “active” transformations, i.e.the coordinate system is fixed while the physical system is being transformed — here, rotated.Accordingly, one should not view the transformation from the axes Ox, Oy, Oz to the directionsOu, Ov, Ow as a change of coordinates, but as the rotation of “physical” axes — which possiblycoincided with the coordinate axes in the initial state, in case the Cartesian coordinates aredenoted x, y, z.

The action of R can be decomposed into the composition of three rotations about axes that aresimply related to Oy, Oz, Ou, and Ow:

• First, one performs a rotation about Oz through an angle 0 ≤ α < 2π, which is actuallydetermined by the second step, and which brings the axis Oy along a new direction On, theso-called line of nodes. This first rotation will be denoted Rz(α).

70 The groups SO(3) and SU(2) and their representations

O

x

y

z

u

v

w

α

β

γ

n

Figure VI.1 – Euler angles

• Second, one performs a rotation Rn(β) about On through an angle 0 ≤ β ≤ π such that itbrings the “old” axis Oz along the “new” direction Ow.

• Eventually, one performs a rotation Rw(γ) about Ow through an angle 0 ≤ γ < 2π whichbrings the line of nodes On along the direction Ov.

All in all, one may label R by the three Euler (v) angles α, β, γ and write

R(α, β, γ) = Rw(γ)Rn(β)Rz(α). (VI.1a)

One can actually express Rw(γ) in terms of Rz(γ) and Rn(β) in terms of Ry(β), and thereby derivethe alternative decomposition

R(α, β, γ) = Rz(α)Ry(β)Rz(γ), (VI.1b)

which only involves rotations about the “old” axes.

Remarks:

∗ The convention for the Euler angles is far from being universal. Here we introduced the choicewhich is the most usual in classical mechanics or quantum mechanics, and which leads to the“z-y-z-convention” of Eq. (VI.1b), which we shall encounter again in § VI.3.2 b.

∗ The three Euler angles take their values in a bounded region of the parameter space R3, whichshows that the group of rotations of R3 is a three-parameter compact group.

:::::::VI.1.1 b

::::::::::::::::::::::::::::::::::::::::::::::::::::::Parameterization through a unit vector and an angle

The other usual description of rotations consists in using a unit vector ~n along the rotation axistogether with the angle ψ of the rotation, where the latter can be taken in the interval [0, 2π] orequivalently [−π, π]. Accordingly, a given rotation is then denoted R~n(ψ).

Remark VI.2. Obviously, a given rotation can be described by two equivalent sets consisting of aunit vector and an angle:

R~n(ψ) = R−~n(−ψ), (VI.2)

as can also be checked on Eq. (VI.3) below. One may reduce the interval in which ψ takes its values

(v)L. Euler, 1707–1783

VI.1 The group SO(3) 71

to 0 ≤ ψ ≤ π to decrease this redundancy, yet it cannot be totally removed since there still remainsR~n(π) = R−~n(π) for any unit vector ~n.

Now, since a unit vector of R3 is parameterized through 2 real parameters — for instance,identifying ~n with a point on the unit sphere S2, by the polar angle θ ∈ [0, π] and azimuthal angleϕ ∈ [0, 2π[ of the spherical-coordinates system —, we recover the fact that the rotations form athree-parameter group.

The effect on a vector ~x ∈ R3 of the rotation through the angle ψ about the direction along theunit vector ~n is given by Rodrigues’(w) formula:

R~n(ψ)~x = (cosψ)~x+ (1− cosψ)(~n ·~x)~n+ (sinψ)~n×~x. (VI.3)

The formula follows from writing ~x = ~x‖ + ~x⊥, where ~x‖ = (~n · ~x)~n is the projection of ~xparallel to ~n, and realizing that R~n(ψ)~x leaves ~x‖ invariant while transforming ~x⊥ = ~x−~x‖ into(cosψ)~x+ (sinψ)~n×~x. 2

In the case of an infinitesimally small angle |δψ| 1, a Taylor expansion of the Rodrigues formulato first order gives

R~n(δψ)~x = ~x+ δψ~n×~x+O(δψ2). (VI.4)

The reader is invited to check the validity of this relation by drawing ~x and R~n(δψ)~x in the planeorthogonal to the direction of ~n.

The description by a unit vector and an angle is especially useful for expressing the rotationconjugate to another rotation R~n(ψ). For any rotation R, one checks the relation

RR~n(ψ)R−1 = R~n′(ψ) with ~n′ = R~n. (VI.5)

Accordingly, two conjugate rotations have the same rotation angle. Conversely, two rotationsthrough the same angle ψ are conjugate, the conjugating element being the rotation that mapsthe unit vector of the first one to the unit vector of the second one. We thus recover the resultalready found in § I.4.3 c, according to which a conjugacy class of the group of rotations in three-dimensional space consists of all rotations with a given angle.

VI.1.2 The group SO(3) and its Lie algebra so(3)

Once a Cartesian coordinate system has been fixed, the rotations in three-dimensional spacepreserving a given point are represented by 3 × 3 real matrices O such that OTO = OOT = 13

and with determinant 1, i.e. by the matrices of the group SO(3). Hereafter, we shall denote thesematrices with the same symbol R as the corresponding rotations

Remark: According to the general formula given in § V.1.2 c, SO(3) is a Lie group of dimension 3,in agreement with what was found above for the group of rotations in three-dimensional Euclideanspace.

:::::::VI.1.2 a

:::::::::::::::::::::Generators of so(3)

As stated in § V.2.3 c, the Lie algebra so(3) consists of the antisymmetric real 3 × 3 matrices.That is, a rotation through an infinitesimally small angle |δψ| 1 about the direction along theunit vector ~n can be written as

R~n(δψ) = 13 − iδψJ~n +O(δψ2) (VI.6)

where the matrix −iJ~n is real and antisymmetric — and where the physicists’ convention (nota-

(w)O. Rodrigues, 1795–1851


tion V.29) was used. In turn, the rotation about ~n through a finite angle ψ reads

R~n(ψ) = e−iψJ~n , (VI.7)

as follows from the identity limp→∞

(1 +

x

p

)p= ex.

Considering the infinitesimal rotations about the three axes of the coordinate system, i.e. aboutthe unit vectors ~e1, ~e2, ~e3, one finds a basis of so(3), namely

−iJ1 =

0 0 00 0 −10 1 0

, −iJ2 =

0 0 10 0 0−1 0 0

, −iJ3 =

0 −1 01 0 00 0 0

. (VI.8)

Considering for instance the well-known matrix of a rotation around the third coordinate axis,

R3(ψ) =

cosψ − sinψ 0sinψ cosψ 0

0 0 1

,

and invoking Eq. (V.33), one indeed finds limψ→0

R3(ψ)− 13ψ

= −iJ3. 2

The corresponding matrices Jk — the generators, in the physicists’ convention(30) — are given by

J1 =

0 0 00 0 −i0 i 0

, J2 =

0 0 i0 0 0−i 0 0

, J3 =

0 −i 0i 0 00 0 0

, (VI.9a)

i.e. the ij entry of the matrix Jk reads

(Jk)ij = −iεijk (VI.9b)

where εijk is the usual totally antisymmetric Levi-Civita(x) symbol with ε123 = 1.

Remark VI.10. The generators (VI.9) obey the relation

Tr(JiJj) = 2δij for all i, j ∈ 1, 2, 3, (VI.10)

which can be viewed as a normalization condition on the generators.

:::::::VI.1.2 b

:::::::::::::::::::::::::::::::::Commutation relations of so(3)

Calculating the various commutators of the generators (VI.9), one finds

[J1, J2] = iJ3 , [J2, J3] = iJ1 , [J3, J1] = iJ2. (VI.11a)

These three equations, as well as the trivial commutator of every matrix Ji with itself, can begathered together in the form of the single equation

[Ji, Jj ] = i

3∑k=1

εijkJk for all i, j ∈ 1, 2, 3. (VI.11b)

One sees that the commutator of two different generators is non-zero, which is equivalent to thefact that rotations about different axes do not commute.

Comparing the Lie bracket (VI.11) with Eq. (V.29) shows that the structure constants of theLie algebra so(3) are f ijk = εijk for generators obeying the normalization condition (VI.10). In the

(30)Yet with the notation Jk instead of T a.(x)T. Levi-Civita, 1873–1941

VI.1 The group SO(3) 73

present case, the Jacobi identity (V.30b) becomes

3∑k=1

(εijkεklm + εjlkεkim + εlikεkjm

)= 0, (VI.12)

as can be checked using the possibly more familiar identity

3∑k=1

εkijεklm = δilδjm − δimδjl

and the property εkij = εijk.

:::::::VI.1.2 c

:::::::::::::::::::::::::::::::::::::::::::::::::::Generator of the rotations about an arbitrary axis

Comparing now the infinitesimal version (VI.4) of Rodrigues’ formula and Eq. (VI.6), one seesthat the generator J~n is such that −iJ~n~x = ~n ×~x for all ~x ∈ R3. That is, −iJ~n is a 3 × 3 matrixwhose ij element obeys

−i(J~n)ij =3∑

k=1

εikjnk = −3∑

k=1

εijknk,

where the second identity uses the antisymmetry of the Levi-Civita symbol while the nk are thecomponents of the unit vector ~n. Invoking Eq. (VI.9b), one may now write

−i(J~n)ij =

3∑k=1

[−i(Jk)ij

]nk

i.e.

J~n =

3∑k=1

nkJk ≡ ~n · ~J. (VI.13)

In the rightmost term of this equation, the dot product is not a scalar product, but merely aconvenient notation, since ~n is a “normal” vector of R3 while ~J is a shorthand notation for thetriplet consisting of the three matrices J1, J2, J3. Equation (VI.13) illustrates the fact that thelatter form a basis of generators, J~n being the generator of rotations about the ~n-direction.

Inserting Eq. (VI.13) into Eq. (VI.7), the rotation through ψ about the direction ~n may bewritten as

R~n(ψ) = e−iψ(n1J1+n2J2+n3J3). (VI.14)

Since the generators Jk do not commute with each other, see Eqs. (VI.11), the exponential of a sumof matrices on the right hand side of this equation cannot be recast as a product of exponentials.

Remarks:

∗ The property R~n(ψ) = R−~n(−ψ) can again be checked on Eq. (VI.14).

∗ Parameterizing a rotation through Euler angles, one can use the decomposition (VI.1b) and writeevery rotation about the z- or y-direction in exponential form, which leads to

R(α, β, γ) = e−iαJ3 e−iβJ2 e−iγJ3 . (VI.14)

This time, one may not rewrite the product of exponentials as the exponential of the sum of theexponents.


VI.2 The group SU(2)The group SU(2), consisting of the 2 × 2 complex matrices U such that U †U = UU † = 12 anddetU = 1, is like SO(3) a three-parameter Lie group. After recalling a few results on SU(2), weshall study its Lie algebra (Sec. VI.2.1), and then discuss the relation between SU(2) and SO(3) infurther detail (Sec. VI.2.2).

VI.2.1 The group SU(2) and its Lie algebra su(2)

:::::::VI.2.1 a

:::::::::::::::::::::::::::::::::::::::::::::::::Parameterization of a generic element of SU(2)

A generic element of SU(2) is of the form

U =

(u4 + iu3 u2 + iu1

u2 − iu1 u4 − iu3

)with

u1, u2, u3, u4 ∈ R

|u1|2 + |u2|2 + |u3|2 + |u4|2 = 1.(VI.15)

Conversely, one checks that every matrix of this form is in SU(2), which shows the existence of abijection between SU(2) and the unit sphere S3 ⊂ R4. With the help of the Pauli(y) matrices

σ1 =

(0 1

1 0

), σ2 =

(0 −i

i 0

), σ3 =

(1 0

0 −1

), (VI.16)

Eq. (VI.15) takes the form

U = u412 + i(u1σ1 + u2σ2 + u3σ3

)≡ u412 + i~u · ~σ, (VI.17)

where ~u resp. ~σ is a “vector” whose entries are the real numbers u1, u2, u3 resp. the three Paulimatrices (VI.16), while their dot product is a shorthand notation — similar to that introduced inEq. (VI.13).

Remark: The identity

σiσj = δij12 + i

3∑k=1

εijkσk for all i, j ∈ 1, 2, 3 (VI.18)

shows that the product of two matrices of the form (VI.17) is again of the same form — i.e. thatthe matrix multiplication is an internal composition law on SU(2).

Introducing the unit vector ~n ≡ −~u/|~u| and the “angle” ψ2 ∈ [0, π] defined by cos ψ2 = u4 and

sin ψ2 = |~u|, the term on the right hand side of Eq. (VI.17) becomes(

cosψ

2

)12 − i

(sin

ψ

2

)~n · ~σ = e−iψ~n·~σ/2,

where the identity follows from the definition of the exponential of a matrix under considerationof the identity (~n · ~σ)2 = 12. Denoting U~\(ψ) this matrix of SU(2), which is obviously entirelydetermined by the angle ψ and the unit vector ~n, we may thus write

U~n(ψ) = e−iψ~n·~σ/2 = e−iψ(n1σ1+n2σ2+n3σ3)/2. (VI.19)

The reader is invited to note the similarity between this expression for a generic matrix of SU(2),parameterized by ψ ∈ [0, 2π] and ~n ∈ S2, and Eq. (VI.14).

:::::::VI.2.1 b

:::::::::::::::::::::Generators of su(2)

The Lie algebra su(2) consists of the traceless antihermitian 2× 2 matrices. A possible basis ofgenerators, which is suggested by Eq. (VI.19), consists of the Pauli matrices. Conventionally, one(y)W. Pauli, 1900–1958

VI.2 The group SU(2) 75

rather takes as generators the matrices T i = 12σi for i = 1, 2, 3:

T 1 =

(0 1

212 0

)=σ1

2, T 2 =

(0 − i

2i2 0

)=σ2

2, T 3 =

(12 0

0 −12

)=σ3

2. (VI.20)

Denoting Ui(ψ) with i ∈ 1, 2, 3 the element of SU(2) obtained by taking the unit vector ~nwith components nj = δij in Eq. (VI.19), one has

i limψ→0

Ui(ψ)− 12ψ

=σi2,

which under consideration of Eq. (V.33) leads to the generators (VI.20). 2

Equation (VI.18) translates at once into

T iT j =δij412 +

i

2

3∑k=1

εijkTk for all i, j ∈ 1, 2, 3,

from which one deduces two properties of the generators (VI.20). On the one hand, they obey the“normalization”

Tr(T iT j) =δij2

for all i, j ∈ 1, 2, 3. (VI.21)

On the other hand, their commutator is given by

[T i, T j ] = i3∑

k=1

εijkTk for all i, j ∈ 1, 2, 3. (VI.22)

The structure constants of the Lie algebra su(2) are thus f ijk = εijk — like the structure constants ofso(3). Accordingly, any calculation carried out with the generators Ji [Eq. (VI.9)] or T i [Eq. (VI.20)]which only involves commutation relations will hold for both Lie algebras. We shall exploit thisresult in Sec. VI.3, when discussing the representations of the Lie groups SO(3) and SU(2).

VI.2.2 Relation between the Lie groups SU(2) and SO(3)

two-to-one group homomorphism from SU(2) to SO(3) — see exercise 38.iiiKernel of the homomorphism is 12,−12 ∼= Z2: isomorphism between SU(2)/Z2 and SO(3)

Both groups are connected. In addition, SU(2) is simply connected,(31) while SO(3) is not.SU(2) is the universal cover of SO(3) — i.e. SU(2) is the spin group Spin(3).

(31)That is, all closed paths on SU(2) can be continuously contracted to a point.


VI.3 Representations of the Lie groups SU(2) and SO(3)The reader has probably already encountered various physical objects whose mathematical repre-sentations transform differently under the rotations of three-dimensional space:(32)

• scalar quantities, which remain invariant under rotations and are represented by a singlenumber, as e.g. the electric charge or a scalar field(33) like temperature;

• “vector” quantities, represented by a triplet of real numbers that transform like the Cartesiancoordinates xi of the position, i.e.

V i′ =

3∑i=1

Ri′iV

i for i′ ∈ 1, 2, 3, (VI.23)

where the Ri′i are the elements of a matrix of SO(3);

• “tensor” quantities, represented by a collection of real numbers T i1...is labeled by two or moreindices taking their values in the set 1, 2, 3, such that each index of the tensor transformslike a “vector” index under rotations:

T i′1...i′s =

3∑i1=1

· · ·3∑

is=1

Ri′1i1· · ·Ri′s

isT i1...is for i′1, . . . , i

′s ∈ 1, 2, 3. (VI.24)

For instance, a tensor of rank 2 like the inertia tensor consists of 9 real numbers.

In each of these cases, the mathematical objects (scalars, vectors, tensors) are elements of variousvector spaces with dimension 1, 3 or more, which are in fact different representation spaces of thegroup of rotations of three-dimensional space — or equivalently of SO(3).

In this section, we systematically investigate the representations of SO(3) and of the companiongroup SU(2). Since both groups are compact Lie groups, we shall only study unitary representations,anticipating on results detailed in ??. After arguing why it is convenient to focus on representationsof the Lie algebra su(2) (Sec. VI.3.1), we derive in Sec. VI.3.2 the irreducible representations ofSU(2) — and in passing of SO(3). We then consider the direct product of such irreps. (Sec. VI.3.3).

Throughout the section, we shall often use the bra-ket notation of quantum mechanics. Moregenerally, the reader will probably recognize results from the formalism of angular momentum a.k.a.“spin” in quantum mechanics.

VI.3.1 Representations of SU(2) and of the Lie algebra su(2)

Since SU(2) and SO(3) are connected Lie groups, they are entirely determined, through expo-nentiation, by their respective Lie algebras. Generalizing this idea, we argue that representationsof the Lie groups can be deduced from representations of the Lie algebras, and more specifically ofsu(2) (§ VI.3.1 a), after which we discuss a few obvious representations (§ VI.3.1 b)

:::::::VI.3.1 a

::::::::::::::::::::::::::::::::::::::::::Representations of the Lie algebra su(2)

Consider the linear mapping φ from su(2) to so(3), seen as vector spaces, that maps eachgenerator T i ∈ su(2) to the generator Ji ∈ so(3) for i = 1, 2, 3. This mapping is clearly bijective, forit maps a basis of the first space to a basis of the second one. Since su(2) and so(3) have the samestructure constants f ijk = εijk, see Eqs. (VI.11) and (VI.22), the mapping φ will also preserve theLie brackets. All in all, ψ is thus a Lie algebra isomorphism, i.e. su(2) and so(3) are isomorphic.

(32)We omit here the discussion of the transformation property under spatial parity, i.e. the point reflection ~r → −~r,since the latter is not in SO(3), but rather in O(3).

(33)In the case of a field, one has a single number at each point in space (and time).

VI.3 Representations of the Lie groups SU(2) and SO(3) 77

Therefore, each representation D of su(2) provides a representation D φ−1 of so(3) — andconversely every representation of so(3) is of this form. Accordingly, we shall from now on talk ofrepresentations of su(2), knowing that they may be used for so(3) as well.

A representation of dimension d of the Lie algebra su(2) is a linear mapping from su(2) to amatrix Lie algebra, whose elements are d×d matrices, such that it preserves the commutator of twoarbitrary matrices. Accordingly, that mapping should also preserve the commutator of the imagesof the generators T i of su(2), which we shall denote Ji. That is, we shall look for a triplet of d× dHermitian matrices J1, J2, J3 obeying

[Ji,Jk

]= i

3∑k=1

εiklJl for all i, k ∈ 1, 2, 3. (VI.25)

By exponentiation, the generators Ji generate unitary d× d matrices

U(θ1, θ2, θ3) = e−i(θ1J1+θ2J2+θ3J3)

that constitute a unitary representation of dimension d of SU(2).

Remark: Beware of the different meanings of “dimension”, which should not be confused with eachother. On the one hand, there is the dimension of the Lie group, or of its Lie algebra, that is thenumber of real continuous parameters of the group (Sec. V.1.1), namely 3 in the case of su(2). Thisis also the number of generators of the Lie group/algebra. On the other hand, there is the dimensionof the representation space for a given representation of the Lie group (see definition II.5), or of itsLie algebra, which was denoted d in the previous discussion.

:::::::VI.3.1 b

::::::::::::::::::::::::::::::::::::::::::::Low-dimensional representations of SU(2)

Before investigating the properties of an arbitrary irreducible representation of SU(2), let usnote that we already know three of them.

First, there is the defining representation, in which every element U ∈ SU(2) is represented byitself! The corresponding generators are then the matrices T i introduced in Eq. (VI.20).

Since SU(2) consists of 2 × 2 matrices, this is a two-dimensional representation, very oftenreferred to as the “2” representation of SU(2) — as we shall see below, it is also called the “spin-1

2representation” and it is irreducible. Anticipating on the terminology introduced in ??, this is alsothe fundamental representation of SU(2).

Remark: As we shall discuss in § VI.3.2 b, the 2-representation of SU(2) is not a linear representationof SO(3).

According to Sec. VI.2.2, another representation of SU(2) is that which maps the unitary matrixU~n(ψ) ∈ SU(2) to the orthogonal matrix R~n(ψ) ∈ SO(3). This three-dimensional irreduciblerepresentation is the “3” representation of SU(2). Following the terminology of ??, this is theadjoint representation of SU(2) — and of SO(3) — since it is generated by the structure constants,see Eq. (VI.9b).

Note that as a representation of SU(2), it is not faithful, since the mapping SU(2) → SO(3) isnot injective.

Remark: From the point of view of SO(3), the 3-representation is also the defining and the funda-mental representation.

Eventually, the “1” representation of SU(2) or SO(3) is the trivial representation, for which thereis little to mention apart from the fact that the three corresponding generators are all equal to the1 × 1 matrix (0) — which is the only way to fulfill the commutation relation (VI.25) with suchcommuting matrices.


VI.3.2 Unitary representations of SU(2)

Let us now come to the general case of a unitary representation of SU(2) of arbitrary dimensiond > 1 generated by Hermitian matrices J1, J2, J3 obeying relation (VI.25). Instead of directlysearching for the generators, we shall follow the classical construction and investigate the eigenvaluesand eigenvectors of one of them (§ VI.3.2 a), and thereafter work in the basis of these eigenvectors.

:::::::VI.3.2 a

::::::::::::::::::::::::::::::::::::::::::::Diagonalization of the generators of SU(2)

Since two different generators Ji, Jk do not commute, they cannot be simultaneously diago-nalizable. That is, in any given basis at most one generator can be diagonal,(34) sayJ3, which weshall also denote Jz.

As is customary, we introduce an auxiliary d×d matrix, function of the generators Ji, namelythe (quadratic) Casimir (z) element

~J 2 ≡J 21 + J 2

2 + J 23 . (VI.26)

Since every generator Ji is Hermitian, so is ~J 2. In addition, one quickly checks that the commu-tation relations (VI.25) lead to [

~J 2,Ji

]= 0 ∀i ∈ 1, 2, 3, (VI.27)

where 0 here denotes the zero d× d matrix.

Remark: As we shall show below, all eigenvalues of the Casimir element are non-negative, so that~J 2 is not traceless. Therefore it does not belong to the Lie algebra generated by the Ji, but to

a larger algebra.

:::::::::::::::::::::::::::::Eigenvectors of ~J 2 and Jz

Since ~J 2 and Jz commute, they are simultaneously diagonalizable, i.e. define a basis of simul-taneous eigenvectors. We momentarily denote the latter |λ,m〉, with

~J 2 |λ,m〉 = λ|λ,m〉 , Jz |λ,m〉 = m|λ,m〉. (VI.28)

Since ~J 2 and Jz are Hermitian, their eigenvalues λ and m are real numbers. In turn, the eigen-vectors are d-dimensional complex vectors, i.e. elements of Cd, which are pairwise orthogonal forthe Hermitian scalar product. We further assume that they are normalized to unity: |||λ,m〉|| = 1.

From definition (VI.26) and the hermiticity of the generators follows that λ is non-negative.Indeed, one may consider scalar products with 〈λ,m| and write on the one hand

〈λ,m | ~J 2 |λ,m〉 =

3∑i=1

〈λ,m |J 2i |λ,m〉 =

3∑i=1

〈λ,m |J †i Ji |λ,m〉 =

3∑i=1

∣∣∣∣Ji |λ,m〉∣∣∣∣2 ≥ 0,

and on the other hand, using Eq. (VI.28) and the assumed normalization,

〈λ,m | ~J 2 |λ,m〉 = λ〈λ,m |λ,m〉 = λ.

Both equations together then yield λ ≥ 0. Since the mapping j ∈ R+ 7→ j(j + 1) ∈ R+ is bijective,one may thus write λ ≡ j(j + 1) with j ≥ 0. Replacing the notation |λ,m〉 by |j,m〉, the definingrelations (VI.28) become

~J 2 |j,m〉 = j(j + 1)|j,m〉Jz |j,m〉 = m|j,m〉

(VI.29a)

(VI.29b)

(34)Anticipating on the terminology of ??, SU(2) is said to be of rank 1.(z)H. Casimir, 1909–2000


with j ∈ R+ and m ∈ R. In turn, the orthonormality of the eigenvectors reads

〈j′,m′ |j,m〉 = δjj′ δmm′ for all j, j′ ∈ R+,m,m′ ∈ R. (VI.30)

::::::::::::::::::::“Ladder operators”

With the help of the non-diagonal generators J1 and J2, one defines two d× d matrices by

J+ ≡J1 + iJ2 and J− ≡J1 − iJ2. (VI.31)

In contrast to J1 and J2, the matrices J± are not Hermitian, but obey

J †+ = J− , J †

− = J+. (VI.32)

Using Eq. (VI.26), one first finds that the matrices J+ and J− commute with ~J 2:[~J 2,J+

]=[~J 2,J−

]= 0, (VI.33)

but do not commute with Jz. Writing [Jz,J±] = [J3,J1] ± i[J3,J2] = iJ2 ± i(− iJ1

), one

eventually obtains [Jz,J±

]= ±J±. (VI.34)

The commutation relations (VI.33), (VI.34) allow one to derive properties of the vectorsJ+ |j,m〉and J− |j,m〉. Thus, one first finds

~J 2J± |j,m〉 = J± ~J 2 |j,m〉 = J±(j(j + 1)|j,m〉

)= j(j + 1)J± |j,m〉.

This identity means that each vector J+ |j,m〉, J− |j,m〉 is either eigenvector of the Casimir element~J 2 with the eigenvalue j(j + 1), or equals the zero vector |∅〉.In addition, Eq. (VI.34) yields

JzJ± |j,m〉 =[Jz,J±

]|j,m〉+ J±Jz |j,m〉 = ±J± |j,m〉+ J±Jz |j,m〉.

Using Eq. (VI.29b), the second term on the right hand side equals mJ± |j,m〉, which leads to

JzJ± |j,m〉 = (m± 1)J± |j,m〉. (VI.35)

When it differs from the zero vector |∅〉, J+ |j,m〉 resp. J− |j,m〉 is eigenvector of Jz with theeigenvalue m+1 resp. m−1. The matrix J+ resp. J− thus increases resp. decreases the eigenvalueassociated to the generator Jz by 1, hence its denomination creation resp. annihilation operator inquantum-mechanical applications — together, they are referred to as the ladder operators.

::::::::::::::::::::::::::::Eigenvalues of ~J 2 and Jz

For any eigenvector |j,m〉, the square norm ||J± |j,m〉||2 must be non-negative. Using theHermitian conjugation (VI.32), this equals 〈j,m |J∓J± |j,m〉. Invoking the definitions (VI.31),the product J∓J± can be rewritten as

J∓J± =(Jx ∓ iJy

)(Jx ± iJy

)= J 2

x + J 2y ± i

[Jx,Jy

].

With the commutation relation (VI.25), this becomes

J∓J± = J 2z + J 2

y ± i(iJz

)= ~J 2 −J 2

z ∓Jz, (VI.36)

so that the matrix element 〈j,m |J∓J± |j,m〉 takes the form

〈j,m |J∓J± |j,m〉 = 〈j,m | ~J 2 |j,m〉 − 〈j,m |J 2z |j,m〉 ∓ 〈j,m |Jz |j,m〉.

Every term on the right hand side can be computed using Eqs. (VI.29) and the normalization ofthe eigenvector |j,m〉:

〈j,m | ~J 2 |j,m〉 − 〈j,m |J 2z |j,m〉 ∓ 〈j,m |Jz |j,m〉 = j(j + 1)−m2 ∓m = j(j + 1)−m(m± 1),


yielding eventually||J± |j,m〉||2 = j(j + 1)−m(m± 1). (VI.37)

This is non-negative if and only if |m| ≤ j.

Let us now fix j ∈ R+ and investigate the possible values of the eigenvalue m. Let mmax bethe largest one, which exists thanks to the condition |m| ≤ j; by definition, the correspondingeigenvector |j,mmax〉 is non-zero. Invoking Eq. (VI.35) (with J+), the vector J+ |j,mmax〉 mustbe the zero vector |∅〉 — otherwise it would be eigenvector of Jz with the eigenvalue mmax + 1,contradicting the definition of mmax. Considering Eq. (VI.37) then yields mmax = j, and thus theexistence of the eigenvector |j, j〉, characterized by

|j, j〉 6= |∅〉 and J+ |j, j〉 = |∅〉. (VI.38a)

Similarly, one shows using J− that the smallest possible eigenvalue of m at given j is −j, with acorresponding eigenvector obeying

|j,−j〉 6= |∅〉 and J− |j,−j〉 = |∅〉. (VI.38b)

Starting now from the eigenvector |j, j〉 and operating recursively with the matrix J−, one findsfor every p ∈ N (

J−)p |j, j〉 ∝ |j, j − p〉, (VI.39)

with the usual convention that(J−)0 is the identity matrix. To ensure that the eigenvaluem = j−p

remains bounded from below, j− p must for some value pmax of p equal the lower bound −j, whichgives 2j = pmax, i.e. 2j ∈ N. Stated differently, the real number j characterizing the eigenvalue ofthe Casimir element must be an integer or a half-integer:

j ∈ 0, 1

2, 1,

3

2, . . . . (VI.40a)

The other eigenvalue m then takes all values between −j and j in steps of 1:

m ∈ −j,−j + 1, . . . , j − 1, j, (VI.40b)

i.e. all in all 2j + 1 different values.

Remark: According to Eq. (VI.35), J+ |j,m〉 resp. J− |j,m〉 is either the zero vector or eigenvectorof Jz with the eigenvalue m+ 1 resp. m− 1, i.e. proportional to |j,m+ 1〉 resp. |j,m− 1〉. In turn,Eq. (VI.37) yields the norms of those eigenvectors, which fixes the proportionality factors to theorthonormal basis vectors up to a phase. A usual choice — in particular in quantum mechanics —for the relative phase between |j,m〉 und |j,m ± 1〉 is the so-called Condon(aa)–Shortley(ab)-phaseconvention, namely

J+ |j,m〉 =√j(j + 1)−m(m+ 1) |j,m+ 1〉 =

√(j −m)(j +m+ 1) |j,m+ 1〉 (VI.41a)

and

J− |j,m〉 =√j(j + 1)−m(m− 1) |j,m− 1〉 =

√(j +m)(j −m+ 1) |j,m− 1〉. (VI.41b)

:::::::VI.3.2 b

::::::::::::::::::::::::::::::::::::::::::::::::Irreducible representations of SU(2) and SO(3)

Consider the 2j+ 1-dimensional space V (j) spanned by the eigenvectors |j,m〉 where m takesall allowed values between −j and j. By definition of the eigenvectors, V (j) is invariant underJz = J3. According to Eq. (VI.39) and to the second of Eqs. (VI.38b), it also invariant under J−.Eventually, V (j) is also invariant under J+, since we have shown J+ |j,m〉 ∝ |j,m+1〉 for all m—unless m = j, in which case J+ |j, j〉 = |∅〉. Being invariant under both J+ and J−, V (j) is thusinvariant under J1 and J2, and therefore under all three generators.(aa)E. U. Condon, 1902–1974 (ab)G. H. Shortley, 1910–??


Denoting J(j)i the restriction of Ji to V (j) for every i = 1, 2, 3,(35) one at once sees that J

(j)1 ,

J(j)

2 , J(j)

3 map V (j) to itself and obey the commutation relation (VI.25), i.e. are the generators ofa representation of the Lie algebra su(2) on V (j), of dimension 2j + 1. Exponentiation then yieldsa (matrix) representation D (j) of the Lie group SU(2) of the same dimension, which is irreducible.

In the basis of V (j) consisting of the eigenvectors |j,m〉, the matrices U (j)(θ3) ≡ e−iθ3J(j)

3 —which represent the elements of SU(2) of the form e−iθ3σ3/2 — are diagonal, which shows thatany proper subspace of V (j) invariant under the U (j)(θ3) is spanned by a subset of that basis.One easily checks that such subspaces are not preserved by either J

(j)1 or J

(j)2 (or both), and

accordingly by the matrices e−iθ1J(j)

1 or e−iθ2J(j)

2 of the representation. 2

D (j) is called the spin j representation, and often simply denoted by its dimension (in thesenotes, in boldface): 2 (spin 1

2), 3 (spin 1), and so on. . . One can actually show that for every integeror half-integer j there exists an irreducible representation of SU(2) of dimension 2j + 1.

Remark: The Casimir operator[~J (j)]2 commutes by definition with all generators J

(j)i , and will

thus commute with all matrices of the form e−i(θ1J(j)1 +θ2J

(j)2 +θ3J

(j)3 ), i.e. all matrices of the spin-j

representation of SU(2). On the other hand, the previous calculations show that[~J (j)]2 is diagonal

on the representation space V (j) — it equals j(j + 1) times the identity matrix. Since the spin-jrepresentation is irreducible, one recognizes here Schur’s lemma.

Let D (j)(U) denote the (2j + 1)× (2j + 1) unitary matrix representing U ∈ SU(2) in the spin-jrepresentation, and D

(j)m′m(U) its matrix elements in the basis of the eigenvectors |j,m〉:

D (j)(U)|j,m〉 =

j∑m′=−j

D(j)m′m(U)|j,m′〉. (VI.42)

Choosing the parameterization U~n(ψ) of the SU(2) matrices introduced in Eq. (VI.19), with ~n aunit vector of R3 and ψ an angle, we denote the corresponding matrix in the spin-j representationD (j)(~n, ψ) and its elements D

(j)m′m(~n, ψ). By definition, the latter is

D(j)m′m(~n, ψ) ≡ 〈j,m′ |D (j)(~n, ψ) |j,m〉. (VI.43)

Consider first the case ~n = ~e3 ≡ ~ez, yielding D (j)(~ez, ψ) = e−iψJ(j)z . Since the basis vectors are

eigenvectors of J(j)z and orthonormal, one obtains at once the matrix elements of D (j)(~ez, ψ):

D(j)m′m(~ez, ψ) ≡ 〈j,m′ |e−iψJ

(j)z |j,m〉 = δm′m e−imψ. (VI.44)

In turn, setting ~n = ~e2 ≡ ~ey in Eq. (VI.43) defines an element of Wigner’s(ac) small d-matrix :

D(j)m′m(~ey, ψ) ≡ 〈j,m′ |e−iψJ

(j)2 |j,m〉 ≡ djm′m(ψ). (VI.45)

Tabulated values of these elements (for the small values of j) can be found in the literature, forinstance in Ref. [2], Chap. 45.(36)

Remarks:

∗ An element of Wigner’s D-matrix is

Djm′m(α, β, γ) ≡ 〈j,m′ |e−iαJ

(j)3 e−iβJ

(j)2 e−iγJ

(j)3 |j,m〉 = e−im′α djm′m(β) e−imγ , (VI.46)

where the matrix e−iαJ(j)3 e−iβJ

(j)2 e−iγJ

(j)3 is the (2j + 1)-dimensional generalization of the repre-

sentation (VI.14) of a rotation in three-dimensional Euclidean space with Euler angles.(35). . . as well as i = +,−, if need be(36)available at http://www-pdg.lbl.gov/2017/reviews/rpp2017-rev-clebsch-gordan-coefs.pdf.(ac)E. P. Wigner, 1902–1995

http://www-pdg.lbl.gov/2017/reviews/rpp2017-rev-clebsch-gordan-coefs.pdf


∗ As the reader can check,(37) the matrix D (j)(~ex, ψ) ≡ e−iψJ(j)1 , obtaining by letting ~n = ~e1 ≡ ~ex

in Eq. (VI.43), can actually be expressed through the matrices D (j)(~ez,±π2 ) and D (j)(~ey, ψ). That

is, D (j)(~ex, ψ) is entirely determined by Eqs. (VI.44) and (VI.45).

Coming back to the matrix elements (VI.44) and setting ψ = 2π, one obtains for every j

D(j)m′m(~ez, 2π) = δm′m e−2imπ = δm′m(−1)2m.

Since j −m ∈ N for every value of m, 2j − 2m is always an even integer, so that (−1)2m = (−1)2j

for all m, leading to

D(j)m′m(~ez, 2π) = δm′m(−1)2j for all m,m′ ∈ −j,−j + 1, . . . , j − 1, j. (VI.47a)

That is, the (2j + 1)-dimensional matrix representing the element U(~ez, 2π) = −12 of SU(2) issimply proportional to the identity matrix:

D (j)(~ez, 2π) = (−1)2j12j+1. (VI.47b)

In particular, for odd j, i.e. for a half-integral spin, D (j)(~ez, 2π) is the negative of the identitymatrix. This is OK as far as SU(2) is concerned, but it means that the matrices D (j)(~n, ψ) withj a half-integer do not form a linear representation of the group SO(3) — since a rotation throughan angle ψ = 2π is the identity transformation, which must be mapped to the identity matrix.(38)

As the reader knows, there do exist physical objects with half-integer spin: elementary par-ticles like electrons or quarks; protons and neutrons; atomic nuclei with an odd number ofconstituents. . .

Remarks:

∗ Conversely, Eq. (VI.47b) also means that every representation D (j) of SU(2) with j ∈ N will notbe faithful, since both 12 and −12 are mapped to the (2j + 1)-dimensional identity matrix.

∗ One can show that every representation D (j) with j ∈ N, i.e. every irreducible representation ofSO(3), is equivalent to a real representation.

This does not hold for the representations D (j) of SU(2) with half-integer spin j. However, thelatter can be shown to be equivalent to their complex conjugate D (j)∗, i.e. to be self-conjugate. Forinstance, one quickly checks the property σ2T

iσ−12 = −(T i)∗ for every generator of the defining

representation 2, which after multiplication by −i and exponentiation gives

σ2 U~n(ψ)σ−12 = U~n(ψ)∗,

illustrating the claim for the spin-12 case.

:::::::VI.3.2 c

:::::::::::::::::::::::::::::::::::::::::Characters of the spin-j representation

One can show that any matrix U~n(ψ) = e−iψ~n·~σ/2 of SU(2) is similar to the diagonal matrixe−iψσ3/2 = diag(e−iψ/2, eiψ/2) with a similarity matrix which is itself an element of SU(2). Thisgives at once the conjugacy classes of SU(2), which are thus entirely characterized by the parameterψ ∈ [0, 2π].

In the spin-j representation, the matrix e−iψ~n · ~J (j) representing U~n(ψ) will then be similar toe−iψJ

(j)3 = D (j)(~ez, ψ), i.e. they have the same trace:

Tr(e−iψ~n · ~J (j))

= Tr[D (j)(~ez, ψ)

]=

j∑m=−j

e−imψ,

(37)The relation generalizes the conjugacy relation (VI.5) between matrices of SO(3).(38)The half-integral spin representations are projective representations of SO(3), i.e. representations “up to a phase

factor”.


where the second identity uses the matrix elements (VI.44). The sum is easily computed and yields

χ(j)(ψ) ≡ Tr(e−iψ~n · ~J (j))

=

2j + 1 if ψ = 0

sin[(j + 12)ψ]

sin ψ2

for ψ 6= 0,(VI.48)

where we have introduced the notation χ(l) for the trace, which is obviously the character of therepresentation. Note that the formula given for ψ 6= 0 tends to 2j + 1 in the limit ψ → 0, so thatthe character is a continuous function of the parameter ψ labeling the conjugacy classes of SU(2).

Remark: Following a similar reasoning, one shows that every matrix e−iψ~n · ~J (j) of the spin-j repre-sentation of SU(2) has determinant 1, which means that the generators J

(j)i are traceless.

VI.3.3 Tensor products of irreducible representations of SU(2)

:::::::VI.3.3 a

:::::::::::::::::::::::::Clebsch–Gordan series

Theorem VI.49. Let D (j1), D (j2) be two irreducible representations of the Lie group SU(2). Theirtensor product is then decomposable, with the Clebsch–Gordan series

D (j1) ⊗D (j2) =

j1+j2⊕J=|j1−j2|

D (J). (VI.49)

Applying both sides of Eq. (VI.49) to an element U~n(ψ) of SU(2) and taking the trace, oneobtains an identity involving characters:

χ(j1)(ψ)χ(j2)(ψ) =

j1+j2∑J=|j1−j2|

χ(J)(ψ) for all ψ, (VI.50)

irrespective of the choice of the unit vector ~n. This result can actually be shown using the expres-sion (VI.48) for χ(j)(ψ) and the usual addition / multiplication theorems for trigonometric functions,which in turn proves the decomposition (VI.49).

The reader interested in the proof can first show the identity

χ(j)(ψ)χ(j′)(ψ) = χ(j+j′)(ψ) + χ(j− 12 )(ψ)χ(j′− 1

2 )(ψ),

and then apply it to (j, j′) = (j1, j2), and iteratively to (j, j′) = (j1− k2 , j2−

k2 ) for the successive

values k = 1, 2, . . . , 2 min(j1, j2)− 1, using eventually χ(0)(ψ) = 1.

Remarks:

∗ In the language of physicists, the tensor product of irreducible representations corresponds tothe “addition of spins” (or “angular momenta”), and theorem VI.49 is the well-known rule for thisaddition, according to which the total spin J of a system consisting of a spin j1 and a spin j2 cantake the values J = |j1 − j2|, |j1 − j2|+ 1, . . . , j1 + j2.

∗ Every irreducible representation D (J) with J ∈ |j1−j2|, . . . , j1+j2 comes up with multiplicity 1(and the others with multiplicity 0) in the Clebsch–Gordan series (VI.49), i.e. SU(2) is simplyreducible (cf. definition III.54).

∗ Using Eq. (VI.49), one sees that every irreducible representation D (J) can be obtained by con-sidering enough tensor products of the defining representation D ( 1

2) with itself, which justifies the

denomination “fundamental”.


Example VI.51. To illustrate the Clebsch–Gordan series (VI.49), one may write down the tensorproduct of two spin-1 representations, which are also irreducible representations of SO(3):

D (1) ⊗D (1) = D (0) ⊕D (1) ⊕D (2). (VI.51a)

Denoting the representations by their respective dimension, this also reads

3⊗ 3 = 1⊕ 3⊕ 5, (VI.51b)

which shows that the dimensions on both sides of the equal sign indeed match.Consider now two 3-dimensional real vectors ~v, ~w with coordinates vi, wj : both transform

under spin-1 representation of SO(3). One can construct a tensor (for three-dimensional rotations)of rank 2 consisting of the 9 entries viwj : the latter transforms under SO(3) under the reduciblerepresentation 3⊗ 3. More precisely, one can write

viwj =1

3

(~v · ~w

)δij +

1

2

(viwj − vjwi

)+

1

2

[viwj + vjwi − 2

3

(~v · ~w

)δij]. (VI.51c)

The first term on the right hand side is invariant under rotations, i.e. transforms under the trivialrepresentation (1). The second, antisymmetric term — in which one recognizes the componentsof the cross product ~v × ~w — transforms under the 3 representation. Eventually, the last term issymmetric and traceless — i.e. the sum over i = j equals 0 —, and can be shown to transformunder the 5 representation of SO(3).

:::::::VI.3.3 b

:::::::::::::::::::::::::::::::Clebsch–Gordan coefficients

The Clebsch–Gordan series (VI.49) involves two different “natural” orthonormal bases. The basisadapted to the left hand side consists of all possible tensor products of basis vectors |j1,m1〉 and|j2,m2〉, which will be denoted |j1, j2;m1,m2〉, where only m1 and m2 are running while j1 andj2 are fixed. In turn, the right hand side rather involves the bases |J,M〉 with J running from|j1 − j2| to j1 + j2 and for each J in that set, M ∈ −J, . . . , J.

As in Sec. III.4.3 [see Eq. (III.52)], one defines the Clebsch–Gordan coefficients CJ ;Mj1,j2;m1,m2

asthose entering the decomposition of the vector |J,M〉 on the basis |j1, j2;m1,m2〉:

|J,M〉 =∑m1,m2

CJ ;Mj1,j2;m1,m2

|j1, j2;m1,m2〉. (VI.52a)

Exploiting fully the bra-ket notation of quantum mechanics, one also writes

CJ ;Mj1,j2;m1,m2

≡ 〈j1, j2;m1,m2 |J,M〉. (VI.52b)

The latter notation allows one to recognize the meaning of various properties of the coefficients atonce, see e.g. Eqs. (VI.54) below.

The Clebsch–Gordan coefficients obey a number of properties, a few of which we now list.(39)

• The Clebsch–Gordan coefficients are real . Accordingly, one may also write

CJ ;Mj1,j2;m1,m2

= 〈J,M |j1, j2;m1,m2〉

besides Eq. (VI.52b).

This “property” is actually a convention for the choice for the relative phase of the basisvectors |j,m〉, when one explicitly constructs them by successive tensor product startingfrom the basis vectors | 12 ,±

12 〉 of the fundamental representation.

(39)Further results on the coefficients can be found in standard textbooks on Quantum Mechanics, as e.g. Ref. [3],chapter X (complement BX) or Ref. [4], chapter 4.3.

VI.4 Behavior of quantum-mechanical observables under rotations 85

• The coefficient CJ ;Mj1,j2;m1,m2

is non-zero if and only if the numbers J,M, j1, j2,m1,m2 obey the“selection rules”

M = m1 +m2 (VI.53a)|j1 − j2| ≤ J ≤ j1 + j2 (VI.53b)J + j1 + j2 ∈ N. (VI.53c)

• The Clebsch–Gordan coefficients obey various “orthogonality relations”, in particular∑m1,m2

CJ ;Mj1,j2;m1,m2

CJ′;M ′

j1,j2;m1,m2= δJJ ′δMM ′ (VI.54a)

and ∑J,M

CJ ;Mj1,j2;m1,m2

CJ ;Mj1,j2;m′1,m

′2

= δm1m′1δm2m′2

. (VI.54b)

VI.4 Behavior of quantum-mechanical observables under rotations

VI.4.1 Rotation of a quantum-mechanical system

Consider a quantum-mechanical system Σ, whose physical state is described by a ket-vector |Ψ〉of a Hilbert space H . Let A denote an observable of the system, i.e. a Hermitian operator on H .Figure VI.2 represents both Σ and an apparatus A for the measurement of the physical quantityassociated to A, for various relative spatial configurations.

(a) (b)

(c) (d)

Σ Σ

Σ Σ

A A

A A

Figure VI.2

(a) In the reference configuration, in which the vector state of the system is |Ψ〉, measurementsof the expectation value of the observable A yield 〈Ψ |A |Ψ〉.

(b) With respect to (a), the system Σ has been rotated: let R denote the corresponding rotation(in three-dimensional Euclidean space). The latter induces a unitary transformation on theHilbert space H of the system (see discussion in § II.3.3 b), such that the state vector becomesR |Ψ〉. Accordingly, the expectation value of A is now 〈Ψ |R†AR |Ψ〉.


(c) The system Σ has the same orientation as in (a), yet now the measurement apparatus has beenrotated, by the same rotation R as that affecting Σ in (b). To account for the fact that therelative orientation of Σ and the apparatus has changed with respect to (a), whereas the statevector |Ψ〉 remains unchanged, one has to consider that the rotation induces a transformationof the observable A to a new Hermitian operator ˆ

R(A)on H . In turn, the expectation value

of measurements performed by the apparatus is now 〈Ψ | ˆR(A)|Ψ〉.

(d) Rotating now both the physical system Σ and the measurement apparatus by R, the statevector becomes R |Ψ〉 and the observable ˆ

R(A), i.e. the expectation of measurements is given

by 〈Ψ |R† ˆR(A)R |Ψ〉.

If space is isotropic, i.e. if physics is invariant under rotations, then the measurements in theconfigurations (a) and (d) should give the same results, since the relative orientations of the physicalsystem and the measurement apparatus are identical in both cases. That is, one has

〈Ψ |R† ˆR(A)R |Ψ〉 = 〈Ψ |A |Ψ〉.

Requiring that this identity hold for any state vector |Ψ〉 ∈H yields R†ˆ

R(A)R = A, and thus

ˆR(A)

= RAR†. (VI.55)

One easily checks that the transformation operator R2R1 acting on observables A which cor-

responds to successive rotations R1 and R2 — i.e. to the rotation R2R1 — is the product of theoperators ˆ

R2 and ˆR1 induced by each rotation:

R2R1 =

ˆR2

ˆR1, (VI.56)

which shows that the operators ˆR form a group.

Remark VI.57. The set of linear operators A on a Hilbert space H form a vector space L, calledLiouville(ad) space. The operators ˆ

R introduced in the preceding discussion are operators on L,and relation (VI.56) means that they form a linear representation of the group of rotations ofthree-dimensional Euclidean space, or equivalently of the Lie group SO(3), on L.

Besides the (cumbersome) notation ˆO with a double hat for the operators on L, one may also

introduce a convenient notation for its elements, in which an operator A ∈ L on H is denoted|A〉〉. The equations involving such “superkets” and “superoperators” then take a similar form asthose for kets of H and the operators acting on them, as e.g. ˆ

O|A〉〉 instead of ˆO(A).

::::::::::::::::::::::Infinitesimal rotations

Let R ~δϕdenote the rotation through an infinitesimal angle δϕ about the direction ~n, where for

the sake of brevity we introduced the shorthand notation ~δϕ ≡ (δϕ)~n. The corresponding inducedunitary operator on the Hilbert space H reads [cf. Eq. (VI.6)]

R ~δϕ= 1H − i ~δϕ · ~J +O

(δϕ2), (VI.58)

where the three operatorsJi with i = 1, 2, 3 are the Hermitian generators of a representation of thegroup SO(3) on H , while 1H is the identity operator on H .Similarly, the operator ˆ

R ~δϕon the Liouville space L induced by the rotation R ~δϕ

takes the form

ˆR ~δϕ

= ˆ1L − i ~δϕ ·

ˆ~J +O

(δϕ2), (VI.59)

where the “superoperators” ˆJi

are the generators of the representation of SO(3) on L.

(ad)J. Liouville, 1809–1882


Using Eq. (VI.55), the operator ˆR ~δϕ

acts on an operator A ∈ L according to

ˆR ~δϕ

(A)

= R ~δϕAR†~δϕ

, (VI.60a)

which under consideration of Eq. (VI.58) gives, up to terms of order δϕ2

ˆR ~δϕ

(A)

=(1H − i ~δϕ · ~J

)A(1H + i ~δϕ · ~J

),

where the hermiticity of the generators Ji was used. Thus, one finds

ˆR ~δϕ

(A)

= A− i[~δϕ · ~J , A

]+O

(δϕ2). (VI.60b)

Comparing this result with the generic form (VI.59) shows that the generator ˆJi of the repre-

sentation of SO(3) on the Liouville space acts on an operator A on H according toˆ

Ji

(A)≡[Ji, A

]for i = 1, 2, 3. (VI.61)

Remark: Using Eq. (VI.61), one can check that the generators ˆJi obey the commutation relation[

ˆJi,

ˆJk

]= i

3∑l=1

εiklˆ

Jl (VI.62)

of the group SO(3).

VI.4.2 Irreducible tensor operators

:::::::VI.4.2 a

::::::::::::::::::::::::::::::::Definition and characterization

Definition VI.63. Let j ∈ N. A set of 2j + 1 operators T (j)m with m ∈ −j,−j + 1, . . . , j on a

Hilbert space H is said to form a spin-j irreducible tensor operator if they constitute the “standard”basis of a spin-j irreducible representation of SO(3) on the Liouville space L of the linear operatorson H .

This definition can be translated into various characterizations of the operators T (j)m . First,

denoting D(j)m′m(R) the entries of the (2j+ 1)× (2j+ 1)-matrix representing a given rotation R [see

Eq. (VI.42)], one hasˆ

R(T (j)m

)=

j∑m′=−j

D(j)m′m(R)T

(j)m′ . (VI.64a)

Invoking Eq (VI.55), one may also write

R T (j)m R† =

j∑m′=−j

D(j)m′m(R)T

(j)m′ , (VI.64b)

where R is the unitary operator on H representing the rotation R.Considering now infinitesimal rotations instead of finite ones, one deduces from the latter equa-

tion the equivalent identity [Ji, T

(j)m

]=

j∑m′=−j

[J(j)i ]m′m T

(j)m′ , (VI.65a)

where the operator Ji is one the three Hermitian generators of the representation of SO(3) on Hwhile the coefficients [J

(j)i ]m′m are its matrix elements in the basis of eigenvectors |j,m〉, which

are easily read off Eqs. (VI.29b) and (VI.41) [under consideration of definitions (VI.31)]. Thus, onemay write [

Jz, T(j)m

]= mT (j)

m (VI.65b)


and [J±, T

(j)m

]=√j(j + 1)−m(m± 1) T

(j)m±1. (VI.65c)

Equations (VI.65) show that an irreducible tensor operator is fully characterized by its commutationrelations with the generators of rotations Ji, which is what we shall exploit in § VI.4.2 b below.

Using Eq. (VI.61), the previous relations can be rewritten with the help of the generators ˆJi of

the representation of SO(3) on the Liouville space L :

ˆJi

(T (j)m

)=

j∑m′=−j

[J(j)i ]m′m T

(j)m′ (VI.66a)

ˆJz

(T (j)m

)= mT (j)

m (VI.66b)

ˆJ±(T (j)m

)=√j(j + 1)−m(m± 1) T

(j)m±1. (VI.66c)

Remark: The reader is invited to rewrite the equations (VI.66) with the superket notation |j,m〉〉(see remark VI.57) in lieu of T (j)

m , so as to bring them in a form similar to that of relations alreadyencountered in Sec. VI.3.

:::::::VI.4.2 b

:::::::::::Examples

Scalar operatorsA scalar operator A is an operator that commutes with all three generators Ji of rotations,[

Ji, A]

= 0 for all i = 1, 2, 3 (VI.67)

corresponding to the case j = 0.A first example of scalar operator is the quadratic Casimir operator ~J 2 [see Eqs. (VI.26) and

(VI.27)]. Equation (VI.67) also characterizes the Hamilton operator of a system which is invariantunder rotations.

Vector operatorsConsidering now the case j = 1, a vector operator ~V is a set of three operators Vx, Vy, Vz that

obey the commutation relation[Ja, Vb

]= i

∑c=x,y,z

εabcVc for all a, b ∈ x, y, z, (VI.68)

where as usual Jx ≡J1, Jy ≡J2, and Jz ≡J3. Introducing the standard components

V(1)

1 ≡ − 1√2

(Vx + iVy) , V(1)

0 ≡ Vz , V(1)−1 ≡

1√2

(Vx − iVy), (VI.69)

one checks that the latter obey Eqs. (VI.65b)–(VI.65c) with j = 1 and m = 1, 0,−1, respectively.

In his lectures on quantum mechanics, the reader should already have encountered a numberof examples of vector operators, in particular the position operator ~r, momentum operator ~p, andorbital angular-momentum operator ~L ≡ ~r × ~p of a particle, as well as the intrinsic-spin operator~S — which, even when its describes the spin degree of freedom of a spin-1

2 particle, consists of threeoperators forming a vector operator!

:::::::VI.4.2 c

::::::::::::::::::::::::::::::::::::::::::::::::Tensor product of irreducible tensor operators

see e.g. exercise 44. More later.


VI.4.3 Wigner–Eckart theorem

Consider an irreducible tensor operator T (j)m . According to the Wigner–Eckart (ae) theorem, its

matrix elements in the basis of the common eigenvectors of the generator Jz and the Casimiroperator ~J 2 take the form

〈j′′,m′′ |T (j)m |j′,m′〉 = Cj

′′;m′′

j,j′;m,m′〈j′′ || T (j) ||j′〉, (VI.70)

where Cj′′;m′′

j,j′;m,m′ is the Clebsch–Gordan coefficient defined by Eqs. (VI.52) while 〈j′′ || T (j) ||j′〉, knownas the reduced matrix element , is now independent of m, m′, and m′′.

Viewing the matrix representation of T (j)m in the basis of eigenvectors |j′,m′〉, it means that

the (2j′ + 1)× (2j′′ + 1) block associated to fixed values j′ and j′′ is proportional to the matrix ofthe same size consisting of the Clebsch–Gordan coefficients Cj

′′;m′′

j,j′;m,m′ .

Remark: Throughout this chapter, we only consider transformations under SO(3) (and SU(2)), i.e.the tensor operators introduced in Sec. VI.4.2 are characterized by their commutation relations withgenerators of a representation of SO(3), while the Clebsch–Gordan coefficients are those of SO(3).The whole construction, including the Wigner–Eckart theorem, can be generalized to other groupsas e.g. the special unitary groups SU(n).

:::::::::::::Applications

In physical applications, the Wigner–Eckart theorem in particular leads to selection rules forthe transitions induced by an operator involving an irreducible tensor operator.Consider for instance an atomic system invariant under rotations, whose energy eigenstates havedefinite values of the orbital and magnetic quantum numbers ` and m ∈ −`, . . . , `. When thesystem is in an external (classical) electric field ~E, its Hamilton operator is perturbed by a termwhich in the dipolar approximation takes the form W = −~P · ~E, where

~P =∑i

qi~ri

is the electric dipole operator, which depends on the charges and positions of the particles (nucleusand electrons) constituting the system. This perturbation can induce a transition between an initialstate |i〉 and a final state |f〉, with a rate proportional (to first order in perturbation theory) to thesquared amplitude |〈f |W |i〉|2.

Expressing the scalar product ~P · ~E through the standard components P (1)m with m = −1, 0, 1

of the dipole operator, one sees that the computation of the transition rate involves that of theamplitude 〈f |P (1)

m |i〉. Invoking the Wigner–Eckart theorem, the latter is proportional to a Clebsch–Gordan coefficient:

〈f |P (1)m |i〉 ∝ C

`f ;mf1,ì;m,mi

where ì,mi resp. `f ,mf are the quantum numbers of the initial resp. final state. Using the selectionrules (VI.53) obeyed by the Clebsch–Gordan coefficients, one deduces the following conditions onthese quantum numbers to ensure a non-vanishing matrix element:

|ì − `f | ≤ 1 , |mi −mf | ≤ 1 , ì + `f ≥ 1.

The first two conditions mean that a dipolar transition can at most change the angular momentumby one unit (of ~), and the third one, that the initial and final states cannot be both s-states (with` = 0).

(ae)C. Eckart, 1902–1973


As second example, consider the case of a spin-j′ system whose precise “polarization”, i.e. valueof m′, is unknown — and where every single polarization is equally probable —, so that in thecalculation of the expectation value of an observable A, one needs to average over the differentpolarization states, i.e.

〈A〉 =1

2j′ + 1

j′∑m′=−j′

〈j′,m′ |A |j′,m′〉.

In the case of a spin-j tensor observable T (j)m , the Wigner–Eckart theorem (VI.70) gives

〈T (j)m 〉 =

1

2j′ + 1

j′∑m′=−j′

〈j′,m′ |T (j)m |j′,m′〉 =

1

2j′ + 1

j′∑m′=−j′

Cj′;m′

j,j′;m,m′〈j′ || T (j) ||j′〉,

where the reduced matrix element 〈j′ || T (j) ||j′〉 can actually be taken out of the sum. Now, one canshow that the sum of the coefficients Cj

′;m′

j,j′;m,m′ over all allowed m′ values actually vanishes unlessj = 0 and m = 0 — in which case it trivially equals 2j′ + 1, yielding

〈T (j)m 〉 = δj0 δm0 〈j′ || T (j) ||j′〉.

That is, the expectation value of an irreducible tensor operator in an “unpolarized state” vanisheswhen the operator is not scalar.

CHAPTER VII

Representations of GL(n,C)

and its continuous subgroups

integer n ≥ 2

VII.1 Low-dimensional irreducible representations of GL(n,C)

VII.1.1 Representations of dimension 1

::::::::VII.1.1 a

::::::::::::::::::::::Trivial representation

As always, the mapping which maps an arbitrary elementM ∈ GL(n,C) to the complex number1 constitutes the trivial representation of the group GL(n,C), denoted 1.

::::::::VII.1.1 b

:::::::::::::::::::::::::::::Determinant representation

Another representation of dimension 1 consists of the mapping which maps an given elementM ∈ GL(n,C) to its determinant detM ∈ C \ 0.

In the case of the subgroups of GL(n,C) consisting of matrices with determinant 1, like SL(n,C),SL(n,R), SU(n) or SO(n), the determinant representations obviously coincides with the trivialrepresentation.

VII.1.2 Vector representation

Definition VII.1. The defining representation of GL(n,C), i.e. the automorphism of GL(n,C) whichmaps every matrix M ∈ GL(n,C) to itself, is called vector representation and denoted n.

The latter notation corresponds to the dimension of the representation, which comes from thefact that the matrices of GL(n,C) operate on complex vectors of dimension n, i.e. on the vectorspace Cn. Since there is no proper subspace of Cn which is invariant under all elements of GL(n,C),the vector representation is irreducible.

VII.1.3 Complex conjugate representation

Theorem & Definition VII.2. The mapping which to every matrix M ∈ GL(n,C) associates thematrix M∗ ∈ GL(n,C) whose entries are complex conjugate to those of M is a faithful irreduciblerepresentation of GL(n,C) of dimension n, called the complex conjugate representation of the vectorrepresentation and denoted nnn.

In the case of the subgroups of GL(n,C) consisting of real matrices, i.e. of GL(n,R) and itssubgroups, the complex conjugate representation nnn coincides with the vector representation n.

92 Representations of GL(n) and its continuous subgroups

VII.1.4 Dual representation

Theorem & Definition VII.3. The mapping which to every matrix M ∈ GL(n,C) associates thematrix (M−1)T ∈ GL(n,C) is a faithful irreducible representation of GL(n,C) of dimension n,called the dual or contragredient representation of the vector representation and denoted n∗.

Considering that the matrix M represents a regular linear map V → W in given bases, whereV and W are n-dimensional complex vector spaces, then M−1 represents a map W → V , and(M−1)T a map V ∗ → W ∗, where V ∗ and W ∗ are the dual spaces to V and W , respectively.

Property VII.4. In the case of the unitary group U(n) and its subgroups SU(n), O(n), SO(n). . . ,the dual representation coincides with the complex conjugate representation.

A unitary matrix U ∈ U(n) obeys U U † = U †U = 1n, i.e. (U −1)T = (U †)T = U ∗. 2

Eventually, yet another irreducible representation of dimension n of the group GL(n,C) is thedual of the complex conjugate, denoted n∗n∗n∗, which maps M ∈ GL(n,C) to the Hermitian conjugateof its inverse, (M−1)†.

VII.2 Tensor representations of GL(n,C)In this section...

VII.2.1 Tensor products of the vector representation

In Sec. VII.1.2, we introduced the vector representation n of GL(n,C): the corresponding n×n-matrices operate on the vector space Cn according to

x ∈ Cn → x′ = Mx ∈ Cn, (VII.5a)

i.e. component-wise

xi → x′i =

n∑j=1

M ijxj ∀i ∈ 1, . . . , n, (VII.5b)

where the xi resp. x′i are the components of x resp. x′ and M ij the entries of the matrix M ,

where as usual the left resp. right index stands for the line resp. column of the matrix.Starting from the vector space Cn, one constructs for every p ∈ N∗ the tensor-product space

(Cn)⊗p ≡ Cn ⊗ · · · ⊗ Cn︸︷︷︸p copies

,

whose elements are tensors of order p, hereafter generically denoted TTT, with components (in a givenbasis) traditionally characterized by p indices:

T i1i2...ip with i1, i2, . . . , ip ∈ 1, . . . , n.

Generalizing Eq. (VII.5), one defines the operation of M ∈ GL(n,C) on such tensors:

Theorem & Definition VII.6. The mapping from GL(n,C) into the group of regular linear applica-tions of the tensor-product space (Cn)⊗p into itself which associates to a matrix M with entries M i

j

the linear operator M⊗p such that

M⊗p : TTT→ T′T′T′ = (M ⊗ · · · ⊗M︸︷︷︸p copies

)TTT, (VII.6a)

corresponding in terms of components to

VII.2 Tensor representations of GL(n,C) 93

T i1i2...ip → T ′i1i2...ip =n∑

j1=1

· · ·n∑

jp=1

M i1j1M

i2j2 · · ·M ip

jpTj1j2...jp , (VII.6b)

is a representation of GL(n,C) of dimension np, called the tensor-product representation

n⊗ · · · ⊗ n︸︷︷︸p copies

. (VII.6c)

This representation is generally reducible when p > 1.

VII.2.2 Operations on the indices of a tensor

::::::::VII.2.2 a

::::::::::::::::::::::::::::::::::::::::Operation of a permutation on a tensor

Definition VII.7. Let σ ∈ Sp be a permutation of p elements. One associates with it an operator σacting on the tensor-product space (Cn)⊗p such that for every tensor TTT of order p, the tensor σ(TTT)has the components (

σT)i1i2...ip = T iσ(1)iσ(2)...iσ(p) . (VII.7)

That is, the permutation acts on the tensor indices.

Remark: One can actually show that this definition is independent of the choice of basis in (Cn)⊗p.

Example VII.8. Letting p = 2 and considering the transposition τ12 = (1 2) ∈ S2, the associatedoperator τ12 is such that for every second-order tensor TTT, the tensor τ12(TTT) has the components(

τ12T)ij

= T ji for all i, j ∈ 1, . . . , n. (VII.8)

Property VII.9. The operator σ associated to a permutation σ ∈ Sp is linear.

Theorem VII.10. For every permutation σ ∈ Sp and every matrix M ∈ GL(n,C), the linearoperators σ and M⊗p on the tensor-product space (Cn)⊗p commute. That is, denoting the image byM⊗p with a ′ as in Eq. (VII.6a), one has

σ(TTT′) =(σ(TTT)

)′ (VII.10)

for every tensor TTT of order p.

::::::::VII.2.2 b

::::::::::::::::::Young operators

Any linear combination of linear operators on the tensor-product space (Cn)⊗p is again a linearoperator. Accordingly, one can associate such operators to the elements of the group algebra CSpin the same manner as in definition VII.7. In particular, to each Young element Y introduced in§ III.5.1 b corresponds a linear operator Y on (Cn)⊗p, generically called Young operators.

For instance, one associates to the symmetrizer (III.55a) and to the antisymmetrizer (III.55b)for p elements the operators

S ≡ 1

p!

∑σ∈Sp

σ (VII.11a)

A ≡ 1

p!

∑σ∈Sp

ε(σ)σ, (VII.11b)

where we recall that ε(σ) denotes the signature of the permutation σ. Invoking Eq. (VII.7), theirrespective actions on a tensor TTT ∈ (Cn)⊗p of order p read


(S T

)i1i2...ip =1

p!

∑σ∈Sp

T iσ(1)iσ(2)...iσ(p) (VII.12a)

(A T

)i1i2...ip =1

p!

∑σ∈Sp

ε(σ)T iσ(1)iσ(2)...iσ(p) (VII.12b)

for every p-tuple (i1, i2, . . . ip). The tensor S (TTT) resp. A (TTT) is then symmetric resp. antisymmetricunder the exchange of any two indices.

In the case of order-2 tensors, these expressions become(S T

)ij=

1

2

(T ij + T ji

)and

(A T

)ij=

1

2

(T ij − T ji

)(VII.13)

for every i, j ∈ 1, . . . , n.

Remark: While the operators σ associated to permutations are regular, this is no longer the case ofthose associated to Young elements. For instance, the operation of the antisymmetrizer (VII.11b)on any fully symmetric tensor(40) — i.e. on an element of the image of (Cn)⊗p by S — is the zerotensor.

VII.2.3 Irreducible representations built from the vector representation

::::::::VII.2.3 a

::::::::::::::::::::::::::::::::::::::::::::::::::Irreducible representations and Young operators

The theorem (VII.10) extends to the Young operators Y , which all commute with every operatoron (Cn)⊗p of the form M⊗p with M ∈ GL(n,C):

for every TTT ∈ (Cn)⊗p , Y(M⊗p(TTT)

)= M⊗p

(Y (TTT)

).

One deduces thereof at once that the image of every Young operator Y is a subspace of thetensor-product space (Cn)⊗p which is invariant under all operators M⊗p of the tensor-productrepresentation n⊗ · · · ⊗ n.

Consider TTT ∈ im Y : there exists SSS ∈ (Cn)⊗p such that TTT = Y (SSS). Then for any M ∈ GL(n,C)one may write

M⊗p(TTT) = M⊗p(Y (SSS)

)= Y

(M⊗p(SSS)

),

which shows that M⊗p(TTT) ∈ im Y . 2

Accordingly, the tensors of order p belonging to the image of any Young operator Y spana subspace of (Cn)⊗p on which the restrictions of the linear operators M⊗p constitute a linearrepresentation of GL(n,C). One can show that this representation is irreducible. More precisely,one has the following result:(41)

Theorem VII.14. Given an integer p and a Young operator Y acting on order-p tensors, theimage by Y of the tensor-product space (Cn)⊗p is the representation space of an irreducible linearrepresentation D (Y ) of the group GL(n,C), which is contained in the Clebsch–Gordan decompositionof the p-fold tensor-product representation n⊗ · · · ⊗ n.

More generally, the representation D (Y ) is also an irreducible representation of the subgroupsSL(n,C), GL(n,R), SL(n,R), U(n), and SU(n). On the other hand, it is in general reduciblewith respect to GL(n− 1,C) and its subgroups, and with respect to O(n), SO(n) and the indefiniteorthogonal groups O(n−m,m), SO(n−m,m) where m ∈ 1, . . . , n− 1.

(40)or more generally, on any tensor which remains unchanged under the exchange of two of its indices.(41)A reference would be welcome!


::::::::VII.2.3 b

::::::::::::::::Young tableaux

Building on the correspondence introduced in Sec. III.5.1 between Young elements of CSp and(standard) Young tableaux with p boxes, we may now associate a specific Young tableau with therepresentation D (Y ) introduced in the previous paragraph.

More precisely, each irreducible representation contained in the Clebsch–Gordan decompositionof the p-fold tensor-product representation n ⊗ · · · ⊗ n built with the vector representation n ofGL(n,C) is associated with a Young diagram with p boxes, such that the number of rows of thediagram equals at most n.

Examples:

∗ The vector representation n is associated with the Young diagram with a single box .

∗ For tensors of order 2, there are two possible Young diagrams. As was mentioned in § III.5.1 b,the diagram with a simple row corresponds to the symmetrizer, while that with a single column isassociated to the antisymmetrizer:

S ≡ 1

2

(Id + (1 2)

)::::I ; (VII.15a)

A ≡ 1

2

(Id− (1 2)

)::::I ; (VII.15b)

∗ With 3 boxes, i.e. associated with tensors of order 3, we have found 4 standard Young tableaux(Fig. III.1). The one with a single line is associated to the symmetrizer, that with a single columnwith the antisymmetrizer, so we want to focus on those corresponding to the Young diagram .Replacing the numbers 1, 2, 3 by the indices i, j, k, we have the consider the two tableaux

i jk

and i kj

.

Let us in particular deal with the left one. Adapting what was found in § III.5.1 b, the correspondingYoung element (of CS3) is Y = Id+(i j)− (i k)− (i j k), up to an irrelevant factor. The associatedYoung operator Y is then such that its action on a tensor TTT gives a tensor with components

(Y T )ijk = T ijk + T jik − T kji − T jki,

i.e. symmetric under the exchange of i and j, which are in the first row of the Young tableau, andantisymmetric under the exchange of i and k, which both sit in the same column of the Youngtableau.

The procedure outlined in the last example can be generalized. Starting from a given standardYoung tableau with p boxes — associated with a Young element Y following the recipe given in§ III.5.1 b —, one replaces one-to-one the numbers 1,. . . ,p in the boxes by indices i1,. . . ,ip. Theexpression of the component (Y T )i1...ip of the tensor Y (TTT) can then be deduced from the expressionof Y in terms of permutations of Sp, by letting the permutations act on the indices.

The antisymmetry with respect to the indices in a given column of the Young tableau explainswhy the latter can have at most n rows. If it had more, there would always be at least two of thecorresponding indices taking the same value, so that the antisymmetrization automatically wouldgive 0: the associated tensor is the zero tensor of order p.

Remark: The above recipe gives the form of the tensors that constitute the representation space ofa given irrep. contained in the p-fold tensor-product of the vector representation, but it does notspecify the actual representation of a matrix M of GL(n,C).


Dimension of the irrep. of GL(n,C) associated with a Young tableauGiven a Young diagram with p boxes, the necessary symmetrization resp. antisymmetrization

over indices in a same row resp. column leads to restrictions on the number of independent p-tuplesof indices which are relevant — corresponding to the dimension of the associated representationspace. To compute this dimension d, one can combine the various combinatorial constraints into asimple recipe, which now give and illustrate on the example of the Young diagram

with 7 boxes.

• As first step, one fills the boxes of the Young diagram, with n in the upper left box and thenincreasing resp. decreasing by 1 for each step further right in a row resp. down in a column:

n n+1 n+2 n+3

n−1 n

n−2

The p numbers thus entered in the Young diagram are multiplied with each other (yieldinghere n2(n2 − 1)(n2 − 4)(n+ 3)).

• In a second step, one enters in another copy of the same Young diagram the respective hooklengths of the boxes, as introduced in § III.5.1 a. As already illustrated on the same examplein Fig. III.3, this gives

6 4 2 1

3 1

1

These hook lengths are then multiplied with each other (yielding here 6 · 4 · 2 · 2).

• Eventually, the wanted dimension d is the ratio of the number found in the first step dividedby that found in the second step.

Remark: If a column contains more than n boxes, then the entry in the (n+ 1)-th box in the firststep would be zero, resulting in a vanishing dimension for the associated representation: the recipeautomatically “knows” about the maximal possible number of boxes in a column.

Using this recipe, one quickly finds the dimensions of the representations associated with thediagram with a single row resp. column with p boxes:

· · ·︸︷︷︸p boxes

: dimensionn(n+1) · · · (n+p−1)

p(p−1) . . . 1=

(n+p−1)!

(n−1)!p!=

(n+p−1

p

); (VII.16)

p boxes

...: dimension

n(n−1) · · · (n−p+1)

p(p−1) . . . 1=

n!

(n−p)!p!=

(n

p

). (VII.17)

Note two interesting special cases of the latter formula (which is only meaningful for p ≤ n).First, if there are p = n boxes, the associated representation has dimension 1, and one easily sees


that the corresponding representation of GL(n,C) is the determinant representation introduced in§ VII.1.1 b.

In the case of a single column with p = n− 1 boxes, Eq. (VII.17) shows that the correspondingirreducible representation has dimension n. If n = 2, this is clear with n − 1 boxes one obtainsthe vector representation n. For n ≥ 3, the result is more interesting, and will discussed at furtherlength in Sec. VII.2.5 hereafter.

::::::::VII.2.3 c

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Tensor products of irreps. built from the vector representation

“outer product” introduced in Sec III.5.2

VII.2.4 Tensor products of the dual and conjugate representations

The construction of Secs. VII.2.1–VII.2.3 can readily be repeated to build representations ofGL(n,C) from tensor products of the three further n-dimensional irreducible representations (com-plex conjugate, dual, and dual to the complex conjugate) introduced in Secs. VII.1.3–VII.1.4.

::::::::VII.2.4 a

::::::::::::::::::::::::::::::::::::::::::::Tensor products of the dual representation

As was mentioned in Sec. VII.1.4 the vector space on which the dual representation n∗ acts isactually the dual space of Cn, i.e. the space of the linear forms Cn → C. In matrix representation,such a linear form is represented by a row vector, which we shall denote xT (where x stands for acolumn vector). In addition, the components of a linear form in a given basis will be denoted withsubscript-indices.

If M is an arbitrary element of GL(n,C), the operation of the matrix (M−1)T of the dualrepresentation on an arbitrary linear form xT reads

xT → x′T = xT(M−1)T, (VII.18a)

which indeed gives a linear form on Cn. In terms of components, this becomes

xi → x′i =

n∑j=1

(M−1)ijxj =

n∑j=1

xj((M−1)T

)ji∀i ∈ 1, . . . , n, (VII.18b)

where xi resp. x′i denote the components of xT resp. x′T and (M−1)ij

=((M−1)T

)jithe entry

in the i-th row and j-th column of the matrix M−1, i.e. in the j-th row and i-th column of thematrix (M−1)T.

In complete similarity with what was done with the vector representation, one can now definefor any p ≥ 2 the p-fold tensor product representation

n∗ ⊗ · · · ⊗ n∗︸︷︷︸p copies

, (VII.19)

such that the action of an element M ∈ GL(n,C) on a tensor of order p built from linear forms, i.e.with components characterized by p subscript indices

Ti1i2...ip with i1, i2, . . . , ip ∈ 1, . . . , n,

reads

Ti1i2...ip → T ′i1i2...ip =n∑

j1=1

· · ·n∑

jp=1

(M−1)i1j1(M−1)i2

j2 · · · (M−1)ipjpTj1j2...jp (VII.20)

for every p-tuple i1, i2, . . . , ip.

Remarks:

∗ Subscript indices, associated with linear forms and their tensor products, are generally referredto as contravariant . In turn, superscript indices, associated with vectors, are called covariant .


∗ A quantity which is traditionally denoted with n contravariant indices is the fully antisymmetricn-dimensional Levi-Civita symbol εi1i2...in , defined by

εi1i2...in =

+1 if i1, i2, . . . , in is an even permutation of 1, 2, . . . , n;

−1 if i1, i2, . . . , in is an odd permutation of 1, 2, . . . , n;

0 if at least two indices take the same value.(VII.21)

Letting a matrix M ∈ GL(n,C) act on it according to Eq. (VII.20) yields

εi1i2...in → ε′i1i2...in =

n∑j1=1

· · ·n∑

jn=1

(M−1)i1j1(M−1)i2

j2 · · · (M−1)injnεj1j2...jn = det(M−1)εi1i2...in .

(VII.22)If the determinant ofM (and thusM−1) differs from 1, the symbol is not invariant under the actionof M , and in particular loses its defining property ε12...n = 1. In that respect, the Levi-Civitasymbol is not considered as a “tensor”.

On the other hand, εi1i2...in remains invariant under the action of all elements of SL(n,C), and thusis a “valid” contravariant tensor for this group and its subgroups — in particular SU(n) or SO(n).

As in Sec. VII.2.3, one can let Young operators act on the contravariant tensors of order p, whichleads to irreducible representations of GL(n,C) that are contained in the Clebsch–Gordan series ofthe p-fold tensor-product representation n∗ ⊗ · · · ⊗ n∗.

In turn, these irreducible representations can be associated with Young tableaux, which specifythe symmetry properties of the indices of the respective tensors. These tableaux are built from afundamental building block which is no longer the vector representation n, but now the dualrepresentation n∗.

Note that the recipe for the dimension of the irrep(s). built from the vector representation andassociated with a given Young diagram still holds for the irreducible representations built from thedual representation.

::::::::VII.2.4 b

::::::::::::::::::::::::::::::::::::::More tensor product representations

Instead of using the vector or dual representation as fundamental building block, one may alsouse the complex conjugate nnn or its dual n∗n∗n∗, and construct tensor-product representations

nnn⊗ · · · ⊗ nnn , n∗n∗n∗ ⊗ · · · ⊗ n∗n∗n∗ (VII.23)

that operate on respective tensors whose components involve a “new type” of indices for eitherpossibility,(42) since they now transform with M∗ or (M−1)† instead of M or (M−1)T.

One may also construct “mixed” tensors, which constitute the objects that transform underrepresentations built from more than one building block n, n∗, nnn, n∗n∗n∗. For instance, the tensor-product representation n⊗ n∗ acts on tensors with one covariant and one contravariant index:

T ij → T ′ij =

n∑k=1

n∑l=1

M ik(M

−1)jlT kl for all M ∈ GL(n,C), i, j ∈ 1, . . . , n,

which combines Eqs. (VII.5b) and (VII.18b) in a natural way.All in all, one can show(43) that the finite-dimensional irreducible representations of GL(n,C)

indeed act on tensors whose components are labeled by 4 types of indices. However, these irreps.do not coincide with the set of all possible tensor products of n, n∗, nnn, n∗n∗n∗ and of the irreduciblerepresentations built from each of these building blocks with the help of Young operators. Forinstance, one can already find that the representation n⊗n∗ is reducible in GL(n,C) — it containsthe trivial representation.(42)A common notation uses “dotted” indices, cf. Sec. ?? on the irreducible representations of SL(2,C).(43)NB: need a reference!


VII.2.5 Representations of SL(n,C) and SL(n,R)

As mentioned in the second part of theorem VII.14, the finite-dimensional irreducible represen-tations of GL(n,C) are also irreducible with respect to its subgroups SL(n,C), GL(n,R), SL(n,R),U(n) and SU(n). However, representations that are inequivalent with respect to GL(n,C) may nowbecome equivalent, which leads to a number of simplifications, as we now sketch in this section andthe following.

We first quickly discuss the representations of the groups SL(n,C), GL(n,R), and SL(n,R).Since these groups are not compact, their respective finite-dimensional irreducible representationsare not unitary, and not equivalent to a unitary representation.

::::::::VII.2.5 a

::::::::::::::::::::::::::::::Representations of SL(n,C)

In the case of the group SL(n,C), the dual representation n∗ is equivalent to the irreduciblerepresentation acting on totally antisymmetric tensors of order n − 1 with covariant indices, i.e.built on on the vector representation n only:

n∗ ∼= ...

n− 1 boxes (VII.24)

More generally, one shows the equivalence between the irreducible representations acting on totallyantisymmetric covariant tensors of order p and on totally antisymmetric contravariant tensors oforder n− p, respectively.

The key to these equivalences is that the n-dimensional Levi-Civita symbol εi1i2...in is invari-ant under SL(n,C), whose matrices all have determinant 1. Therefore, one can introduce acorrespondence

T i1...ip 7→ Ti1...in−p≡

n∑in−p+1=1

· · ·n∑

in=1

εi1i2...inTin−p+1in−p+2...in

between both type of tensors which is preserved under the action of any matrix of SL(n,C).

Obviously, similar equivalences hold between pairs of representations acting on totally antisymmetrictensors built from the complex conjugate representation nnn and acting on its dual n∗n∗n∗.

Accordingly, the finite-dimensional irreducible representations of SL(n,C) act on tensors whosecomponents are labeled by two sets of indices only, instead of 4. Moreover, each of these sets isassociated with a Young tableau — which characterizes the symmetry properties of the respectiveindices — with at most n − 1 rows. A column with n boxes is indeed equivalent to the trivialrepresentation, and thus irrelevant.

We shall illustrate these ideas on the example of SL(2,C) in a later chapter.

::::::::VII.2.5 b

:::::::::::::::::::::::::::::::::::::::::::::Representations of GL(n,R) and SL(n,R)

As already mentioned in Sec. VII.1.3, the vector and complex-conjugate representations n andnnn of GL(n,R) coincide. Similarly, the dual representation n∗ and its complex conjugate nnn∗ areidentical.

Accordingly, the tensors constituting the representation spaces of the finite-dimensional irre-ducible representations of GL(n,R) are characterized by two sets of indices, covariant and con-travariant, i.e. which transform respectively like a vector or like a linear form.

Since SL(n,R) is a subgroup of GL(n,R), the equalities of the n and nnn and of the n∗ and nnn∗

representations further hold. In addition, a further simplification comes from the fact that SL(n,R)is also a subgroup of SL(n,C), so that the results given in § VII.2.5 a apply. In particular, the dualrepresentation n∗ is related to the vector representation n, see Eq. (VII.24).


All in all, one thus finds that the finite-dimensional irreducible representations of SL(n,R) acton tensors whose components involve indices of a single type, i.e. which can be built from the vectorrepresentation n only.

VII.2.6 Representations of SU(n)

Let us now consider the representations of the special unitary group SU(n). A first importantproperty is that since SU(n) is compact, its finite-dimensional irreducible representations are alwaysequivalent to a unitary representation — as is obvious in the case of the defining representation n.

::::::::VII.2.6 a

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Relations between the vector representation and its dual and complex-conjugate

Since SU(n) is a subgroup of SL(n,C), the results of § VII.2.5 a hold, in particular the equiva-lence (VII.24) relating the dual representation n∗ to the vector representation n. In addition, theproperty VII.4 valid for unitary groups gives a further relation between the dual and the complex-conjugate representation nnn, which are identical. Altogether, this gives

nnn = n∗ ∼= ...

n− 1 boxes, (VII.25)

where stands for the vector representation n, and all four n-dimensional irreducible represen-tations listed in Sec. VII.1.2–VII.1.4 are related.

In the case n = 2, relation (VII.25) means that the complex-conjugate representation 222 of SU(2)is equivalent to the defining representation 2. This is indeed the case, since one can check startingfrom Eq. (VI.17) that a generic matrix U ∈ SU(2) obeys U∗ = σ2Uσ

−12 , as already noted in the last

remark of § VI.3.2 b.Starting from n = 3, the defining representation and its complex-conjugate (or dual) are in-

equivalent.

::::::::VII.2.6 b

::::::::::::::::::::::::::::::::::Adjoint representation of SU(n)

Besides the vector representation n and its complex conjugate nnn, another irrep. of SU(n), theadjoint representation, is often encountered in physical applications,(44) which we now introduce.

Consider the tensor product n ⊗ nnn of the vector representation and its complex conjugate.Representing them by Young diagrams and using relation (VII.25), one finds the Clebsch–Gordanseries

⊗ ...=

... ⊕ ... ,

n− 1

n

(n−1)+1 boxes

(VII.26a)

where the first Young diagram on the right-hand side stands in fact for the trivial representation —for which there is no simpler diagram. Denoting the representations by their respective dimensions,this reads

n ⊗ nnn = 1 ⊕ n2 − 1, (VII.26b)

where the reader is invited to check the dimension (n2−1) of the rightmost irreducible representationin Eq. (VII.26a) with the help of the recipe given in § VII.2.3 b. That is, the tensor product(44)See Sec. IX.3 for examples.


representation n ⊗ nnn decomposes into the direct sum of two irreducible representations, one ofwhich is the trivial representation. The second one, with dimension n2 − 1, is called the adjointrepresentation.

One can show that the adjoint representation is self-conjugate, i.e. that it is equivalent to the(n2 − 1)-dimensional representation in which every matrix is replaced by its complex conjugate.

Here are two possible lines of arguments to prove that result:First, looking at the Young diagram, the column with n − 1 boxes, corresponding to the anti-symmetrization of n − 1 covariant indices, is equivalent to a single contravariant index, whilethe column with only one box, which stands for a single contravariant index, is equivalent to aset of n − 1 antisymmetrized contravariant indices. That is, the Young diagram of the adjointrepresentation is the same as that of its dual — which is also its complex conjugate.

Alternatively, one may realize that the tensor product n ⊗ nnn is “obviously” self-conjugate, andthat in its Clebsch–Gordan series (VII.26b) the trivial representation is also self-conjugate, sothat the remaining adjoint representation must be self-conjugate as well.

As a consequence, the tensor product of the adjoint representation with itself contains the trivialrepresentation in its Clebsch–Gordan series, as can be readily checked by performing the outerproduct of the associated Young diagrams.

Example: In the case of SU(2), for which we know the Clebsch–Gordan series (see theorem VI.49),the adjoint representation is the 3, which indeed appears in the tensor product of the self-conjugate2-representation with itself. In a more physics-oriented language, the addition of two spins 1

2(defining representation, 2) gives a triplet (spin 1, adjoint representation 3) and a singlet (spin 0,trivial representation 1).

Eventually, the addition of two spins 1 can give rise to a spin 0 (besides a spin 1 and a spin 2),i.e. the tensor product of two adjoint representations contains the trivial representation.

:::::::::::::::::::::::::::::::::::::::::Generators of the adjoint representation

The dimension n2 − 1 of the adjoint representation is also the number of real parameters ofthe Lie group SU(n). As a matter of fact, the adjoint representation of SU(n) can be deducedby the exponentiation from a representation of the Lie algebra su(n), whose n2 − 1 generatorsTAA=1,...,n2−1 are directly related to the structure constants fabc of the group. More precisely,the BC-entry of the generator TA is

(TA)BC = −ifABC for all A,B,C ∈ 1, . . . , n2 − 1. (VII.27)

The Jacobi identity (V.30b) then expresses the property that the commutators of these generatorsobey the characteristic relation (V.29) involving the structure constants.

Remark: In Eq. (VII.27), we used the convenient (but far from universal!) convention in whichlabels running from 1 to n2 − 1 are denoted with capital letters, while those running from 1 to nare denoted with lowercase indices.

For example, the 3 generators of the defining representation of SO(3), which is also the adjointrepresentation of SU(2), given in Eq. (VI.9) are precisely such that their matrix elements are relatedto the structure constants of SU(2) by in the above equation.

CHAPTER VIII

Lorentz and Poincaré groups

Consider the four-dimensional real vector space R4. Its vectors will generically be denoted in asans-serif font, as e.g. x. Assuming a basis has been chosen, the components of a vector will bedenoted with Greek letters, running from 0 to 3: xµ. The components x1, x2, x3 will often becollectively represented by a three-vector ~x.

The introduction of a pseudo-Euclidean metric with signature (−,+,+,+), corresponding to ametric tensor ηηη with components

ηµν ≡

−1 for µ = ν = 0

+1 for µ = ν ∈ 1, 2, 30 for µ 6= ν

(VIII.1a)

turns R4 into a pseudometric space, called Minkowski (af) spacetime and hereafter denoted M 4. Itis convenient to associate with the metric tensor a 4× 4 matrix η defined as

η = diag(−1, 1, 1, 1) (VIII.1b)

i.e. such that its entries are precisely the components ηµν .

VIII.1 Definitions and first propertiesWith the metric tensor (VIII.1), the pseudo-distance between two infinitesimally close points withrespective coordinates xµ and xµ + dxµ is the so-called line element

ds2 ≡ dxµηµν dxν = −(dx0)2 + d~x2, (VIII.2a)

where d~x2 denotes the squared Euclidean norm of the three-vector with components (dx1, dx2,dx3).Introducing the (column) four-vector dx with components dxµ and the transposed row-vector dxT,one can equivalently write

ds2 = dxTηdx, (VIII.2b)

where η is the matrix introduced in Eq. (VIII.1b).

VIII.1.1 Poincaré group

Theorem & Definition VIII.3. The transformations x ∈ M 4 → x′ ∈ M 4 that leave the infinitesimalline element (VIII.2) invariant, i.e. the Minkowski spacetime isometries, form a group, called thePoincaré(ag) group.

Requiring that the transformations should be at least twice continuously differentiable, one canshow that they are necessarily affine,(45) i.e. of the form(45)A proof is presented in Appendix VIII.A to this chapter.(af)H. Minkowski, 1864–1909 (ag)H. Poincaré, 1854–1912

VIII.1 Definitions and first properties 103

xµ → x′µ = Λµνxν + aµ ∀µ ∈ 0, 1, 2, 3, (VIII.4)

where the real numbers aµ are arbitrary, while the coefficients Λµν obey the relation

Λµρ ηµνΛνσ = ηρσ ∀ρ, σ ∈ 0, 1, 2, 3. (VIII.5a)

Equation (VIII.4) leads to dx′µ = Λµν dxν , which in turn gives for the infinitesimal line elementds′2 = dx′µηµνdx′ν

dx′µηµν dx′ν = Λµρ dxρηµν Λνσ dxσ = dxρΛµρ ηµν Λνσ dxσ.

By definition this should equal ds2 = dxρηρσ dxσ, which is only possible for every dxµ if andonly if Eq. (VIII.5a) holds. 2

Viewing the numbers Λµν as the entries of a 4×4 matrix Λ, Eq. (VIII.5) is equivalent to the matrixidentity

ΛTηΛ = η, (VIII.5b)

where η was introduced in Eq. (VIII.1b). This condition can directly be derived by using theinfinitesimal line element in its form (VIII.2b) and writing dx′ = Λdx.

Theorem VIII.6. The spacetime translations x→ x′ = x + a, i.e. component-wise

xµ → x′µ = xµ + aµ ∀µ ∈ 0, 1, 2, 3 (VIII.6)

with aµ ∈ R, form a normal subgroup of the Poincaré group.

Another important subgroup of the Poincaré group, which however is not a normal subgroup,is that introduced in the following section.

VIII.1.2 Lorentz group

Theorem & Definition VIII.7. The linear transformations

x ∈ M 4 → x′ = Λx ∈ M 4 (VIII.7a)

or equivalently

xµ → x′µ = Λµνxν ∀µ ∈ 0, 1, 2, 3 (VIII.7b)

that preserve the infinitesimal line element (VIII.2) are called Lorentz(ah) transformations and forma group, the Lorentz group, denoted O(3, 1).

Remarks:

∗ As the notation suggests, the Lorentz group is one of the indefinite orthogonal groups introducedin § V.1.2 d.

∗ Obviously, the Lorentz transformations are the Minkowski spacetime isometries that leave a pointof M 4 invariant, i.e. such that aµ = 0 in Eq. (VIII.4). Accordingly, the Lorentz group is a subgroupof the Poincaré group.

(ah)H. A. Lorentz, 1853–1926

104 Lorentz and Poincaré groups

::::::::VIII.1.2 a

:::::::::::::::::::::::::::::::::::::::Examples of Lorentz transformations

The matrices of the form

ΛR =

1 0 0 0

00 R0

with R ∈ SO(3) (VIII.8)

represent a first class of Lorentz transformations, whose action on a four-vector is a rotation of thespatial components while keeping the time-component unchanged: (x0,~x) → (x′ 0 = x0,~x ′ = R~x).For instance, the rotation through ψ around the x1-axis is represented by the matrix

ΛR(~e1,ψ) =

1 0 0 00 1 0 00 0 cosψ − sinψ0 0 sinψ cosψ

. (VIII.9)

Theorem VIII.10. The matrices of the form (VIII.8) form a subgroup of the Lorentz group.This subgroup is actually a (faithful and reducible) representation of SO(3).

A second large class of Lorentz transformations consists of the so-called Lorentz boosts (or specialLorentz transformations). For such a boost with (reduced) velocity β ∈ ]− 1, 1[ along the directionwith unit vector ~n ∈ S2, the corresponding transformation reads(46)

x0 → x′ 0 = x0 cosh ξ + (~n ·~x) sinh ξ

~x→ ~x ′ = ~x+ (cosh ξ − 1)(~n ·~x)~n+ (sinh ξ)x0~n,(VIII.11)

where ξ ≡ artanhβ ∈ R is the rapidity of the boost. For a Lorentz boost along the x1-direction,the associated transformation matrix is

Λ =

cosh ξ sinh ξ 0 0sinh ξ cosh ξ 0 0

0 0 1 00 0 0 1

. (VIII.12)

The Lorentz boosts do not form a group — successive boosts along non-parallel directions do notyield a boost, but the combination of a boost and and spatial rotation. However, the Lorentz boostsalong a fixed (arbitrary) direction ~n do form a subgroup of the Lorentz group, which is isomorphicto (R,+).

Let us eventually mention three particular Lorentz transformations, which will help us charac-terize the connected components of the Lorentz group in the following paragraph.• The space inversion transformation (x0,~x) → (x′ 0 =x0,~x ′=−~x) reads in matrix representa-

tion

ΛP =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (VIII.13)

• The time reversal transformation (x0,~x)→ (x′ 0 =−x0,~x ′=~x) is represented by the matrix

ΛT =

−1 0 0 00 1 0 00 0 1 00 0 0 1

. (VIII.14)

(46)The reader is invited to note the similarity between the form of this transformation and Rodrigues’ formula (VI.3)for rotations.

VIII.1 Definitions and first properties 105

• Eventually, the composition of the previous two transformations transforms x into x′ = −x,i.e. the corresponding matrix is the negative of the identity, ΛPΛT = ΛTΛP = −14.

These three transformations are obviously their own inverses, and one easily checks that the matrices14,ΛP ,ΛT ,ΛPΛT form a subgroup which is isomorphic to the Klein group V4.

:::::::::VIII.1.2 b

:::::::::::::::::::::::::::::::Structure of the Lorentz group

Taking the determinant of both sides of Eq. (VIII.5b) gives (detΛ)2 detη = detη for everymatrix Λ of the Lorentz group, that is, since detη is non-zero,

detΛ = ±1. (VIII.15)Theorem & Definition VIII.16. The Lorentz transformations with determinant equal to 1, calledproper Lorentz transformations, form a subgroup of the Lorentz group denoted SO(3, 1).

In contrast, the Lorentz transformations with determinant −1 are called improper , as e.g. thespatial-parity and time-reversal transformations (VIII.13)–(VIII.14).

Consider now the property (VIII.5a) obeyed by the coefficients of a Lorentz transformation forthe case ρ = σ = 0:

Λµ0 ηµνΛν0 = η00 = −1.

On the right hand side, only the terms with µ = ν are relevant, and one obtains(Λ0

0

)2= 1 +

3∑i=1

(Λi0)2 ≥ 1, (VIII.17)

valid for all Λ ∈ O(3, 1). Accordingly, the coefficient Λ00 is either greater than or equal to +1, or

smaller than or equal to −1. More precisely, one easily shows the following two results:

Theorem & Definition VIII.18. The Lorentz transformations such that Λ00 ≥ 1, called orthochronous

Lorentz transformations, form a subgroup of the Lorentz group denoted O+(3, 1).

Theorem & Definition VIII.19. The proper, orthochronous Lorentz transformations, i.e. such thatdetΛ = +1 and Λ0

0 ≥ 1, form a normal subgroup of the Lorentz group denoted SO+(3, 1) and calledrestricted Lorentz group.

Theorem VIII.20. The most general element of the restricted Lorentz group SO+(3, 1) is a productof a Lorentz boost (VIII.11) and a rotation transformation (VIII.8).

Property VIII.21. SO+(3, 1) is the connected component of the identity in O(3, 1).

In fact, the Lorentz group — like every indefinite orthogonal group — consists of four connectedcomponents, which can be characterized with the help of the spatial inversion (VIII.13) and timereversal transformation (VIII.14):

• The connected component of the identity is the restricted Lorentz group SO+(3, 1) introducedin definition (VIII.19). Note that this component contains the subgroup of rotations (VIII.8).

• The connected component of the space inversion transformation (VIII.13) consists of thematrices of the form Λ = ΛPΛ

↑+ with Λ↑+ ∈ SO+(3, 1). These matrices Λ are precisely the

orthochronous (Λ00 ≥ 1) improper (detΛ = −1) Lorentz transformations.

• The connected component of the time reversal transformation (VIII.14) consists of the matricesof the form Λ = ΛTΛ

↑+ with Λ↑+ ∈ SO+(3, 1). These are the Lorentz transformations with

detΛ = −1 and Λ00 ≤ −1 (“antichronous”).

• Eventually, the Lorentz transformations with detΛ = 1 and Λ00 ≤ −1 are of the form −Λ↑+

with Λ↑+ ∈ SO+(3, 1) and constitute the connected component of −14.


Besides the topological meaning of the four subsets SO+(3, 1), ΛP SO+(3, 1), ΛT SO+(3, 1), andΛPΛT SO+(3, 1) mentioned above, one can also note that these are the cosets of the restrictedLorentz group in O(3, 1), i.e.

O(3, 1)/SO+(3, 1) ∼= 14,ΛP ,ΛT ,ΛPΛT ∼= V4, (VIII.22)

where the second isomorphism was already mentioned at the end of § VIII.1.2 a.

VIII.2 The Lorentz and Poincaré groups as Lie groupsAccording to theorem (VIII.20), any element of the restricted Lorentz group SO+(3, 1) can bewritten as the product of a Lorentz boost and a three-dimensional rotation. As the reader knowsfrom Sec. VI.1, the latter are characterized by three real parameters. In turn, a general Lorentzboost (VIII.11) also depends on three real parameters, namely the rapidity ξ and the two parametersassociated with the unit three-vector along which the boost takes place.

All in all, the Lorentz group O(3, 1) is thus a 6-parameter group. Since the rapidity can takeany value in R, this is a non-compact group.

Turning to the Poincaré group, one deduces from the generic form (VIII.4) of a Minkowski space-time isometry that it is a non-compact 10-parameter group. Thus, 6 real parameters correspond tothe Lorentz transformations (Λµν), and 4 to the spacetime translations (aµ).

In this section, we shall investigate the generators of both groups, starting with those ofthe (restricted) Lorentz group (Sec. VIII.2.1), and dealing afterwards with the Poincaré group(Sec. VIII.2.2). Note, however, that since O(3,1) is a subgroup of the Poincaré group, one mayinstead begin with the latter and consider the former afterwards.

VIII.2.1 Generators of the Lorentz group

Starting from the explicit form of spatial rotation transformations (VIII.8)–(VIII.9) and Lorentzboosts (VIII.11)–(VIII.12), and invoking Eq. (V.33), one can derive an expression for the associatedgenerators.

Thus, the spatial rotations around the x1, x2 and x3 axes will be respectively generated by

J1 =

0 0 0 00 0 0 00 0 0 −i0 0 i 0

, J2 =

0 0 0 00 0 0 i0 0 0 00 −i 0 0

, J3 =

0 0 0 00 0 −i 00 i 0 00 0 0 0

, (VIII.23)

which trivially generalizes the three-dimensional generators (VI.9a). In turn, the generators of theLorentz boosts along the x1, x2 and x3 directions read

K1 =

0 i 0 0i 0 0 00 0 0 00 0 0 0

, K2 =

0 0 i 00 0 0 0i 0 0 00 0 0 0

, K3 =

0 0 0 i0 0 0 00 0 0 0i 0 0 0

. (VIII.24)

Remark VIII.25. While the generators of rotations (VIII.23) are Hermitian, i.e. lead after expo-nentiation to unitary — in fact, orthogonal — matrices, those of Lorentz boosts (VIII.24) are not:they are antihermitian, and thus generally yield non-unitary matrices.

The generators (VIII.23)–(VIII.24) obey the commutation relations

[Ji, Jj ] = i3∑

k=1

εijkJk ∀i, j ∈ 1, 2, 3, (VIII.26a)

VIII.2 The Lorentz and Poincaré groups as Lie groups 107

which merely expresses the fact that the matrices Ji are generators of a representation of SO(3),as well as

[Ki,Kj ] = −i3∑

k=1

εijkJk ∀i, j ∈ 1, 2, 3, (VIII.26b)

and

[Ji,Kj ] = i3∑

k=1

εijkKk ∀i, j ∈ 1, 2, 3. (VIII.26c)

The latter relation means that the matrices Kii=1,2,3 form a vector operator under SO(3) rotations,cf. § VI.4.2 b. In turn, the commutation relation (VIII.26b) means that the product of Lorentz boostsalong different directions is not a pure Lorentz boost, but also involves a rotation — i.e. that theLorentz boosts do not form a subgroup of the Lorentz group.

VIII.2.2 Generators of the Poincaré group

::::::::VIII.2.2 a

:::::::::::::::::::::::::::::::::::::::::::::::::::::Infinitesimal Lorentz and Poincaré transformations

Consider the generic expression (VIII.4) of a Minkowski spacetime isometry. For a transforma-tion lying infinitesimally close to the identity, one may write

x→ x′ = (14 +ω)x + ε, (VIII.27a)

or equivalently, in terms of components

xµ → x′µ =(δµν + ωµν

)xν + εµ for all µ ∈ 0, 1, 2, 3, (VIII.27b)

where the (real) numbers ωµν , εµ are much smaller than 1 in norm. In particular, the entries ofan infinitesimal Lorentz transformation — which is necessarily an orthochronous, proper Lorentztransformation — take the form

Λµν = δµν + ωµν , (VIII.28)where the properties of the numbers ωµν , equivalent to Eq. (VIII.5a), still need to be specified.

With the above ansatz, one finds

Λµρ ηµνΛνσ =(δµρ + ωµρ

)ηµν(δνσ + ωνσ

)' δµρ ηµνδνσ + δµρ ηµνω

νσ + ωµρ ηµνδ

νσ,

where in the second identity the quadratic contribution in ω was neglected. The first term on theright hand side is precisely δµρ ηµνδνσ = ηρσ, so that Eq. (VIII.5a) becomes

ωρσ + ωσρ = 0 for all ρ, σ ∈ 0, 1, 2, 3. (VIII.29)

That is, the 4×4 matrix with entries ωρσ —which is not the matrixω introduced in Eq. (VIII.27a) —is antisymmetric, and thus entirely determined by the entries in the triangle above the diagonal,i.e. the 6 elements with σ > ρ. One thus recovers the fact that a generic matrix of the restrictedLorentz group depends on 6 real parameters — and in turn, adding the 4 extra parameters εµ, thatthe Poincaré group is a 10-parameter group.

:::::::::VIII.2.2 b

::::::::::::Generators

Equation (VIII.27b) together with the condition (VIII.29) allow one to easily re-derive the setof six 4× 4 generators of the Lorentz group already found in Sec. VIII.2.1, yet leave the generatorsof translations aside.

To find a set of generators that also includes the translations, we look at operators acting onfunctions of the spacetime coordinates x — i.e. physically on fields. Considering an infinitesimaltransformation (VIII.27) of the coordinates, a trivial Taylor expansion to first order gives for anysufficiently differentiable function f defined on M 4

f(x′) = f(xµ + εµ + ωµνx

ν)' f(x) +

(εµ + ωµνx

ν)∂f(x)

∂xµ,


which can be recast in the form

f(x′) =

(Id + iεµPµ −

i

2ωµνJµν

)f(x)

where we have defined differential operators

Pµ ≡ −i∂

∂xµ≡ −i∂µ (VIII.30a)

and

Jµν ≡ −i(xµ∂ν − xν∂µ

)(VIII.30b)

for all µ, ν ∈ 0, 1, 2, 3. These operators are the generators of the action of the Poincaré group onthe functions on M 4. One then easily finds that they obey the commutation relations

[Pµ,Pν ] = 0 for all µ, ν ∈ 0, 1, 2, 3, (VIII.31a)

which expresses that commutativity of translations, together with

[Jµν ,Pρ] = i(ηµρPν − ηνρPµ

)for all µ, ν, ρ ∈ 0, 1, 2, 3 (VIII.31b)

and eventually

[Jµν ,Jρσ] = i(ηµρJνσ − ηνρJµσ + ηνσJµρ − ηµσJνρ

)for all µ, ν, ρ, σ ∈ 0, 1, 2, 3.

(VIII.31c)

Let us for instance check Eq. (VIII.34b): given a function f on M 4, one has

[Jµν ,Pρ]f = −i(xµ∂ν − xν∂µ

)(−i∂ρ)f + i∂ρ

[−i(xµ∂ν − xν∂µ

)]f

= −xµ∂ν∂ρf + xν∂µ∂ρf + ∂ρ(xµ∂νf

)− ∂ρ

(xν∂µf

).

Using the product rule in the last two terms together with ∂ρxσ = ηρσ, one finds

[Jµν ,Pρ]f = −xµ∂ν∂ρf + xν∂µ∂ρf + ηµρ∂νf + xµ∂ρ∂νf − ηνρ∂µf − xν∂ρ∂µf.

The first and fourth terms compensate, as do the second and sixth, leaving

[Jµν ,Pρ]f = ηµρ∂νf − ηνρ∂µf = iηµρ(−i∂νf)− iηνρ(−i∂µf) = i(ηµρPν − ηνρPµ

)f. 2

Remark VIII.32. Restricting oneself to the components i, j = 1, 2, 3, the definitions (VIII.30) readPi = −i∂i and Jij = −i

(xi∂j−xj∂j

), which are up to a factor ~ the linear-momentum and angular-

momentum operators acting on wave functions in spatial representation in non-relativistic quantummechanics.

In turn, P0 = −P0 = i∂0 can be interpreted — up to irrelevant factors of ~ and c — as theHamilton operator. Accordingly, we shall use P0 rather than P0 in equations which will be givenhereafter.

Let us now define for i = 1, 2, 3 the operators

Ji ≡1

2

3∑j,k=1

εijkJjk , Ki ≡ J0i, (VIII.33a)

where the first definition can be “inverted” as

Jij =

3∑k=1

εijkJk. (VIII.33b)

VIII.3 Representations of the Lorentz group 109

With these definitions, the commutation relations (VIII.31) become on the one hand

[Ji, Jj ] = i3∑

k=1

εijkJk , [Ki,Kj ] = −i3∑

k=1

εijkJk , [Ji,Kj ] = i3∑

k=1

εijkKk ∀i, j ∈ 1, 2, 3,

(VIII.34a)

which are precisely the relations (VIII.26) already found for generators of the Lorentz group. Inaddition, one also finds

[Ji,Pj ] = i3∑

k=1

εijkPk ∀i, j ∈ 1, 2, 3, (VIII.34b)

which means that the three generators P1,P2,P3 form a vector representation of the subgroupSO(3) of the Poincaré group, as well as

[Ki,Pj ] = iδijP0 ∀i, j ∈ 1, 2, 3, (VIII.34c)

where we used P0 = −P0. One further has

[P0, Ji] = 0 and [P0,Pi] = 0 ∀i ∈ 1, 2, 3, (VIII.34d)

which may be viewed as expressing momentum and angular-momentum conservation. Eventually,one has

[P0,Ki] = −iPi ∀i ∈ 1, 2, 3. (VIII.34e)

VIII.3 Representations of the Lorentz group

Appendix to Chapter VIII

VIII.A Proof of the linearity of Minkowski spacetime isometriesIn this appendix, we show that the transformations xµ → x′µ that preserve the infinitesimal lineelement ds2 = ηµν dxµ dxν are affine transformations.

Let xµ → x′µ(xν) be such a transformation. As we shall see hereafter, the functions x′µ shouldbe at least twice continuously differentiable to ensure the validity of the proof — physically, thisrequirement is however quite reasonable. Using the chain rule, the differential of the coordinatefunction x′ρ reads

dx′ρ =∂x′ρ

∂xµdxµ, (VIII.35)

which allows one to write the infinitesimal line element expressed in terms of the primed components:

ds′2 ≡ ηρσ dx′ρdx′σ = ηρσ∂x′ρ

∂xµ∂x′σ

∂xνdxµ dxν . (VIII.36)

The condition ds′2 = ds2 with ds2 = ηµν dxµ dxν thus translates into

ηµν = ηρσ∂x′ρ

∂xµ∂x′σ

∂xνfor all µ, ν ∈ 0, 1, 2, 3. (VIII.37)

One can now differentiate both sides of this equations with respect to the coordinate xλ withλ ∈ 0, 1, 2, 3, which thanks to the product rule yields

0 = ηρσ

(∂2x′ρ

∂xµ∂xλ∂x′σ

∂xν+∂x′ρ

∂xµ∂2x′σ

∂xν∂xλ

)∀λ, µ, ν ∈ 0, 1, 2, 3. (VIII.38a)

Exchanging the dummy indices λ and µ resp. λ and ν, this equation gives

0 = ηρσ

(∂2x′ρ

∂xλ∂xµ∂x′σ

∂xν+∂x′ρ

∂xλ∂2x′σ

∂xν∂xµ

)∀λ, µ, ν. (VIII.38b)

resp.

0 = ηρσ

(∂2x′ρ

∂xµ∂xν∂x′σ

∂xλ+∂x′ρ

∂xµ∂2x′σ

∂xλ∂xν

)∀λ, µ, ν. (VIII.38c)

One can now add Eqs. (VIII.38a) and (VIII.38b) and subtract Eq. (VIII.38c). The second term ofEq. (VIII.38a) cancels out with that of Eq. (VIII.38c) while, thanks to the identity ηρσ = ησρ, thefirst terms of Eq. (VIII.38b) and (VIII.38c) compensate. Eventually, the two remaining terms areequal:

0 = 2ηρσ∂x′σ

∂xν∂2x′ρ

∂xµ∂xλ∀λ, µ, ν ∈ 0, 1, 2, 3. (VIII.39)

This equation can be interpreted as representing the product from left to right of a row vector∂2x′ρ/∂xµ∂xλ (with four components ρ = 0, 1, 2, 3) with two 4× 4-matrices with respective entries2ηρσ and ∂x′σ/∂xν . If the latter matrix is regular, i.e. if the transformation xµ → xµ

′ can be inverted,one can multiply the above equation with the inverse matrix, which leads to the condition

∂2x′ρ

∂xµ∂xλ= 0 ∀λ, µ, ρ ∈ 0, 1, 2, 3. (VIII.40)

VIII.A Proof of the linearity of Minkowski spacetime isometries 111

After twofold integration, this equation leads first to ∂x′ρ/∂xµ = Λρµ with Λρµ ∈ R, then to theaffine transformation

x′ρ = Λρµ xµ + aρ, (VIII.41)

i.e. to the announced result (VIII.4).

CHAPTER IX

Symmetries in classical field theory

intro

IX.1 Basics of classical field theoryConsider independent scalar fields ϕa(x), a = 1, . . . , N , defined at each point x of a D-dimensionalspace-time M D on which a metric with signature (−,+,+, · · · ,+) is defined — the obvious case,which we consider in the following, being that of the 4-dimensional Minkowski(ai) spaceM 4 of specialrelativity. As we shall illustrate on a few examples, the fields ϕa(x) may be real- or complex-valued,and may represent the components in a given reference frame of a vector or tensor field.

IX.1.1 Definitions

The dynamical behavior of the fields is described by a Lagrange density

L[ϕa(x), ∂µϕa(x)

], (IX.1a)

which is a functional of the fields and of their derivatives ∂µϕa(x) with µ ∈ 0, 1, 2, 3, where ∂µdenotes the partial derivative with respect to the coordinate xµ of the space-time point x in a givenreference frame.

Integrating the Lagrange density over the spatial coordinates yields the Lagrangian

L =

∫L[ϕa(x), ∂µϕa(x)

]d3~r. (IX.1b)

In turn, integrating the Lagrangian over a time interval [t1, t2] gives the action

S =

∫ t2

t1

Ldt =

∫L[ϕa(x), ∂µϕa(x)

]d4x (IX.1c)

with d4x the 4-volume element, d4x = dx0 dx1 dx2 dx3.

Remarks:

∗ If the Lagrange density of a system of fields is invariant under the transformations of the Lorentzor Poincaré groups, then this also holds for the resulting action, since d4x is Lorentz-invariant.

∗ The Lagrange density for a given physical system is not unique. Anticipating on next section,two Lagrange densities L and L ′ that only differ by a 4-divergence ∂µMµ(x), withM a 4-vectorfield, are equivalent: their integrals give respective actions S, S′, which lead to the same equationsof motion.(ai)H. Minkowski, 1864–1909

IX.1 Basics of classical field theory 113

IX.1.2 Euler–Lagrange equations

According to the principle of the stationary action, or Hamilton’s principle, the equations ofmotion obeyed by a system of fields can be obtained by demanding that the action S shouldbe stationary when the field configuration is varied, assuming that the fields do not vary at theboundaries of the space-time region over which the action is computed.

Considering a (small) variation δϕa of each field ϕa, which results in a variation δ∂µϕa = ∂µδϕaof its partial derivatives, the ensuing variation of the action (IX.1c) reads

δS =

∫ ∑a

[∂L

∂ϕaδϕa +

∂L

∂(∂µϕa)δ∂µϕa

]d4x,

where the second term involves a non-written sum over the Lorentz indices µ. When integratingthat term by partial integration, the contribution from the integrated term is zero, thanks to theassumption that δεa vanishes on the boundaries of the integration region. One thus obtains

δS =

∫ ∑a

[∂L

∂ϕa− ∂µ

∂L

∂(∂µϕa)

]δϕa d4x.

The stationarity of the action is realized when δS = 0 for arbitrary variations δϕa, i.e. wheneach field ϕa and its derivatives obey the Euler–Lagrange equations

∂L[ϕa, ∂νϕa

]∂ϕa

= ∂µ

[∂L

[ϕa, ∂νϕa

]∂(∂µϕa)

]. (IX.2)

These represent the equations of motion of the fields.

Example: The simplest possible example is that of a single real scalar field ϕ with the Lagrangedensity

L [ϕ, ∂µϕ] = −1

2

(∂µϕ

)(∂µϕ

)− 1

2m2ϕ2. (IX.3)

Straightforward calculations, where one should not forget the identity ∂µϕ = ηµν∂νϕ, yield

∂L

∂ϕ= −m2ϕ and

∂L

∂(∂µϕ)= −∂µϕ,

so that the Euler–Lagrange equation gives −m2ϕ = −∂µ∂µϕ, that is, restoring the x-dependenceof the field

∂µ∂µϕ(x)−m2ϕ(x) = 0 (IX.4)

which is known as the Klein(aj)–Gordon(ak) equation.

Remark: In analogy to the construction in analytical mechanics, one associates to each field ϕagoverned by the Lagrange density L a conjugate momentum

πa ≡∂L

∂(∂0ϕa). (IX.5)

With its help, one defines the Hamilton density

H [πa, ϕa] =∑a

πa ∂0ϕa −L , (IX.6)

whose integral over the spatial coordinates gives the Hamilton function of the system.For the real scalar field of the previous example, one at once finds π = −∂0ϕ = ∂0ϕ, which

yields H = 12

(∂0ϕ

)2+ 1

2

(~∇ϕ

)2+ 1

2m2ϕ2.

(aj)O. Klein, 1894–1977 (ak)W. Gordon, 1893–1939

114 Symmetries in classical field theory

IX.2 Symmetries in field theories and conserved quantitiesAn important feature of classical field theory lies in its relation to symmetries. On the one hand,implementing a symmetry into a given theory is rather simple, as we shall see in the followingsection. On the other hand, if a continuous symmetry is present, then there a exist correspondingconserved quantity (or set of conserved quantities), as we shall now see.

IX.2.1 Symmetries in a field theory

Using the Lagrange formalism, one can easily study the behavior of a system of classical fieldsunder various types of transformations. For physical applications, two big classes of such transfor-mations are of interest, which may possibly be applied simultaneously.

On the one hand, one can consider space-time transformations, affecting the “position” of thefield system. These are generically of the form

x→ x′ = f(x, θc

), (IX.7a)

where f denotes a sufficiently regular function depending on real parameters θc. IntroducingMinkowski coordinates, the transformation reads

xµ → x′µ = fµ(xν, θc

)(IX.7b)

for µ = 0, 1, 2, 3 (or up to D − 1 in the more general case of a D-dimensional space-time M D).On the other hand, one may also consider “internal” transformations of the fields, of the form

ϕa(x)→ ϕ′a(x) = ga(ϕb(x), Λc(x)

), (IX.8)

where ga is some function depending on real parameters Λc(x). An important possibility, cor-responding to so-called gauge transformations, is that the latter parameters indeed depend on thespace-time position of the fields. In case of space- and time-independent parameters, one also talksof global transformations.

A group of transformations is said to be a symmetry of a physical system described by fields ifthe associated action (IX.1c) is invariant under the transformations.

IX.2.2 Noether theorem

Let us now show that every continuous group of symmetry transformations of the action S of asystem leads to the existence of a continuous quantity, which constitutes the Noether (al) theorem.

:::::::IX.2.2 a

::::::::::::::::::::::::Result and discussion

More precisely, we shall find that if the integral of a Lagrange density L[ϕa(x), ∂µϕa(x)

]over a 4-volume Ω — which will be further specified hereafter — is invariant under simultaneousinfinitesimal transformations

xµ → x′µ = xµ + δxµ ∀µ = 0, 1, 2, 3, (IX.9a)ϕa(x)→ ϕ′a(x

′) = ϕa(x) + δϕa(x) ∀a (IX.9b)

of the space-time coordinates and the fields, then the Noether current with components

Jµ ≡∑a

∂L

∂(∂µϕa)δϕa + Tµνcan.δxν (IX.10)

obeys ∫Ω∂µJ

µ(x) d4x = 0, (IX.11)

(al)E. Noether, 1882–1935

IX.2 Symmetries in field theories and conserved quantities 115

where the canonical energy-momentum tensor is defined by

Tµνcan. ≡ ηµνL −∑a

∂L

∂(∂µϕa)∂νϕa. (IX.12)

In the above, the 4-volume Ω is delimited in the temporal direction by two constant-time hyper-planes t = t1 and t = t2, while it extends to infinity in the spatial directions, in agreement withEqs. (IX.1b) and (IX.1c). That is, the boundary ∂Ω of Ω consists of the two hyperplanes t = t1 andt = t2 and of the hypersurface ∂R3 at spatial infinity for every t ∈ [t1, t2] — where one assumes thatevery field vanishes together with its derivatives. Accordingly, an integral over ∂Ω actually reducesto integrals over the spatial coordinates ~r ∈ R3 at t = t1 and t = t2. In particular, the divergencetheorem reads for any 4-vector-field Bµ∫

Ω∂µB

µ(x) d4x =

∮∂ΩBµ(x) d3σµ =

[∫R3

B0(t,~r) d3~r

]t=t2t=t1

,

where we used the fact that on the hyperplane t = t1 resp. t = t2, the outwards-oriented hypersurfaceelement d3σµ is purely along the time direction with d3σ0 = d3~r resp. −d3~r.

Applying the latter result to the relation (IX.11) obeyed by the Noether current (IX.10) yields∫Ω∂µJ

µ(x) d4x = 0 =

[∫R3

J0(t,~r) d3~r

]t=t2t=t1

,

which shows that the Noether charge

Q ≡∫R3

J0(t,~r) d3~r (IX.13)

is a conserved quantity, a constant of motion.

Remarks:

∗ Using η00 = −1 and ∂0ϕa = −∂0ϕa, definition (IX.12) shows that the 00-component of thecanonical energy-momentum tensor coincides with the Hamilton density (IX.6): T 00

can. = H .

∗ Strictly speaking, Jµ is a current density.

∗ Instead of the invariance of the action, one often finds in the literature the stronger requirementof having an invariant Lagrange density L — which of course automatically leads to an invariantaction. This stronger condition has the advantage that it leads to the “local conservation equation”∂µJ

µ(x) = 0 in lieu of Eq. (IX.11), yet it is not needed for the existence of the conserved Noethercharge.

:::::::IX.2.2 b

:::::::Proof

Let us first rewrite the expression of action of the fields

S =

∫ΩL[ϕa(x), ∂µϕa(x)

]d4x (IX.14)

valid for the transformed fields and coordinates (IX.9). When transforming the latter, one simulta-neously modifies the integration 4-volume: Ω→ Ω′. Accordingly, the transformed action reads

S′ =

∫Ω′

L[ϕ′a(x′), ∂µϕ′a(x′)

]d4x′.

Introducing the boundary ∂Ω of the 4-volume Ω — which is also, up to terms of order δxµ, theboundary of Ω′ —, one may decompose the integral over Ω′ into the sum of an integral over Ω andof an integral over the layer of thickness δxµ that corresponds to the difference Ω′ \ Ω:


S′ =

∫ΩL[ϕ′a(x), ∂µϕ′a(x)

]d4x +

∮∂Ω

L[ϕ′a(x), ∂µϕ′a(x)

]δxµ d3σµ, (IX.15)

where d3σµ is a normal (hyper)surface 4-vector at each point of the boundary ∂Ω. Note that everyspace-time position is now expressed in terms of the “old” positions x, which allows us to suppressit from now on for the sake of brevity. Accordingly, we introduce the variation of the field ϕa at afixed point:

δϕa(x) ≡ ϕ′a(x)−ϕa(x),

which will generally differ from δϕa(x) if x′ and x are different, i.e. if the space-time coordinates aretransformed: to first order in the variation δx, one has

δϕa = ϕ′a(x′ − δx)−ϕa(x) ' ϕ′a(x′)− δxν∂νϕa(x)−ϕa(x)

i.e.δϕa ' δϕa(x)− δxν∂νϕa(x). (IX.16)

Coming back to Eq. (IX.15), one may write ϕ′a(x) = ϕa(x) + δϕa(x) in the arguments of theLagrange density and expand to first order in the variations δx, δϕa, and δ(∂µϕa) = ∂µ(δϕa):

S′ '∫

Ω

[L[ϕa, ∂µϕa

]+∑a

(∂L

[ϕa, ∂µϕa

]∂ϕa

δϕa +∂L

[ϕa, ∂µϕa

]∂(∂µϕa)

∂µδϕa

)]d4x

+

∮∂Ω

L[ϕa, ∂µϕa

]δxµ d3σµ.

Note that in the integral over the boundary ∂Ω, we have suppressed terms stemming from theexpansion of the Lagrange density that yield contributions of at least second order in the variations.From now on, we shall also omit the argument ϕa, ∂µϕa of the Lagrange density and its(functional) derivatives, since it is the same in every term.

One may now subtract the action (IX.14) from the new expression of S′, which is straightforward.Performing an integration by parts for the term in the 4-volume integral involving the partialderivatives ∂µ(δϕa), one obtains

δS ≡ S′ − S =

∫Ω

∑a

[∂L

∂ϕa− ∂µ

(∂L

∂(∂µϕa)

)]δϕa d4x +

∮∂Ω

[∑a

∂L

∂(∂µϕa)δϕa + L δxµ

]d3σµ.

Thanks to the Euler–Lagrange equations (IX.2), the integrand of the 4-volume integral identicallyvanishes, since we assume that the field configuration ϕa(x) is a solution of the equations ofmotion. In turn, expressing δϕa through δϕa [Eq. (IX.16)] in the integral over ∂Ω yields

δS =

∮∂Ω

[∑a

∂L

∂(∂µϕa)

(δϕa − δxν∂νϕa

)+ L δxµ

]d3σµ.

After some straightforward rewriting, this becomes

δS =

∮∂Ω

[∑a

∂L

∂(∂µϕa)δϕa +

(ηµνL −

∑a

∂L

∂(∂µϕa)∂νϕa

)δxν

]d3σµ. (IX.17)

Using the definitions of the Noether current (IX.10) and the canonical energy-momentum ten-sor (IX.12), this becomes

δS =

∮∂ΩJµ d3σµ.

The action is stationary for any infinitesimal transformation of the fields and space-time co-ordinates if its variation δS vanishes, which invoking the divergence theorem precisely gives theannounced result (IX.11) — which concludes the proof.

IX.2 Symmetries in field theories and conserved quantities 117

IX.2.3 Examples

:::::::IX.2.3 a

::::::::::::::::::::::::::::::::::::::::::Energy and momentum of a field system

Considering a Lagrange density which is invariant under space-time translations

xµ → x′µ = xµ + aµ (IX.18)

with aµ ∈ R for every µ ∈ 0, 1, 2, 3, one finds by setting δϕa = 0 and δxν = aν in the expres-sion (IX.10) of the Noether current that the latter becomes proportional to the canonical energy-momentum tensor. Accordingly, the four components

P ν ≡∫R3

T 0νcan.(t,~r) d3~r for ν = 0, 1, 2, 3,

are conserved quantities, which corresponds to the energy (P 0) and the momentum of the fieldsystem, and form a four-vector.

:::::::IX.2.3 b

::::::::::::::::::::::Complex scalar field

Let ϕ(x) denote a complex scalar field, with the Lagrange density

L [ϕ,ϕ∗, ∂µϕ, ∂µϕ∗] = −

(∂µϕ

)(∂µϕ

)∗ −m2ϕϕ∗, (IX.19)

where ϕ and ϕ∗ are to be considered as independent fields.The Euler–Lagrange equations

∂L

∂ϕ= ∂µ

∂L

∂(∂µϕ)and

∂L

∂ϕ∗= ∂µ

∂L

∂(∂µϕ)∗

are easily computed and lead to respective Klein–Gordon equations

∂µ∂µϕ∗(x)−m2ϕ∗(x) = 0 and ∂µ∂

µϕ(x)−m2ϕ(x) = 0, (IX.20)

which are naturally equivalent if m ∈ R.One sees at once that the Lagrange density (IX.19) is invariant under the “global U(1) transfor-

mation” ϕ(x)→ ϕ′(x) = e−iΛϕ(x)

ϕ∗(x)→ ϕ′∗(x) = eiΛϕ(x)(IX.21)

where Λ ∈ R is invariant of the space-time position. In the case of an infinitesimally small Λ, thesetransformations become

ϕ′(x) ' ϕ(x)− iΛϕ(x) , ϕ′∗(x) ' ϕ∗(x) + iΛϕ∗(x),

corresponding to infinitesimal variations δϕ = −iΛϕ, δϕ∗ = iΛϕ∗ of the independent fields. In-serting the latter in the expression (IX.10) of the Noether current yields

Jµ = i(ϕ∗∂µϕ−ϕ∂µϕ∗

), (IX.22)

where we have divided by Λ. One easily checks with the equations of motion (IX.20) that thiscurrent, whose spatial components are reminiscent of the probability current of non-relativisticquantum mechanics, obeys the local conservation equation ∂µJµ = 0.

Remarks:∗ One can add to the Lagrange density (IX.19) a “potential term” −V (|ϕ|2) depending only on thesquared modulus |ϕ|2 of the field. This modifies the equations of motion, yet the theory remainsinvariant under the global U(1) symmetry (IX.21), and the corresponding Noether current is stillgiven by Eq. (IX.22).

∗ The reader is encouraged to go through the above example by replacing ϕ by 1√2(ϕ1+iϕ2), where

ϕ1 and ϕ2 are now real scalar fields. The symmetry transformations (IX.21) are then replaced bySO(2) transformations, i.e. rotations in the (ϕ1,ϕ2)-plane — which are of course totally equivalent.


IX.3 Gauge field theories

APPENDIX A

Elements of topology

A.1 Topological spaceopen / closed set; continuity; (path-)connectedness; compacity; homotopy group(?)

homeomorphism: bijective map such that f and f−1 are continuousdiffeomorphism: homeomorphism such that f and f−1 are infinitely differentiableanalytic diffeomorphism:

A.2 ManifoldsCharts, atlas

Definition A.1. A manifold M is a Hausdorff topological space with an atlas such that the opensets Ui provide a covering of M and that the mappings ϕj ϕ−1

i are infinitely differentiable.

The dimension n of the manifold is the dimension of the vector space into which the charts ϕimap.

120 Elements of topology

Bibliography

[1] D. Rim, An Elementary Proof That Symplectic Matrices Have Determinant One,[arXiv:1505.04240].

[2] Particle Data Group Collaboration [C. Patrignani et al.], The Review of Particle Physics,Chin. Phys. C 40 (2016) 100001 available online at http://www-pdg.lbl.gov/.

[3] C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum mechanics, vol. 1 (Wiley-VCH, Weinheim,1977).

[4] S. Weinberg, Lectures on Quantum Mechanics, 2nd ed. (University Press, Cambridge, 2015).

http://arXiv.org/abs/arXiv:1505.04240

http://www-pdg.lbl.gov/

122 Bibliography

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