Symmetries of Particles and Fields 2021/22

Contents

1 Introduction 1

2 Lie Groups 1

2.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 One Dimensional Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Lie Algebras 7

3.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Properties of Structure Constants . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.6 Invariant Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.7 Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Simplicity 14

4.1 Non semi-simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Cartan’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Casimir Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4 Cartan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Compact Groups I 17

5.1 The su(2) algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Space-time Symmetries 19

6.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2 Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.3 Lorentz algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6.3.1 Relation to sl(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.4 Representations of sl(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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6.4.1 Euclidean Rotation Group . . . . . . . . . . . . . . . . . . . . . . . . 24

6.4.2 sl(2,C) Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.4.3 Dirac Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.5 The Poincare Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.6 Classification of Irreducible Representations . . . . . . . . . . . . . . . . . . 28

6.7 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Symmetry in Action 31

7.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.3 Noether’s Theorem (1918) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.4 Conservation of Energy and Momentum . . . . . . . . . . . . . . . . . . . . 34

7.5 Real Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.6 Complex Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.7 Actions for Spinor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Compact Groups II 38

[ Last updated on Thursday 28th October, 2021 at 18:50 ]

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1 Introduction

The laws of physics are invariant under many different symmetries:

• Parity, time reversal, charge conjugation (P, T, & C) - discrete symmetries

• Translations and rotations (Galilean invariance)

• Lorentz transformations (special relativity)

• Internal symmetries (e.g. isospin, flavour SU(3))

• Gauge symmetries (Electrodynamics (Maxwell)), the Standard Model of particle physics:SU(3)⊗ SU(2)⊗ U(1))

We shall study each of these in turn. The overall structure of the course is:

• Introduction to Lie groups and Lie algebras

• Space-time symmetries; constructing actions with symmetries

• Compact groups – representations

• Applications in physics: Noether’s theorem; isospin; quark model; spontaneous sym-metry breaking; chiral symmetry; gauge theories; QCD; Higgs mechanism; electroweaktheory

This is a theoretical physics course, not a mathematics course. We shall introduce the nec-essary mathematics as we go along, mostly without mathematical rigour and often withoutproof. A more formal treatment is provided by the School of Mathematics course Introduc-tion to Lie Groups (MATH11053), which is unfortunately not running this year!

2 Lie Groups

Definition 1 A group is a set of elements G = {g} with a composition law G×G→ G.

For a multiplicative group:1 the product of any two group elements g1, g2 is itself a groupelement: ∀ g1, g2 ∈ G, g1 g2 = g3 for some g3 ∈ G, with the properties

1. Associativity: (g1g2) g3 = g1 (g2g3)

2. Unit element (or identity element) 1: g1 = 1g = g

3. Unique inverse: gg−1 = g−1g = 1

1Can also have addititive groups. We will concentrate on multiplicate groups.

1

Definition 2 A finite group is a group with a finite number of elements.

Example.

• Z2, elements {+1,−1}. C, P , T (charge conjugation, parity, time reversal) are all Z2

groups.

• ZN , elements {e2πin/N : n = 1, . . . , N}, the N th roots of unity.

• Dihedral groups DN : the group of symmetries of regular polygons with N sides, etc.

[See the Symmetries of Quantum Mechanics course.]

Definition 3 An infinite group is a group with an infinite number of elements, which wewill assume to be labelled by a set of continuous parameters {α} = {α1, α2, . . .} (continuousgroup). We write G = {g(α)}. The dimension of the group is the number of independentparameters. 2

Example. In ordinary three-dimensional (real) space:

• Spatial translations in R3: xi 7→ xi + ai; dimG = 3.

• Spatial rotations in R3: xi 7→ Rij

(n, φ

)xj;

RTR = 1; dimG = 3 – an angle (φ) and an axis n (2 parameters).

Definition 4 A Lie Group is a continuous group, for which the group multiplication hasan analytic structure: if g(αi) = g(βi) g(γi) then αi = φi(βi, γi) where the φi are analyticfunctions of the βi and γi.

Continuous groups may be compact (the group space is bounded) or non-compact (the groupspace is not bounded), and connected or not connected.3

Example.

• Translations: connected but not compact – the magnitude of a translation is notbounded

• Rotations + reflections: compact but not connected: detR = ±1.

Rotations give detR = 1, reflections give detR = −1, we can’t get from +1 to −1continuously, so rotations and reflections are not connected.

2Recall that the order of a group is the total number of elements in the group. This is different from thedimension of the group, which is the number of independent parameters {α}.

3We’ll be a bit more precise when we study specific examples! In previous years, we used topologicaldefinitions, which were unfamilar to many students.

2

2.1 Metric spaces

Metrics will feature in many parts of this course; let’s define (or revise) their properties here.

Consider a real vector space with a basis {xµ}, µ = 1, . . . , N . Define a real symmetricN×N matrix called the metric gµν , then define x2 ≡ gµνx

µxν (using the Einstein summationconvention). Now choose a basis in which:

1. gµν is diagonal, gµν = diag(λ1, . . . , λN) where λi are the (real) eigenvalues of gµν .

2. The eigenvalues are either zero or ±1. Under the rescaling xµ 7→ sxµ, gµν 7→ s−2gµν ,with s ∈ R, then x2 stays fixed. We can rescale each λi separately. Since s2 > 0 thesigns of the eigenvalues cannot change, so in this basis gµν = diag(1, 1, . . . , 0, 0, . . . ,−1,−1).We call this the canonical basis.

For example, in three dimensional space gij = δij, with i, j ∈ {1, 2, 3}, x2 = x21 + x22 + x23(Euclidean space).

In four dimensional space-time gµν = diag(1, 1, 1,−1), x2 = x21 + x22 + x23︸ ︷︷ ︸space

− x24︸︷︷︸time

. Clearly, x2

can be positive, negative or zero.

We will generally write x4 ≡ x0 and use a space-time metric with the opposite signature:gµν = diag(1,−1,−1,−1) whereupon x2 = x20︸︷︷︸

time

− (x21 + x22 + x23)︸ ︷︷ ︸space

.

A metric with a mix of +1 and −1 elements is called an indefinite metric.

Metrics with zero eigenvalues occur in the theory of Lie algebras - see later.

Raising and lowering indices: if det gµν 6= 0, i.e., the metric has no zero eigenvalues, thengµν is invertible. We can then use the metric to raise and lower indices: xµ ≡ gµνx

ν and soforth. Then

x2 = gµνxµxν = xµx

µ = xµxµ = δµνxµxν .

where we used xµ = δµνxν (relabelling of indices), with

δµν = δνµ =

{1 µ = ν0 µ 6= ν

We can also raise indices on the metric: gµν = gµα gνβ gαβ ⇒ gνβ gαβ = δνα, i.e., gµν is theinverse of gµν .

Similarly for the totally antisymmetric Levi-Civita symbol ε

N indices︷ ︸︸ ︷µν···αβ

εµν···αβ = gµµ′gνν

′ · · · gαα′gββ′εµ′ν′···α′β′ ,

and derivatives with respect to xµ

∂µ ≡∂

∂xµ= gµν

∂

∂xν= gµν ∂

ν (exercise: check that∂xα

∂xβ= δαβ, etc.)

If the metric is definite (all the elements have the same sign, with no zeros) we can choose abasis in which gµν = δµν . In this case we can ignore the distinction between upper and lowerindices.

3

2.2 Matrix Groups

A useful way of defining groups is through the linear transformation

xµ 7→ Dµν(α)xν µ, ν = 1, . . . , N.

Dµν(α) is the (µν)th element of the invertible (detD 6= 0), N ×N matrix D(α).4 This gives:

• The general linear groups GL(N,R) or GL(N,C) (for real and complex vector spaces,respectively), and the

• The special linear groups SL(N,R) or SL(N,C) if detD = 1.

• If D ∈ GL(N,R) and DTD = DDT = 1, we get the orthogonal groups, O(N).

• If D ∈ SL(N,R) and DTD = DDT = 1, we get the special orthogonal groups, SO(N).

• If D ∈ GL(N,C) and D†D = DD† = 1, we get the unitary groups, U(N).

• If D ∈ SL(N,C) and D†D = DD† = 1, we get the special unitary groups, SU(N).

• We also find the “pseudo-orthogonal” groups (e.g., SO(3, 1), the Lorentz group: themetric on the space {xµ} is indefinite – with 3 plus signs and 1 minus sign, or viceversa).

• The “pseudo-unitary” groups. These preserve an indefinite sesquilinear metric (involv-ing complex conjugation). [We will not mention them again.]

• The symplectic groups Sp(2N,R), Sp(2N,C) and Sp(2N) – the real, complex andcompact sympectic groups, respectively. [For Sp(2N,R), see tutorial question (1.5).]

• The exceptional groups G(2), F (4), E(6), E(7), and E(8).

The orthogonal and unitary groups leave the metric δµν invariant [tutorial]:5

δµνG7→ Dµ

αDνβδαβ = δµν .

Similarly the special groups leave the totally antisymmetric Levi-Civita tensor εµν···α invari-ant,

εµν···αG7→ Dµ

µ′Dνν′ · · ·Dα

α′εµ′ν′...α′ = detD εµν...α = εµν...α (since detD = 1)

2.3 One Dimensional Groups

Example. Translations on the real line, x 7→ x+ a.

Clearly if g(a)g(b) = g(a + b) then the group is Abelian: g(a)g(b) = g(b)g(a), i.e. allelements of the group commute with each other. There is a simple unitary representationg(a) = eiaλ with λ a fixed parameter (because g(a)g(b) = eiaλeibλ = g(a+ b)).

4Think of xµ as the µth component of a N -component column vector x.5We shall see later that gµν = δµν for compact groups.

4

Theorem 1 All one dimensional Lie groups are such that the parameter a can be chosen sothat g(a)g(b) = g(a+ b) for all a and b.

Proof. The proof, originally due to Aczel, is sketched below.

Using associativity and analyticity of the composition law, we may write:

g(x) (g(y) g(z)) = (g(x) g(y)) g(z) (associativity)

⇒ g(x) g(φ(y, z)) = g(φ(x, y)) g(z) (analyticity)

⇒ φ(x, φ(y, z)) = φ(φ(x, y), z) (analyticity again)

Take the partial derivative of the last expression with respect to z

∂φ(x, Y )

∂Y

∣∣∣∣Y=φ(y,z)

∂φ(y, z)

∂z=

∂φ(φ(x, y), z)

∂z

where we used the chain rule on the left hand side (LHS).

Set z = 0 and choose g(0) = 1, then φ(y, 0) = y, which gives

∂φ(x, y)

∂yψ(y) = ψ(φ(x, y)) where we defined ψ(y) ≡ ∂φ(y, z)

∂z

∣∣∣∣z=0

We can solve this differential equation by rewriting it as

1

ψ(φ(x, y))

∂φ(x, y)

∂y=

1

ψ(y), (1)

and integrating with respect to y, to obtain

ρ(φ(x, y)) = ρ(y) + c(x) (2)

where

ρ(x) =

∫ x

0

dt

ψ(t), ρ(0) = 0

and c(x) is an arbitrary function of x, an integration ‘constant’.

To verify that (2) is indeed the correct solution, differentiate it with respect to y, whichrecovers equation (1) directly.

We can determine c(x) by setting y = 0, which gives ρ(x) = 0 + c(x), and hence

ρ(φ(x, y)) = ρ(x) + ρ(y)

Finally, if we change the parameterisation of the group element from g(x) to g(ρ(x)), wehave

g(ρ(x)) g(ρ(y)) = g(x) g(y) = g(φ(x, y)) = g(ρ(φ(x, y))) = g(ρ(x) + ρ(y))

Thus the group composition law is additive when written in terms of the new (barred)parameters.

5

Corollary 1 All one dimensional Lie groups are Abelian.

Example. The rotation group SO(2)(xy

)7→(

cosφ − sinφsinφ cosφ

)(xy

).

Let z = x+ iy, then z 7→ eiφz. But eiφ ∈ U(1). So U(1) and SO(2) are isomorphic.6

These groups are distinct from the translation group. The parameter φ ∈ [0, 2π) so U(1)and SO(2) are compact. But the translation parameter a ∈ R, i.e. a ∈ (−∞,∞) so thetranslation group is not compact.

Theorem 2 All compact Abelian Lie groups are isomorphic to U(1)⊗ U(1)⊗ · · · ⊗ U(1).

It follows that in this course most of the groups will be non-Abelian: g1g2 6= g2g1.

Corollary 2 The (finite) cyclic groups Cn (or Zn) = {e2mπi/n,m = 0, . . . , n − 1} are sub-groups of U(1).

2.4 Representations

Any set of matrices Dµν(g) satisfying the group composition law,

Dµν(g1)Dν

ρ(g2) = Dµρ(g1 g2)

constitutes a representation of G.7

Representations may be reducible (a direct sum of irreducible representations) or irreducible,just as for finite groups.

Note that if D is a representation, so is its complex conjugate D∗.

A representation for which D = D∗ is a real representation.

Any quantity which transforms under G may be classified according to the representationunder which it transforms:

• Scalars: s 7→ s (i.e., it transforms trivially).

• Vectors: vµ 7→ Dµν(g) vν where D(g) is the defining or fundamental representation.

• Tensors: t µν...︸︷︷︸α

7→ Dµµ′Dν

ν′ · · · tµ′ν′...︸︷︷︸β

or, more succinctly, tα 7→ Dαβ(g) tβ,

where Dαβ is some larger (possibly reducible) representation of G.

Invariant Tensors : tµν... 7→ tµν... under G. The components of an invariant tensor are un-changed by the action of the group.

6A group isomorphism is a function between two groups that sets up a one-to-one correspondence betweenthe elements of the groups in a way that respects the given group operations. If there exists an isomorphismbetween two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphicgroups have the same properties and need not be distinguished. (Source: Wikipedia!)

7We are using the summation convention, pairs of repeated indices are summed over implicitly.

6

Theorem 3 If G is compact, then for every representation there is an equivalent unitaryrepresentation (just as for finite groups: Maschke’s Theorem).

Proof. We limit ourselves to a sketch of the proof.

Starting from the usual inner product (x, y) on a complex (or real) vector space, we definea new inner product8

〈x|y〉 ≡∑g∈G

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

where D(g)x is a representation D(g) of the group G acting on x, and the ‘summation’(integration for continuous groups) is over all elements of G. Let h ∈ G be an arbitrarygroup element. Using the rearrangement theorem9 we can write the new inner product as

〈x|y〉 =∑g∈G

(D(gh)x,D(gh) y) =∑g∈G

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

= 〈D(h)x|D(h) y〉 = 〈x|D†(h)D(h) |y〉

so D†D = 1 with this inner product.

The proof breaks down if G is not compact because the sum/integral over all the groupelements is not well defined.

The action of a group on a Hilbert space must be through a unitary representation, toconserve probability: 1 = ψ∗iψ

i = ψ∗iψ`δi` = (Di

jψj)∗D`

kψkδi` = ψ∗jψk(Di

j∗D`kδi`) =

ψ∗jψk(D†D)jk ⇒ Di

j∗Dik = δjk, i.e., D†D = 1, (where as usual we have denoted the unit

matrix by 1).

3 Lie Algebras

3.1 Generators

Consider a Lie group G = {g(α)}, and choose αa, where a ∈ [1,DimG], such that g(0) = 1.When the αa are small, g(α) is close to 1 (the unit/identity element), and we can use Taylor’stheorem to write10

g(α) = 1 + iαaTa +O(α2) (3)

(this uses analyticity), where

Ta ≡1

i

∂g(α)

∂αa

∣∣∣∣α=0

. (4)

8〈x|y〉 is not the usual inner product in Dirac notation!9Roughly speaking, the rearrangement theorem states that if we multiply all the elements gi in the group

by a fixed element h, we recover all the elements gi once (and once only), but in a different order, i.e., a“rearrangement”.

10NB We’re using the summation convention here. Take care to contract only upper indices with lowerindices.

7

The quantities {Ta} are the generators of the group. The factor of i is a useful convention:if D(g) is a unitary representation, the corresponding representation of the generators ishermitian when αa are real:

1 = D†D = 1− iαa(T †a − Ta) +O(α2)⇒ T †a = Ta .

The number of generators = dimG, i.e. the same as the number of parameters.

Example. SU(N): g†g = gg† = 1, det g = 1, so T †a = Ta, trTa = 0 (the latter follows fromthe constraint det g = 1 and detA = exp tr lnA); the generators are hermitian and traceless.

For SU(2), we generally choose the Pauli matrices:

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

).

For SU(3), the generators are the Gell-Mann matrices {λa : a = 1, . . . , 8} [see later].

Since U(N) doesn’t have the constraint detD = 1, the generators of U(N) are {1, Ta} whereTa are the SU(N) generators.

Example. SO(N): gTg = ggT = 1, det g = 1. Since SO(N) is real, we generally omit thefactors of i in the expansion of g(α) in equations (3) and (4), so that

g(α) = 1 + αaTa +O(α2) and Ta ≡∂g(α)

∂αa

∣∣∣∣α=0

.

whereupon T Ta = −Ta, trTa = 0, i.e., the generators are real antisymmetric. For SO(2)

T1 =

(0 1−1 0

).

For SO(3), we may choose

T1 =

0 1 0−1 0 0

0 0 0

T2 =

0 0 10 0 0−1 0 0

T3 =

0 0 00 0 10 −1 0

.

The additional condition trTa = 0 is automatic as the generators are antisymmetric, soO(N) and SO(N) share the same generators.11

3.2 Commutators

Consider f = ghg−1h−1 with f, g, h ∈ G. If either g = 1 or h = 1 then f = 1. Take g, hinfinitesimal and expand: 12

g = 1 + iα · T + 12(iα · T )2 +O(α3)

h = 1 + iβ · T + 12(iβ · T )2 +O(β3)

then f = 1 + iγ · T + 12(iγ · T )2 +O(γ3)

where γ = O(αβ) and α · T ≡ αaTa, etc.11If we choose hermitian generators for SO(N) and O(N), their elements become purely imaginary. Sim-

ilarly, the generators of SU(N) become anti -hermitian T †a = −Ta if we omit the factors of i in equations (3)

and (4) (exercise). It can be argued that antihermitian generators are more natural as they avoid all factorsof i in all these definitions!

12Strictly, we take the parameters |αa|, |βa| � 1.

8

How did we get this result?

(a) We’ve already discussed the linear terms in the expansion of g (and h).

The quadratic terms follow from Taylor-expanding g to second order in α, and usingg(−α) = g(α)−1 (or g(α)2 = g(2α)) to express the terms with second derivatives inthe form shown. This is a useful exercise – do it!

(b) In fact, by substitution (exercise)

[α · T , β · T ] = − i2γ · T +O(α3, β3)

All terms of O(α), O(β), O(α2), and O(β2) cancel (exercise). Since this is true forarbitrary αa, βb, we must have a relation γc = −2ccab α

a βb, for some set of realnumbers ccab. The factor of −2 is for notational convenience. Hence

[Ta, Tb] = iTc ccab (5)

where there is an implicit sum over the index c ∈ [1,DimG].

The quantities ccab are called the structure constants of the group: they define the Liealgebra LG of the Lie group G.

(c) We use lower case letters to denote the Lie algebras of SU(N), SO(N), etc, by su(N),so(N), etc.

(c) If we omit the factors of i in equations (3) and (4), there is no factor of i in equation (5).

3.3 Exponentiation

Consider a one dimensional subgroup G1 ⊂ G, {g(t)} = {g(α(t))} labelled by a singleparameter t. We can choose t ∈ R such that

g(s) g(t) = g(s+ t); g(0) = 1 (by Theorem 1)

Differentiate the equation above with respect to s:

g′(s) g(t) = g′(s+ t)

and set s = 0, which gives a differential equation for g(t), which we can solve easily:

d

dtg(t) = g′(0) g(t) ⇒ g(t) = exp (tg′(0)) .

But, we can also write

g′(0) =∂g

∂αa

∣∣∣∣α=0

dαa

dt

∣∣∣∣t=0

= iγa Ta where γa ≡ dαa

dt

∣∣∣∣t=0

,

so if g ∈ G1 ⊂ G, theng(t) = exp(i(tγa)Ta) ,

9

therefore every element of G that can be connected (continuously) to the identity has thisform. But the set {eiβaTa} has dimension equal to dimG and coincides with G for βa

infinitesimal. So if we know the generators we know all parts of the group connected to theidentity.

Furthermoregh = exp{iα · T} exp{iβ · T} = exp{iγ · T}

for some γ , because gh ∈ G. But the Baker–Campbell–Hausdorff formula tells us that forany two matrices A and B,

exp(A) exp(B) = exp(A+B + 12[A,B] + 1

3!12([A, [A,B]] + [[A,B], B]) + · · ·)

so if we know LG we can evaluate all the commutators and thus γ(α, β) (at least in principle).

To summarise:

• Lie group ⇒ Lie algebra by linearisation.

• Lie algebra ⇒ connected part of a Lie group by exponentiation.

Note that G ⇒ LG uniquely, but not vice versa: several groups may have the same Liealgebra. These groups are identical locally (in the neighbourhood of 1), but not globally.Fortunately, for a given LG there is only one connected group G which is simply connected(no holes): this is called the universal covering group (not proven here).

Example. O(N), SO(N) have the same generators, so o(N) ∼= so(N) – the Lie algebras areisomorphic.

However, for O(N), since DDT = 1, (detD)2 = 1, which implies detD = ±1, while forSO(N) we also have detD = +1 (by definition). We can’t get from detD = −1 to det = +1by a continuous transformation, so O(N) is not connected, whereas SO(N) is connected (butnot simply connected – see later). We write SO(N) ∼= O(N)/Z2. For rotations, N = 2, 3,think of Z2 as the parity.

For example, in the case of SO(2), choosing the generator to be real antisymmetric,

T =

(0 1−1 0

), T 2 = −1, and hence eφT = 1 cosφ+ T sinφ (exercise).

The matrices

(1 00 −1

)and

(−1 0

0 1

)are in O(2), but they’re not in SO(2) because

they’re not continuously connected to the identity.

• The simply connected covering group of SO(N) is called Spin(N).

• The simply connected covering group of SO(3) is SU(2). This is a special case (seetutorial question (1.4)).

10

3.4 Properties of Structure Constants

Recall the Lie algebra LG of a Lie group G:

[Ta, Tb] = iTc ccab

1. Clearly ccab = −ccba by anti-symmetry of the commutator: [Ta, Tb] ≡ TaTb − TbTa =−[Tb, Ta].

2. Under rescaling Ta 7→ λTa, ccab 7→ λccab.

3. The ccab are real.

Proof. We know already that if D(g) is a representation of G, then D∗(g) is also arepresentation. So if Ta are a representation of LG, −T ∗a are also a representation. But[−T ∗a ,−T ∗b ] = i(−T ∗c )ccab

∗, so ccab∗ = ccab.

4. The Jacobi identity :

[Ta, [Tb, Tc]] + [Tb, [Tc, Ta]] + [Tc, [Ta, Tb]] = 0 .

Note that the Jacobi identity holds for any three square matrices Ta, Tb, Tc (exercise).

Substituting from the Lie algebra, we find (exercise)

cdae cebc + cdbe c

eca + cdce c

eab = 0 .

5. The structure constants form a representation of LG, known as the adjoint represen-tation. If we define Dim-G (Dim-G)×(Dim-G) matrices Aa with elements

(Aa)bc ≡ icbac

then the Jacobi identity can be written

(Aa)de (Ab)

ec − (Ab)

de (Aa)

ec = i(Ae)

dc c

eab,

where we used the anti-symmetry of the structure constants, i.e. ceca = −ceac in thesecond term on the left-hand side (LHS), and similarly for the first term on the right-hand side (RHS). In matrix notation

[Aa, Ab] = iAe ceab.

3.5 The Adjoint Representation

If a vector v transforms under G : v 7→ gv, then ∀h ∈ G, (hv) 7→ g(hv) = (ghg−1)(gv). So ifv 7→ gv, h 7→ ghg−1 gives the action of the group on itself.

Now consider h′ = ghg−1 with h infinitesimal,13 then h′ is also infinitesimal:

h = 1 + iαaTa + · · ·h′ = 1 + iβaTa + · · · where βa = Da

b(g)αb for some numbers Dab(g) (by linearity)

= 1 + iTaDab(g)αb + · · · = 1 + iTbD

ba(g)αa + · · ·

13i.e. h differs from the identity element by the infinitesimal quantity, which in this case is iαaTa

11

where we simply relabelled indices (a ↔ b) to get the last expression. It follows from thefirst and third lines above that

gTag−1 = TbD

ba(g).

Pictorially:

vg−→ gv

↓ h

adg=⇒ ↓ ghg−1

hvg−→ ghv

The (Dim-G)×(Dim-G) matrices Dba(g) form a representation (the adjoint representation)

of the group G:

Dba(1) = δba, TbD

bc(g)Dc

d(h) = (g Tc g−1)Dc

d(h) = h (g Td g−1)h−1 = h g Td (hg)−1 = TbD

bd(hg)

so D(g)D(h) = D(hg) as required. (Note that with the definitions used here, D acts to theleft if g acts to the right.)

The generators in this representation are just the structure constants described above. Toverify this, set g = 1 + iαbTb, so g−1 = 1− iαbTb, and write Dc

a = δca + iαb(Ab)ca . Then

gTag−1 = Ta + iαb (Tb Ta − Ta Tb) +O(α2)

= Ta + iαb Tc iccba +O(α2)

= Tc(δca + iαb iccba

)+O(α2) .

So (Ab)ca = iccba as claimed.

It follows that the adjoint representation is a ‘real’ representation. (Recall that the structureconstants are real, so the elements of Ab are i times a real number, which we call a ‘real’representation). Note that the elements of Ab would actually be real if we used anti -hermitiangenerators!

Example. SU(2) and SO(3).

Take the generators of SU(2) as 12σi. Then since σ1σ2 = iσ3 etc., [ 1

2σi, 1

2σj] = iεijk 1

2σk, so the

structure constants are just εijk. Consider g = 1 + i2ω · σ +O(ω2). Then

gσkg−1 = (1 + i

2ω · σ)σk(1− i

2ω · σ)

= σk + i2ωi[σi, σk]

= σk + ωiεkimσm,

= σk +(ω × σ

)k,

i.e. g σ g−1 = σ + ω × σ +O(ω2)

This is clearly an infinitesimal rotation ∈ SO(3), i.e. an infinitesimal rotation through angleω ≡ |ω| about the axis ω in R3.

It follows that the adjoint representation of su(2) is the defining representation of so(3), thusthat their Lie Algebras are the same: su(2) ∼= so(3). In fact SO(3) ∼= SU(2)/Z2: SU(2) isthe connected covering group of SO(3) (see tutorial question (1.4)).

12

3.6 Invariant Tensors

An invariant tensor of rank n da1a2...an is any tensor whose components are invariant underthe action of the group, i.e.,

d′a1...an = da′1...a′n Da′1a1(g)Da′2

a2(g) · · ·Da′nan(g) ≡ da1...an

The simplest way to construct invariant tensors is to consider traces of products of generators

da1...an ≡ tr(Ta1Ta2 · · · Tan)

G7→ d′a1...an

= tr(Ta′1Ta′2 · · · Ta′n)Da′1a1 · · · Da′n

an = tr((gTa1g

−1) (gTa2g−1) · · · (gTang

−1))

= tr(Ta1(g

−1g)Ta2(g−1g)) · · · Tan(g−1g)

)= da1 ··· an

i.e., da1 ··· an is an invariant tensor for any representation Ta, including the adjoint represen-tation Aa.

Example. cdbc is invariant (use the Jacobi identity).

3.7 Killing Form

An important invariant tensor is the Killing form. This is a metric on the group spacedefined by

gab ≡ tr (AaAb) = (Aa)cd (Ab)

dc = −ccad cdbc = gba (cyclic property of the trace)

where the Aa are the generators in the adjoint representation.

Since the structure constants are real, gab are the elements of a real symmetric DimG×DimGmatrix - this is the advantage of using the adjoint representation. Thus we may choose abasis in which it is diagonal

gab = µa δab (no sum on a.)

Since gab is real symmetric, the eigenvalues µa of are real, but may be positive, negative,or zero. Indeed, for Abelian groups gab = 0 because the structure constants are all zero.(Convince yourself of this.)

Under the rescaling Ta 7→ λTa, with λ ∈ R, then gab 7→ λ2gab (with λ2 > 0). By rescalingeach Ta separately we may reduce gab to the canonical form we introduced earlier

diag(+1, . . . ,+1, 0, . . . , 0,−1, . . . ,−1), i.e., µa = 0, ±1, ∀a.

Example. SU(2):gij = −εikl εjlk = 2δij.

so there’s no difference between upstairs and downstairs indices in the case of SU(2). If thegenerators were rescaled to −1√

2εijk we would have gij = δij,

13

We can use the Killing form to lower indices: for example, we may define from the structureconstants cabc a set of quantities cabc with all lower indices:

cabc ≡ gad cdbc.

This is also an invariant tensor because we can use the Lie algebra to write it in terms ofthe trace of products of generators:

tr(Aa[Ab, Ac]) = i tr(AaAd) cdbc = igad c

dbc = icabc.

Since14

tr(Aa[Ab, Ac]) = tr([Aa, Ab]Ac) = tr([Ac, Aa]Ab) .

it follows that cabc is totally antisymmetric on all three indices.

A second invariant tensor with three indices is dabc = tr(Aa{Ab, Ac}) which is totally sym-metric.15

We need consider only one tensor da1···an for each n > 3, the one which is totally symmetricbecause antisymmetric parts can always be expressed in terms of cabc and tensors of lowerdegree, by using the commutation relations.

Definition 5 The number of independent invariant tensors (excluding cabc) is called therank of the group.

4 Simplicity

We may consider a Lie algebra L = {Ta : a = 1, . . . , l} as a vector space spanned by thegenerators. Then

Definition 6 The subspace S ⊂ L is a subalgebra of L if the commutator [s, s′] ∈ S for alls, s′ ∈ S.

In words, the commutator of any two elements in the subspace also lies in the subspace.

S will be spanned by a new set of generators {Si : i = 1, . . . , l′}. S is a proper subalgebraprovided 0 < l′ < l.

Definition 7 The subalgebra S ⊂ L is an invariant subalgebra (or ideal) if [s, t] ∈ S forall s ∈ S, t ∈ L, i.e., if S “absorbs” L or “S knocks L into S”.

This is analogous to the definition of a normal subgroup: ghg−1 ∈ H, ∀h ∈ H, g ∈ G,H ⊂G.

Definition 8 The Lie algebra L is simple if it has no proper invariant subalgebras, whereasL is semi-simple if it has no proper Abelian invariant subalgebras.

Simple and semi-simple Lie groups are those with simple or semi-simple algebras. Clearlyall simple groups/algebras are also semi-simple.

14trX[Y, Z] = tr(XY Z−XZY ) = trXY Z−trXZY = trXY Z−trY XZ = tr(XY Z−Y XZ) = tr[X,Y ]Z,etc.

15The anti-commutator {A,B} ≡ AB +BA.

14

4.1 Non semi-simple algebras

Consider an algebra which is not semi-simple. Then it has a proper Abelian invariantsubalgebra S ⊂ L.

Let’s take a basis {Si} for S, and extend it to a basis {Si, Pp} = {Ta} for L; where theindices a, b, c ∈ T , i, j, k ∈ S, and p, q, r ∈ P , i.e. the set {Pp} contains all the generatorsthat are in L, but not in S. Then

1. Since S is Abelian: [Si, Sj] = iTa caij = 0 (by definition), therefore caij = 0;

2. Since S is invariant: [Si, Ta] = iTb cbia ⊂ S implies cpia = 0 and, by anti-symmetry of

the commutator, cpai = 0.

Then, using 2. (twice) and 1. (once),

gia = tr(AiAa) = (Ai)cd (Aa)

dc = −ccid cdac = −ckid cdak = −ckij cjak = 0,

so gab has the form

(0 00 gpq

). It follows that gab has zero eigenvalues, and det gab = 0.

4.2 Cartan’s Criterion

A Lie algebra is semi-simple iff 16 the Killing form is invertible, i.e., det gab 6= 0.

We denote the inverse by gab, so that gacgcb = δab. gab is now a pseudo-Riemannian metric.

For semi-simple groups we may choose a basis in which gab and gab are diagonal and bothtake the form diag(1, . . . , 1,−1 . . . ,−1), i.e., all eigenvalues are ±1.

4.3 Casimir Operators

When L is semi-simple, we may define the quadratic Casimir operator

C(2) ≡ gab Ta Tb

Generalising, higher Casimir operators are defined in terms of rank-n invariant tensors andproducts of n generators

C(n) ≡ da1···anTa1 · · ·Tan where da1···an = ga1a′1 · · · gana′n da′1···a′n , n ≥ 3.

There are as many independent Casimir operators as there are independent symmetric in-variant tensors, i.e., the rank of the algebra.

The Casimir operators are useful because they commute with the generators: [C(n), Ta] = 0.To show this, note that

gC(n)g−1 = da1···an g Ta1 · · ·Tan g−1 = da1···an (gTa1g−1) · · · (gTang−1)

= da1···an Ta′1 Da′1a1 · · ·Ta′n D

a′nan

= da1···anTa1 · · ·Tan = C(n)

16iff means ‘if and only if’.

15

where the first expression in the last line follows because da1···an is an invariant tensor. Takingg = 1 + iαaTa with infinitesimal αa gives the result:

(1 + iαaTa)C(n) (1− iαaTa) = C(n) ⇒ C(n) = C(n) + iαa

[Ta, C

(n)]

Since Casimir operators are constructed from products of generators, it follows that theyalso commute with each other: in fact they constitute a maximally commuting set. Theyare thus useful for labelling representations. This is analogous to using a complete set ofmutually commuting operators to label states in quantum mechanics.

For semi-simple Lie algebras it is possible to show using Schur’s lemma that

gab ≡ trTaTb = C(R) gab

for some real number C(R) which depends only on the representation R of the generatorsTa. Taking the trace of the expression for the definition of the quadratic Casimir operatorabove, and using this result gives

trC(2) = gab tr (Ta Tb) = gabC(R) gab = δaaC(R) = (DimG)C(R)

So C(R) = trC(2)/ dimG.

For the adjoint (Ad) representation, C(Ad) = 1 with our normalisation. One often seesinstead the normalisation C(Fun) = 1 or 2 or 1

2for the fundamental (Fun) representation.

4.4 Cartan Decomposition

Consider a semi-simple Lie algebra L that is not simple: it has a proper invariant subalgebraS ⊂ L.

Use the metric gab to define an othogonal subspace P , again using indices i, j, k ∈ S, andp, q, r ∈ P :

gip = tr (AiAp) = 0 for all Ai ∈ S, Ap ∈ P .Now tr (Ai [Aj, Ap]) = tr ([Ai, Aj]Ap) = 0 because [Ai, Aj] ∈ S

Thus [Aj, Ap] ∈ P , i.e. [S, P ] ⊂ P .

Since S is invariant, we already have [S, P ] ⊂ S.

But S ∩ P = 0, so [S, P ] = 0 (all elements of S commute with all elements of P ). We saythat L decomposes into a direct sum L = S ⊕ P .

It is easy to show that P is also a subalgebra of L:

tr ([Ap, Aq]Ai) = tr (Ap [Aq, Ai]) = 0 ([Aq, Ai] = 0 because L = S ⊕ P )

so [P, P ] ⊂ P as required.

It is now easy to prove the following theorem:

Theorem 4 A semi-simple Lie algebra L is a direct sum of mutually commuting simple Liealgebras:

L = L1 ⊕ L2 ⊕ · · · ⊕ Ln,where the Li are simple.

16

Proof. If L is not simple, use the above result to decompose it as

L = S ⊕ P.

Both S and P are semi-simple (otherwise L would not be). If either is not simple, repeatthe procedure. Since L is finite dimensional, this procedure obviously stops.

Example.

• so(4) ∼= su(2)⊕ su(2) (see tutorial question (3.2))

• sl(2,C) ∼= su(2)⊕ su(2) (see later).

The decomposition of the algebra induces a similar decomposition of a semi-simple group

G ∼= (G1 ⊗G2 ⊗ · · · ⊗Gn) /Z ,

where the Gi are simple, and Z is the centre of G, namely the set of all group elements g′ thatcommute with themselves and with all other group elements: Z = {g′ : gg′ = g′g, g ∈ G}.Clearly Z is discrete, as otherwise G would not be semi-simple.

Example: SO(4) ∼= SU(2)⊗ SU(2) /Z2.

5 Compact Groups I

If the Lie group G is compact, the generators of LG may be taken to be hermitian (becausethere are unitary representations). This has far-reaching consequences.

Taking the adjoint representation hermitian, Aa = A†a, so (Aa)bc = (A∗a)

cb, and hence

(Aa)bc = icbac = −iccab .

Thereforegab = tr (AaAb) = (Aa)

dc (Ab)

cd = (−i ccad) iccbd = ccad c

cbd .

Theorem 5 If G is compact LG may be decomposed as

LG ∼= L0 ⊕ (L1 ⊕ · · · ⊕ Ln)

where L0 is Abelian (so L0∼= u(1)⊕ · · · ⊕ u(1)) and the Li are simple.

Proof. If LG is not semisimple, it has a proper Abelian invariant subalgebra S ⊂ L, sogia = 0 for Si ∈ S, Ta ∈ L (derived on page 15).

But we have just shown that for hermitian Aa, gia = cbic cbac, so gii = (cbic)

2 = 0 (notsummed over i); therefore cbic = 0 ∀i, b, c, whence [S, L] = 0. It follows that LG = L0 ⊕ L′where L0 is Abelian and L′ is semisimple, and thus can be decomposed into simple algebrasas previously shown.

So we need consider only semisimple (or indeed simple) compact groups. NB, this decom-position does not apply to non-compact groups because it relies on the generators beinghermitian.

17

Theorem 6 If G is compact, the Killing form gab is positive semi-definite (has non-negativeeigenvalues). If LG is also semisimple, we may choose a basis in which gab = δab, anddistinctions between upper and lower indices can thus be ignored.

Proof. For any real vector xa

gabxaxb = ccad c

cbd x

axb = (xaccad)2 ≥ 0

Therefore all eigenvalues of gab ≥ 0. If LG is semisimple, det gab 6= 0, so gabxaxb > 0 and all

eigenvalues > 0. Finally, rescale the generators so that gab = diag(1, . . . , 1).

Theorem 7 If G is compact and semisimple, trTaTb = C(R)δab for all irreducible represen-tations. The positive real number C(R) is “the Casimir” of the representation R.

Proof. Omitted (uses Schur’s lemma).

5.1 The su(2) algebra

This is the “angular momentum algebra” familiar from quantum mechanics. For su(2) ∼=so(3) [Ti, Tj] = iεijkTk, gij = 2δij, so it is compact and semisimple. Define the “step opera-tors” T± = T1 ± iT2, then

[T3, T±] = ±T±[T+, T−] = 2T3

[T+, T+] = [T−, T−] = 0.

The quadratic Casimir is T 2 = T 21 + T 2

2 + T 23 = T−T+ + T 2

3 + T3, and [T 2, Ti] = 0. There areno other Casimir operators (so su(2) has rank 1): T 2 and T3 form a complete commutingset. We shall show that:

• T 2 has eigenvalues j(j + 1), 2j ∈ N: use to label representations.

• T3 has eigenvalues m = j, j− 1, . . . ,−j, 2m ∈ Z: use to label states within a represen-tation.

Let T3 |jm〉 = m |jm〉 for some (unknown) real numbers j and m. Let the maximum value ofm be j and the minimum value be m, so m ≤ m ≤ j. Then T3T± |jm〉 = T±(T3 ± 1) |jm〉 =(m ± 1)T± |jm〉, while T 2 |jj〉 = (T−T+ + T 2

3 + T3) |jj〉 = j(j + 1) |jj〉 because T+ |jj〉 =0. |jj〉 is the “state of greatest weight”. Since we are interested in finite dimensionalrepresentations there is also a “state of least weight” T− |jm〉 = 0, for which T 2 |jm〉 =(T+T− + T 2

3 − T3) |jm〉 = m(m− 1) |jm〉. Since T 2 is a multiple of the unit matrix we musthave m(m− 1) = j(j + 1), so m+ j = m2 − j2 = (m+ j)(m− j), and m = −j since m ≤ j.

Representations:

-6w0

-6w w− 1

212

-6w w w−1 0 1

-6w w w w− 3

2− 1

212

32

1 2 (fundamental) 3 (adjoint) 4

Combining representations: “start with the state of largest m and work down”: 2⊗2 = 3⊕1,3⊗3 = 5⊕3⊕1, etc. We will show later that all compact semisimple algebras contain su(2)subalgebras which allow for similar constructions of their representations, and compositionof representations.

18

6 Space-time Symmetries

In this section we shall study the symmetry groups of the space-time of special relativity,namely the Lorentz and Poincare groups.

6.1 Kinematics

Let us choose a set of space-time coordinates xµ = (x0, xi), with x0 = ct and i = 1, 2, 3.

We have a metric tensor17 ηµν which is real, symmetric, invertible, det ηµν 6= 0, with inverseηµν , so that ηµν ηνσ = δµσ.

We can choose an orthonormal frame in which ηµν = diag(1,−1,−1,−1). Space-time isnon-compact.

Define xµ ≡ ηµν xν , so xµ = ηµνxν etc. xµ = (x0,−xi).

We use the metric to define “length” of xµ: x2 ≡ ηµνxµxν = ηµνxµxν = xµxµ = xµx

µ =x20 − |x|2.

6.2 Lorentz Group

Under a change of frame xµ 7→ x′µ = Λµν x ν (so xµ 7→ x′µ = Λµ

ν xν).

The postulates of Special Relativity are

• The speed of light c is the same in all frames.

• Space-time is isotropic – the laws of physics are the same in all reference frames.

These lead to:

x′2

= ηµν x′µ x′ν = ηµν Λµ

ρ Λνσ x

ρxσ = ηρσ xρ xσ = x2 ≡ x20 − |x|2

Since this expression is true for all xµ then

ηµν Λµρ Λν

σ = ηρσ,

thus the transformation preserves the metric.

Since Λµρ = (ΛT )ρµ, we may write this in matrix notation as ΛTηΛ = η.

Taking the determinant gives: det η = det ΛT det η det Λ⇒ det Λ = ±1.

In an orthonormal frame, choosing ρ = σ = 0 gives

(Λ00)

2 − (Λi0)

2 = 1

Since (Λi0)

2 ≥ 0, we must have (Λ00)

2 ≥ 1, so either Λ00 ≥ 1 or Λ0

0 ≤ −1. Thus theLorentz group O(3, 1) has 4 disconnected pieces, connected by time reversal T : x0 7→ −x0and parity P : xi 7→ −xi.

17NB the space-time metric tensor ηµν is the not the Killing form of the Lorentz group – see later.

19

If we take Λ ∈ SO+(3, 1) (i.e. det Λ = +1,Λ00 ≥ 1), we have a proper orthochronous Lorentz

transformation.18 We get the other three pieces of the group by taking the products: PΛ,TΛ, PTΛ.

The proper orthochronous transformations are:

• Rotations: Λ =

(1 00 R

), where R is 3 × 3 and satisfies RTR = 1, detR = 1, so

R ∈ SO(3).

• Boosts: e.g., Λ =

cosh η − sinh η− sinh η cosh η

0

0 1

∈ SO+(1, 1).

We get the usual form of the Lorentz boost along the x1 axis (in units where the speed

of light c = 1) if we take cosh η = (1− v2)−12 , sinh η = v (1− v2)−

12 . The most general

orthochronous Lorentz transformation is specified by 3 rotations about orthogonal axes, and3 Lorentz boosts in orthogonal directions. Therefore SO(3, 1) has dimension 6.

6.3 Lorentz algebra

Consider an infinitesimal Lorentz transformation on the (four-)vector 19 V µ, i.e. in thedefining or fundamental or vector representation of the Lorentz group:

V ′µ = ΛµνV

ν = (δµν + ωµν)Vν , (6)

Now write δV µ = ωµν Vν = ωµν Vν where ωµν = −ωνµ is real and anti-symmetric, so it has

6 real parameters:

• ωij = −ωji, ∀ i, j = 1, 2, 3, give rotations (ωij = εijk θ nk)

• ω0i ≡ νi give boosts.

In a general representation, D, of the Lorentz group, we write an infinitesimal transformationas

D(1 + ω) = 1 +i

2ωαβMαβ (7)

where Mαβ = −Mβα with α, β = 0, 1, 2, 3, are a set of six hermitian matrices, the generatorsof the Lorentz algebra. (It may help to think of a = (α, β) as a single index a = 1, 2, . . . 6,which labels the six generators Mαβ = Ma of the Lorentz group, i.e. similar to Ta in ourprevious discussions.)

In the defining (fundamental, vector) representation, from equations (6) and (7), we find

(Mαβ)µν = −i(δµα ηβν − δ

µβ ηαν

)(exercise) (8)

In words, this is the (µν) element of the (α, β)th (matrix) generator Mαβ in the definingrepresentation of the Lorentz group.

18An orthochronous Lorentz transformation preserves the direction of time.19For example V µ = xµ.

20

To find the commutation relations [Mµν , Mαβ], we note that (Λ−1)µν

=(ΛT)µν

for an in-finitesimal Lorentz transformation, and use the group properties in an arbitrary representa-tion D(Λ) to write

D(Λ)D(1 + ω)D(Λ)−1 = D(1 + ΛωΛT )

⇒ D(Λ)

(1 +

i

2ωαβMαβ

)D(Λ)−1 = 1 +

i

2

(Λα

µ ωµν Λβ

ν

)Mαβ

⇒ D(Λ)ωαβMαβD(Λ)−1 = ωαβΛµα Λν

βMµν

In the last two lines, we raised and lowered some pairs of indices and relabelled dummyindices on the RHS (α↔ µ and β ↔ ν) to give ωαβ on both sides.

This holds for all infinitesimal parameters ωαβ, so

D(Λ)MαβD(Λ)−1 = Λµα Λν

βMµν .

Now take Λ = 1 + ε, i.e. Λµν = δµν + εµν and expand through O(ε). Substituting D(Λ) =

1+ i2εµνMµν into the above expression, and cancelling the leading Mαβ from both sides, gives

i

2εµν [Mµν , Mαβ] = εµα δ

νβMµν + δµα ε

νβMµν

= εµαMµβ − εβνMαν

=1

2εµν (Mµβ ηνα −Mνβ ηµα −Mαν ηµβ +Mαµ ηνβ)

where the quantity in parentheses is explicitly anti-symmetric under µ ↔ ν and α ↔ β,and we used the metric ηµν to raise and lower indices. We also used the anti-symmetryεµν = −ενµ, several times. The result holds for all εµν , which is also anti-symmetric, so

[Mαβ, Mµν ] = i (Mαµ ηβν −Mαν ηβµ −Mβµ ηαν +Mβν ηαµ) (9)

which defines the algebra, so(3, 1), of the Lorentz group SO(3, 1)

The generators Mij, i, j = 1, 2, 3, generate an so(3) subalgebra: so(3) ⊂ so(3, 1). To seethis, we write it in a more familiar form by first defining

Ji ≡ − 12εijkMjk .

Then

[Ji, Jj] =1

4εimn εjpq [Mmn, Mpq]

=i

4εimn εjpq (Mmp ηnq −Mmq ηnp −Mnp ηmq +Mnq ηmp)

= − i4εimn εjpq (Mmp δnq −Mmq δnp −Mnp δmq +Mnq δmp) (as ηnq = −δnq, etc)

= −i (δijδmp − δipδmj)Mmp

= −i (δijMmm −Mji) = −iMij

= i εijk Jk

21

where in the last line we inverted the definition of Ji to obtain Mij = − εijk Jk. So, finally,

[Ji, Jj] = iεijkJk,

which is indeed the familiar algebra of su(2) ∼= so(3). [See also tutorial question (3.2).] SoJi are just the usual rotation generators.

Similarly let Ki = M0i, then Ki are the boost generators. We find

[Ki, Kj] = [M0i,M0j] = iMij = −iεijk Jk

and

[Ji, Kj] = −12εimn [Mmn,M0j] = −1

2εimn i (Mm0 ηnj −Mn0 ηmj) = iεijkM0k = iεijkKk.

Thus the algebras satisfied by the generators Ji and Ki are coupled or entangled :

[Ji, Jj] = iεijkJk, [Ki, Kj] = −iεijk Jk, [Ji, Kj] = iεijkKk.

We can disentangle them by defining new generators Ni, which are linear combinations of Jiand Ki

Ni ≡ 12

(Ji + iKi) .

Note that these are not hermitian, N †i = 12(Ji − iKi) 6= Ni.

We find

[Ni, Nj] = 14

([Ji, Jj] + i [Ji, Kj] + i [Ki, Jj]− [Ki, Kj])

= 14

(i εijk Jk − εijkKk + εjikKk + i εijk Jk)

= iεijk12

(Jk + iKk)

= iεijkNk

Similarly[Ni, N

†j ] = 1

4([Ji, Jj]− i [Ji, Kj] + i [Ki, Jj] + [Ki, Kj]) = 0.

and[N †i , N

†j ] = iεijkN

†k

Altogether[Ni, Nj] = iεijkNk [N †i , N

†j ] = iεijkN

†k [Ni, N

†j ] = 0

i.e., the six generators Ni and N †i generate two separate su(2) algebras.

These two su(2) algebras are not completely independent, since they are related by hermitianconjugation, or under parity (Ji 7→ Ji, Ki 7→ −Ki, so Ni 7→ N †i ).

Casimirs: we can construct two Casimir operators by taking the product of generators withinvariant tensors ηµν and εµνρσ

12MµνM

µν = J2 −K2 14εµνρσM

µνMρσ = −J ·K.

The commutation relations show that so(3, 1) has rank 2 and is semi-simple, but it’s notcompact: the Killing form is diag(1, 1, 1,−1,−1,−1): rotations and boosts give oppositesigns.

22

6.3.1 Relation to sl(2,C)

The six (Pauli) matrices {12σi,− i

2σj} satisfy the same algebra as {Ji, Kj}, so they may be

chosen as a basis of generators that generate rotations and boosts respectively.

These six 2 × 2 complex traceless (but not all hermitian) matrices are the generators ofsl(2,C) in the fundamental representation. Therefore sl(2,C) and so(3, 1) are isomorphic:sl(2,C) ∼= so(3, 1).

In fact SL(2,C) is the covering group of the Lorentz group

SO(3, 1) ∼= SL(2,C)/Z2 ,

just as SU(2) is the covering group of SO(3). [See tutorial question (3.3).]

6.4 Representations of sl(2,C)

The generators N3 and N †3 commute. (Recall that they’re not hermitian: sl(2,C) is notcompact.)

Since the Ni and N †i satisfy decoupled su(2) algebras, we can construct two pairs of stepoperators N± and N †± as we did for a single su(2) algebra.

There are two Casimir operators:

1. N2 ≡ NiNi with eigenvalues m(m+ 1)

2. N †2 ≡ N †iN

†i with eigenvalues n(n+ 1)

where m,n = 0, 12, 1, . . ., just as for a single su(2).

Since we have two su(2) algebras, we label each representation by (m,n), and we label thestates within the representation by the eigenvalues of N3 and N †3 , which run from −m,−m+1, . . . ,m, and −n, . . . , n respectively; i.e., there are (2m+ 1)(2n+ 1) states. The spin of thestates within the representation (m,n) varies from |m − n| to m + n, since Ji = Ni + N †i .This is just like adding angular momenta in quantum mechanics.

Example.

1. (0, 0) spin zero: scalar.

2. ( 12, 0) and (0, 1

2) spin 1

2: spinors (the fundamental representation of sl(2, C)).

3. ( 12, 1

2) spin one: representation of a vector, Vµ = (V0, V ) (see below).

4. (1, 0) and (0, 1) spin one antisymmetric tensor Fµν (self-dual and anti-self-dual – seelater). The adjoint representation is (1, 0)⊕ (0, 1) (Fµν has 6 components).

5. (1, 1) spin two: a symmetric traceless tensor Tµν .

We will explore these in more detail as the course progresses.

23

6.4.1 Euclidean Rotation Group

If we consider “imaginary time”, i.e. let t 7→ it, we recover “Euclidean space”, −ηµν 7→δµν , and the symmetry group is SO(4) ∼= SU(2) ⊗ SU(2)/Z2 - see tutorial questions (3.2)and (3.3). Unlike SO(3, 1), SO(4) is compact: gab = diag(1, 1, 1, 1, 1, 1) . The two su(2)subalgebras of so(4) are completely independent, and each state is characterised by two“spins”, n,m, both of which are “observables” (correspond to hermitian generators).

6.4.2 sl(2,C) Spinors

The spinor representations ( 12, 0) and (0, 1

2) of the Lorentz group (or more precisely of

SL(2,C)) are realised by two-component complex Weyl spinors, ψL and ψR respectively.The choice of labels L,R is a convention.

Spinor indices are usually suppressed (although one sometimes sees ψa ∈ ( 12, 0), ψa ∈ (0, 1

2),

with a, a = 1, 2 ). (Note the “dotted index” a in the second case.)

Under an SL(2,C) transformation

ψL 7→ ψ′L = ΛL ψL

ψR 7→ ψ′R = ΛR ψR.

where ΛL and ΛR are 2× 2 complex matrices.

Under a rotation in SU(2) ⊂ SL(2,C), taking generators Ji = 12σi,

ΛL,R = exp{i12σ · ω

},

where ωi = − 12εijkω

jk are the rotation parameters.

The representation of boosts is not unitary. If we take boost generators Ki = − i2σi with

boost parameters νi = ω0i, then

ΛL = exp

{i

2σ · (ω − iν)

}ΛR = exp

{i

2σ · (ω + iν)

}Since the ( 1

2, 0) and (0, 1

2) representations are related by parity, we construct ΛR from ΛL by

changing the sign of the boost parameters.

So, under a parity transformation, ΛL ↔ ΛR and ψL ↔ ψR.

Properties of ΛL,R:

1. Λ−1L = Λ†R (NB, in Euclidean space ΛL and ΛR are completely independent.)

2. Since σ2σiσ2 = −σ∗i , σ2 ΛL σ2 = Λ∗R, σ2Λ†Lσ2 = ΛT

R.

3. Since σ2 σi σ2 = −σTi , σ2ΛTLσ2 = Λ−1L , and hence ΛT

Lσ2ΛL = σ2.

Similarly ΛTRσ2ΛR = σ2.

Exercise: check all of these.

24

Consider a left handed (LH) spinor ψL ∈ ( 12, 0). Under SL(2,C)

σ2ψ∗L 7→ σ2Λ

∗Lψ∗L = ΛR(σ2ψ

∗L)

using result 2. above and σ22 = 1. So σ2 ψ

∗L ∈ (0, 1

2) is a right handed spinor, which we call

the charge conjugate of ψL. Similarly σ2 ψ∗R is a left handed spinor.

Define charge conjugation C : ψL 7→ σ2 ψ∗L, which is right handed, and ψR 7→ σ2 ψ

∗R, which

is left-handed.

From two left handed spinors ψL and χL we can construct a scalar (0, 0) under SL(2,C),since ( 1

2, 0) ⊗ ( 1

2, 0) = (0, 0) ⊕ (1, 0) (just as spin- 1

2times spin- 1

2gives spin-0 plus spin-1 in

quantum mechanics). Explicitly:

χTL σ2 ψL 7→ χTL ΛTL σ2 ΛL ψL = χTL σ2 ψL

where we used result 3. above.

In particular, if we choose χL such that ψR = σ2 χ∗L , and hence ψ†R = χTL σ2 , then ψ†R ψL is

a scalar. Similarly, ψ†L ψR is a scalar.

We can also build a 4-vector from two spinors: ( 12, 0)⊗ (0, 1

2) = ( 1

2, 1

2). Under an infinitesimal

Lorentz transformation

ψ†LψL 7→ ψ†L

(1− i

2σ · (ω + iν)

)(1 +

i

2σ · (ω − iν)

)ψL

= ψ†L ψL − ω0i ψ†L σi ψL (since νi = ωi0 = −ω0i)

while

ψ†LσiψL 7→ ψ†L

(1− i

2σ · (ω + iν)

)σi

(1 +

i

2σ · (ω − iν)

)ψL

= ψ†L

(σi +

i

2[σi, σj]ωj + 1

2{σi, σj} νj

)ψL

= (δij − ωij)ψ†L σj ψL + ωi0 ψ†L ψL.

where we recalled that ωi = − 12εijk ωjk, which we inverted to get ωij = −εijk ωk.

It follows that if we define σµ ≡ (1, σi), then

ψ†LσµψL 7→ (δµν + ωµ

ν)ψ†L σν ψL

We conclude that ψ†L σµ ψL is a 4-vector because it transforms in the same way as xµ underLorentz transformations.

Similary, if we define σµ ≡ (1,−σi), then ψ†R σµ ψR is also a 4-vector.

6.4.3 Dirac Spinors

As we saw earlier, the spinors ψL and ψR transform into each other under parity, ψL ↔ ψR.If we want invariance under parity, we need both ψL and ψR. We construct a 4-component

Dirac spinor ψ ≡(ψLψR

)= ψL⊕ψR with four components, transforming as ( 1

2, 0)⊕ (0, 1

2)

under SL(2,C).

ψ 7→(

ΛL 00 ΛR

)(ψLψR

)≡ Λψ.

(Note: 4 = 2 + 2 here, whereas 4 = 2× 2 in our previous construction of a four-vector.)

25

Under parity, the Dirac spinor transforms as

ψ =

(ψLψR

)7→(ψRψL

)=

(0 11 0

)ψ ≡ γ0 ψ.

where γ0 is a 4 × 4 matrix, with each 1 representing a 2 × 2 unit submatrix. Note that γ0interchanges ψL ↔ ψR, which have opposite parity.

We can project ψ onto L and R spinors using the projection operators PL,R = 12(1 ± γ5),

where

γ5 ≡(

1 00 −1

)PL,R satisfy P 2

L = PL, P 2R = PR, PLPR = PRPL = 0, as required for projection operators.

NB Conventions for these projection operators with the opposite signs are perhaps morecommon!

From Dirac spinors we can construct two scalars under SL(2,C):

ψ†LψR + ψ†RψL = ψ†γ0ψ ≡ ψψ even under parity (scalar)

and ψ†LψR − ψ†RψL = −ψ†γ0γ5ψ = −ψγ5ψ odd under parity (pseudoscalar),

where we defined ψ ≡ ψ†γ0.

Similarly there are two 4-vectors:

ψ†LσµψL + ψ†RσµψR = ψγµψ odd under parity (vector)

ψ†LσµψL − ψ†RσµψR = ψγµγ

5ψ even under parity (axial vector)

where γµ =

(0 σµσµ 0

), i.e., γi =

(0 −σiσi 0

).

These are the Dirac matrices (in the Weyl representation). Remembering that {σi, σj} =2δij, it’s easy to check that they form a Clifford algebra

{γµ, γν} = 2ηµν , γ5 = iγ0 γ1 γ2 γ3 , {γ5, γµ} = 0 , 㵆 = γ0 γµ γ0 , γ5

†= γ5 .

These expressions should be familiar if you’re taking the Quantum Field Theory course.Note however that we obtained them here using group theory only. We didn’t input anyquantum field theory or indeed any quantum mechanics.

6.5 The Poincare Group

Space-time is isotropic and homogeneous: this corresponds to invariance under Lorentztransformations and space-time translations (respectively)

xµ 7→ x′µ = Λµν x

ν + aµ

These are called inhomogeneous Lorentz transformations.

Translations and Lorentz transformations do not commute, because the translation param-eters are ‘rotated’ by a Lorentz transformation. Under two successive transformations, thesecond having the tilde

xµ 7→ x′′µ = Λµν x′ ν + aµ = (Λµ

ν Λνα)xα + (Λµ

ν aν + aµ).

26

If we denote a general Poincare transformation as U(a,Λ), then

U(a,Λ) = U(a, 1)U(0,Λ) ≡ U(a)U(Λ) (shorthand notation)

and, from the result for two successive transformations above, we find

U(a, Λ)U(a,Λ) = U(Λa+ a, ΛΛ) .

Let us define the generators Mµν and Pµ of the Poincare group by

U(Λ) = 1 + i2ωµνMµν with ωµν infinitesimal

U(a) = 1 + iP µ aµ with aµ infinitesimal(10)

We can deduce the commutation relations by setting:

1. Λ = Λ = 1 : U(a)U(a) = U(a+ a) = U(a)U(a), so [Pµ, Pν ] = 0.

2. a = a = 0 : Mµν satisfy the so(3, 1) commutation relations (the Lorentz algebra) asbefore.

3. a = 0, a infinitesimal, Λ = Λ−1, so U(Λ−1)U(a)U(Λ) = U(Λ−1a, 1). Then

U−1(Λ)P µ U(Λ) = P ν Λνµ,

i.e., Pµ transforms as a 4-vector (8). If Λ is infinitesimal, we find (exercise)

[Mµν , Pσ] = −i(Pµ ηνσ − Pν ηµσ).

This holds for any 4-vector.

In terms of Ji = − 12εijkMjk, and Ki = M0i, and writing the four translation generators as

Pµ ≡ (H,Pi), the commutation relations become

[Ji, Pj] = i εijk Pk [Ji, H] = 0

[Ki, Pj] = iH δij [Ki, H] = −iPi .

Together with [Pµ, Pν ] = 0, these commutation relations form the algebra of the Poincaregroup. The Poincare group has dimension 6 + 4 = 10, but it’s neither compact nor semi-simple, the latter because Pµ form an Abelian invariant subalgebra.

Despite this, we can still find two Casimir operators:

1. P 2 = PµPµ : [Pµ, P

2] = [Mµν , P2] = 0

2. The other Casimir operator is not so obvious, but the square of any 4-vector thatcommutes with the generators and is linearly independent of Pµ will suffice.

Define the Pauli–Lubanski vector

Wµ ≡ 12εµνρσM

νρP σ ,

where εµνρσ (with ε0123 = +1) is totally antisymmetric.

Then WµPµ = 0 by anti-symmetry, and [Wµ, Pν ] = 0 using the commutation relations

between Mρσ and P σ. Wµ is a 4-vector, so

[Mµν ,Wσ] = −i(Wµηνσ −Wνηµσ) .

Thus W 2 ≡ WµWµ is a scalar, and [Pµ,W

2] = [Mµν ,W2] = 0, so W 2 is a Casimir

operator.

27

6.6 Classification of Irreducible Representations

Denote the eigenvalues of Pµ by pµ = (p0, p) (which we interpret as energy and momentumin physics applications), and of P 2 by p2. Note that pµ are continuous (not discrete) becausewe can rescale Pµ 7→ αPµ for any real α (α 6= 0) without affecting any of the commutationrelations. This is always true for generators of invariant Abelian sub-algebras.

Let’s consider all possible classes of values of the eigenvalues pµ:

1. p2 = 0, pµ = 0, whence W 2 = 0 in all states. This corresponds to the vacuum (thelowest energy state) in physics.

2. p2 > 0 (timelike): set p2 = m2, where m is the mass, so p20 = m2 + |p2|. We then haveeither p0 > m (positive energy) or p0 < −m (negative energy) states (the sign of p0is an invariant for orthochronous Lorentz transformations). If we go to the rest framepµ = (m, 0), then

Wµ = 12εµνρσM

νρmδσ0

so W0 = 0 and Wi = 12mεijkM

jk = −mJi, where Ji satisfy the su(2) algebra.Therefore W 2 (remember this is a Poincare invariant) has eigenvalues −m2s(s + 1)for s = 0, 1

2, 1, . . ..

We conclude that all representations with p2 > 0 are labelled by pi (momentum,continuous), spin s, and s3 = −s,−s + 1, . . . , s − 1, s, a total of 2s + 1 degrees offreedom). Physically, a state corresponds to a particle of mass m, spin s, 3-momentumpi, and spin projection s3. Note: this is pure group theory [Weyl & Wigner (1939)],no quantum mechanics is involved!

3. p2 = 0: these are massless states, m = 0, , so there is no “rest frame”. Choose insteada frame in which pµ = (p0, 0, 0, p0). Only W 2 = 0 seems relevant for physics (otherwisewe would have infinite-dimensional “continuous spin” (Wigner) representations - seetutorial and the books by Ramond and Weinberg).

Since WµPµ = 0 and W 2 = 0, we have W3 = W0 and W1 = W2 = 0, so

Wµ = λPµ.

Since Wµ and Pµ are four vectors, λ is an invariant, called the helicity. Let ηµ = (1, 0) ,then

λ =Wµη

µ

Pµηµ=

12εijkM

ijP k

P0

=J · P|P |

which is the usual definition of helicity, i.e. the component of J along P .

One can show (tutorial) that: λ = ±s, where s is the (integer or half-integer) spin ofthe representation, i.e. λ = 0 or ± 1

2or ±1, . . . . So massless representations are labelled

by λ (left-handed or right-handed states), states within a representation are labelledby pi.

For example, the massless Weyl fermion has 2 polarisation states λ = ±1/2, as onewould naively expect, but the massless spin-1 photon has only 2 polarisation statesλ = ±1, and the massless spin-2 graviton has only 2 polarisation states λ = ±2.

4. p2 < 0: these are tachyonic states and are unphysical. Since they move faster than thespeed of light, they are acausal.

28

6.7 Fields

Motivation: properties of Casimir operators (Casimirs):

• Semi-simple or compact groups: the Casimirs have a discrete spectrum: states corre-spond to finite dimensional vectors (or spinors or tensors). Our standard example isSU(2) with states |j,m〉.

• Non-semi-simple groups: some Casimirs have a continuous spectrum: states correspondto continuous functions – fields. Example: Poincare group.

Because the eigenvalues of Pµ are continuous, we need to introduce fields φs(xµ), where slabels the spin components. This is sufficient to construct massive and massless particlestates by superposition of states of the form

|x, s〉 ≡ φs(xµ) |0〉 ,

where |0〉 is the vacuum state.

Under a Poincare transformation, a state | 〉 7→ U−1(a,Λ) | 〉, so the field transforms asφ(x) 7→ U−1(a,Λ)φ(x)U(a,Λ).

For a scalar fieldU−1φ (x)U = φ (Λx+ a) . (11)

For an infinitesimal Poincare transformation, φ(xµ) 7→ φ(xµ + δxµ) = φ(xµ) + δφ(xµ), there-fore δφ(xµ) = δxµ

∂∂xµ

φ(xµ) ≡ δxµ ∂µ φ(x) with δxµ = aµ + ωµ

νxν , where we defined the

four-vector operators ∂µ =(

∂∂x0,∇)

, ∂µ =(

∂∂x0,−∇

).

Note that under a Lorentz transformation

∂

∂xµ7→ ∂

∂x′µ=

∂xν∂x′µ

∂

∂xν= Λµ

ν∂

∂xν,

so if φ is a scalar, ∂µφ and ∂µφ are vectors, and therefore ∂µ is a vector operator.

Similarly ∂µ∂νφ is a tensor, and ∂2φ ≡ ∂µ∂µφ is a scalar. The operator ∂2 = = ∂2

∂x20−∇2

is called the d’Alembertian, the relativistic generalisation of the Laplacian.

In terms of generators, from equations (10) & (11),

δφ = −iaµ [P µ, φ]− i

2ωµν [Mµν , φ] = aµ ∂

µ φ + ωµνxν ∂

µφ

as δxµ = aµ + ωµνxν . Since this holds for all (infinitesimal) aµ and ωµν = −ωνµ,

[P µ, φ] = i∂µφ

[Mµν , φ] = i (xν∂µ − xµ∂ν)φ = (xνP µ − xµP ν)φ.

Writing Pµ = (H,Pi),

[H,φ] = i ∂∂tφ Energy (Heisenberg equation)

[P , φ] = −i∇φ Momentum

[J, φ] = −ix×∇φ Angular momentum

where Ji = − 12εijkMjk.

29

NB, we are using units where ~ = c = 1. to restore ~ remember that ~ has dimensions ofangular momentum, so let Pµ 7→ Pµ/~, Mµν 7→Mµν/~.

Since the vacuum has Pµ |0〉 = 0, it follows that

H∣∣x, t⟩ = i

∂

∂t

∣∣x, t⟩ P∣∣x, t⟩ = −i∇

∣∣x, t⟩etc. (Schrodinger equation).

A Fourier transformation to momentum space is often useful:

φ(pµ) ≡∫d4x e−ipµx

µ

φ(xµ)

etc., then [Pµ, φ(pµ)] = pµ φ(pµ) etc.

For fields which are not scalars under the Lorentz group, there is an ‘extra’ factor of D(Λ) =exp( i

2ωµνSµν) in the transformation law

U−1(Λ, a)φ(x)U(Λ, a) = D(Λ)φ(Λx+ a)

whence [Pµ, φ] = i∂µφ as before, while

[Mµν , φ] = i(xν∂µ − xµ∂ν)︸ ︷︷ ︸Lµν

φ+ Sµνφ

where Lµν is the ‘angular momentum’ (generalised to 4 dimensions) we obtained above forthe scalar field, while Sµν is the (intrinsic) ‘spin’ angular momentum (in the appropriaterepresentation of the Lorentz group, D(Λ) = exp i

2ωµνSµν).

For example, for a scalar Sµν = 0, for a Weyl spinor ωµνSµν = i2ωµνσµν , and for a vector

(i.e. the defining or fundamental representation) (ωµνSµν)αβ = ωαβ.

Note that:

1. Lµν satisfy the so(3, 1) algebra. (See tutorial question (4.1) for the so(3) case.)

2. The Pauli–Lubanski vector reduces to Wµ = 12εµνρσS

νρP σ, because εµνρσLνρP σ =

2εµνρσxνP ρP σ = 0, as [P ρ, P σ] = 0. Therefore, only the intrinsic angular momen-

tum (or “spin”) contributes to the eigenvalues of W 2 — single particle states haveno angular momentum. (Colloquially, they can have spin, but not “orbital” angularmomentum.)

30

7 Symmetry in Action

We assume that our dynamics (classical or quantum — here mainly classical) may be derivedfrom an action principle. To construct suitable actions we follow a set of fundamentalprinciples:

1. Locality : we assume the action has the form

S =

∫d4xL

where d4x = dx0 dx1 dx2 dx3 is the Lorentz invariant integration measure in Minkowskispace, and L is the Lagrange density (usually informally called the “Lagrangian”). Weassume that L is a function of local fields Φ (which could be scalars, spinors, vectors,. . .) and their derivatives at one point xµ only - so there’s no “action at a distance”).Locality is necessary to ensure causality in the quantum theory.

2. Unitarity : we assume that the action (and thus the Lagrangian) is real : this is essentialto obtain quantum theories in which probability is conserved (note in classical physicscomplex potential ⇒ absorption).

3. Causality : we require that S leads to classical equations of motion that involve nohigher than second derivatives: higher order differential equations generally have acausalsolutions. This means that L contains terms with at most two derivatives ∂µ.

4. Symmetry : we assume that S is Poincare invariant up to surface terms: since d4x isinvariant, this means that L must be a Lorentz invariant function of the fields Φ(x)and ∂µΦ(x) (terms involving xµ explicitly will not be translation invariant).

We will also consider further symmetries (internal symmetries, gauge symmetries, etc.)to further restrict the action.

5. Renormalisability : terms in the action with coefficients with dimension M−n, n > 0,generally imply that the quantum theory loses all predictive power at energies of orderM . So we either exclude such terms or assume they are small [technical requirement].

7.1 Examples

Real scalar field:L = 1

2∂µφ ∂µφ− V (φ)

where V (φ) doesn’t contain derivatives.

Complex scalar fields:L = ∂µφ† ∂µφ− V (φ†φ)

where φ = (φ1 + iφ2)/√

2 (with φ1 and φ2 real).

Weyl spinor:

L = ψ†Lσµi∂µψL or L = ψ†Rσ

µi∂µψR

Terms in the action of the form mψTRσ2ψR, where m is the mass, lead to fermion numberviolation. They can appear in extensions of the Standard Model with right-handed neutrinos.

31

Dirac spinor:L = ψi∂µγµψ −mψψ

Gauge field:

L = −14FµνF

µν where Fµν = ∂µAν − ∂νAµ

7.2 Equations of Motion

Consider the action for some generic field Φ:

S[Φ] =

∫d4xL(Φ, ∂µΦ).

Under an arbitrary change Φ 7→ Φ + δΦ

δS =

∫d4x δL =

∫d4x

(∂L∂Φ

δΦ +∂L

∂(∂µΦ)δ(∂µΦ)

)=

∫d4x

[∂L∂Φ− ∂µ

(∂L

∂(∂µΦ)

)]δΦ +

∫d4x ∂µ

(∂L

∂(∂µΦ)δΦ

)Noting that δ (∂µΦ) = ∂µ (δΦ) by linearity, we split the second term into two parts, one ofwhich is a total derivative. The last term is a surface term: it vanishes if we insist thatδΦ = 0 on the boundary of space-time. Then Hamilton’s principle δS = 0 for arbitrary δΦimplies

∂µ

(∂L

∂(∂µΦ)

)− ∂L∂Φ

= 0 .

These are the Euler-Lagrange equations for the field, which we call the classical equationsof motion (EoMs), or classical field equations.

For a scalar field with L = 12∂µφ ∂µφ− 1

2m2φ2 we obtain

∂2φ+m2φ = 0 the Klein-Gordon equation for a particle of mass m

For a Dirac field with L = ψi∂µγµψ −mψψ, varying ψ gives

γµi∂µψ −mψ = 0 the Dirac equation

Varying ψ gives an equivalent equation for ψ.

The classical equations of motion are unchanged if

1. L 7→ L+ constant

2. L 7→ L + ∂µKµ, i.e., a term which is a total derivative. The contribution of such a

term to the action can be written using the divergence theorem as an integral over asurface at infinity, which we assume vanishes.

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7.3 Noether’s Theorem (1918)

Consider Φ 7→ Φ + δΦ, with δΦ 6= 0 on the boundary of space-time, and δΦ = α∂Φ

∂αwhere

α is some infinitesimal parameter.

We call this a classical symmetry of the theory if it leaves the classical field equationsinvariant. [It turns out that this is often (but not always) sufficient to ensure that thequantum theory is invariant too].

If d4x is invariant under the symmetry, we require that L must be invariant up to a totalderivative, i.e.,

L 7→ L+ δL = L+ α ∂µKµ (12)

for some Kµ. Of course, for symmetries which leave L invariant, we have Kµ = 0.

Under variation of the fields

δL =∂L∂Φ

δΦ +∂L

∂(∂µΦ)δ(∂µΦ) = α∂µ

(∂L

∂(∂µΦ)

∂Φ

∂α

)+ α

[∂L∂Φ− ∂µ

(∂L

∂(∂µΦ)

)]︸ ︷︷ ︸

= 0 by equations ofmotion

∂Φ

∂α(13)

again using δ (∂µΦ) = ∂µ (δΦ). It follows from equations (12) and (13) that if we define thecurrent density

Jµ(x) =∂L

∂(∂µΦ)

∂Φ

∂α−Kµ the Noether current

then this current is conserved locally, i.e. at every point xµ, it satisfies

∂µJµ(x) = 0.

If Jµ = (J0, J) then∂J0

∂t+∇ · J = 0.

To see what this means, define the charge

Q(t) ≡∫V

d3x J0(x, t)︸ ︷︷ ︸chargedensity

.

Thend

dtQ(t) =

∫V

d3x∂J0(x, t)

∂t= −

∫V

d3x∇ · J = −∫S

dS · J .

The last term is (minus) the flux of the current leaving the surface S, so the charge isconserved.

For every classical continuous symmetry we have a (classically) conserved charge! Thisimportant result is known as Noether’s Theorem.

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7.4 Conservation of Energy and Momentum

Under an infinitesimal translation xµ 7→ xµ + aµ, by Taylor’s theorem

Φ(x) 7→ Φ(x+ a) = Φ(x) + aν∂νΦ

andL(x) 7→ L(x) + aµ∂

µL = L+ aν∂µ (δµνL)

so∂Φ

∂aν= ∂νΦ, and as always δ (∂µΦ) = ∂µ (δΦ).

By Noether’s theorem, defining

T µν ≡∂L

∂(∂µΦ)∂νΦ− δµν L the energy-momentum tensor density

we have four conserved currents T µν , i.e. one for each of ν = 0, 1, 2, 3, which satisfy

∂µTµν = 0 .

The conserved charge associated with time translations is the integral over all space of thelocal energy density

H =

∫V

d3x T 00(x, t)︸ ︷︷ ︸

Hamiltoniandensity

(corresponds to generator of time translations)

where

T 00 =

∂L∂Φ

Φ− L

is the usual Legendre transform between the Lagrangian and the Hamiltonian, but for therespective “densities”.

The conserved quantity associated with spatial translations is the momentum carried by thefields

pi =

∫V

d3x T 0i(x, t)︸ ︷︷ ︸

momentumdensity

where T 0i =

∂L∂Φ

∂iΦ .

NB The momentum density T 0i(x, t) is distinct from the canonical momentum π(x, t) intro-

duced in the QFT course.

Local conservation of energy and momentum is a direct consequence of the homogeneity ofspace-time.

Similarly, conservation of angular momentum follows from the fact that L is a Lorentz scalar(isotropy of space-time). [The derivation is more difficult, see tutorial problem (5.4).]

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7.5 Real Scalar Field

The most general Lagrangian for one scalar field φ(x) consistent with fundamental principles(1, 2, 4) is

L = 12∂µφ∂

µφ︸ ︷︷ ︸kineticterm

− V (φ)︸ ︷︷ ︸potential

+1

Λ21

∂2φ ∂2φ +1

Λ22

(∂µφ ∂µφ)2 +

1

Λ23

(∂µφ ∂νφ)2 + · · ·︸ ︷︷ ︸suppress according to principles 3 and 5

.

The coefficient of the kinetic term is a convention. The field has (mass) dimension 1. (If~ = c = 1 then [M ] = [L−1] = [T−1]). The classical field equation is

∂2φ+ V ′(φ) = 0.

For renormalisable theories in 3+1 dimensions (condition 5)

V (φ) = 12m2φ2︸ ︷︷ ︸

“mass”term

+g

3!φ3 +

λ

4!φ4︸ ︷︷ ︸

“interaction”terms

(nonlinear)

.

where g and λ are called coupling constants. (See the QFT course for the technical detailson renormalisability.)

For simplicity we assume further a discrete Z2 symmetry under φ 7→ −φ, so g = 0.

Now the classical field equation is(∂2 +m2

)φ(x) = − λ

3!φ3(x).

When λ = 0 (∂2 +m2

)φ(x) = 0 (Klein-Gordon equation)

This can be solved by Fourier transformation: its solutions are superpositions of plane wavese−ipµx

µ, where pµp

µ = m2, so φ(x) corresponds to scalar particles of mass m.

Conserved currents:

Tµν = ∂µφ ∂νφ− ηµν L = ∂µφ ∂νφ − 12ηµν ∂ρφ ∂

ρφ + ηµν V (φ) the stress-energy tensor

Then, by the field equations,

∂µ Tµν =(∂2φ+ V ′(φ)

)∂νφ = 0

The energy density isE = T00 = 1

2φ2 + 1

2(∇φ)2 + V (φ)

which is positive definite when V (φ) > 0. In the lowest energy state or vacuum, E isminimised, so φ = constant and the vacuum is given by the minimum of V (φ).

In the example above, when (g = 0, m2 > 0, λ > 0) the minimum is at φ0 = 0.

When (g = 0, m2 < 0, λ > 0) we have φ0 6= 0. This is spontaneous symmetry breaking ofthe Z2 symmetry (see later).

35

The momentum density isPi = T0i = φ ∂iφ .

It vanishes in the vacuum. 20

The conserved “angular momenta” are

Mµν =

∫d3x (xµ T0ν − xν T0µ)

Exercise: check this, noting that Tµν = Tνµ.

7.6 Complex Scalar Field

If φ(x) is complex, φ = φ1 + iφ2, with φ1 and φ2 real, then

L = 12∂µφ

∗ ∂µφ− V (φ∗φ) = 12∂µφ1 ∂

µφ1 + 12∂µφ2 ∂

µφ2 − V (φ21 + φ2

2).

We thus have two scalar particles with the same mass, coupled together through interactionterms. For V (φ∗φ) = 1

2m2φ∗φ+ 1

2λ(φ∗φ)2 the field equations are(∂2 +m2

)φ = −λφ (φ∗φ)

etc. Now, in addition to Poincare invariance, we have a further invariance under φ 7→ eiαφ.Using Noether’s theorem we see that this corresponds to a conserved current

jµ(x) = i∂L

∂(∂µφ)φ − i

∂L∂(∂µφ∗)

φ∗ =i

2(φ ∂µφ

∗ − φ∗ ∂µφ) ,

and thus a charge density j0 = i2

(φφ∗ − φ∗φ

).

This is our first example of an internal symmetry, here it’s U(1) ∼= SO(2). Complex fieldsoften carry such a conserved charge; later we will associate it with electric charge.

7.7 Actions for Spinor Fields

For two-component Weyl spinors, we choose to use four-component Dirac spinor notationwith

ψ =

(ψLψR

)but we project onto left- and right-handed spinors using the projection operators PL,R =

12(1± γ5), with γ5 ≡

(1 00 −1

). Then

ψL = PL ψ =

(ψL0

)and ψR = PR ψ =

(0ψR

)This can be a bit confusing because we have the same symbol ψL (say) for 2 and 4-componentspinors on the LHS and RHS of the first equation, but it means we can use 4-componentspinors for both massless (Weyl) and massive (Dirac) cases.

20NB the canonical momentum π = φ is a different quantity.

36

For a single left-handed Weyl fermion we have the real 4-vector ψLγµψL but no real scalar.21

It follows that (in four-component notation) the only action satisfying principles 1–5 is

LL = iψLγµ∂µψL

Similarly for a right-handed Weyl fermion, again in four-component notation,

LR = iψRγµ∂µψR.

The field equations may be obtained by varying with respect to ψL and ψR respectively:

∂µ

(∂L

∂(∂µψL

)− ∂L∂ψL

= 0 , and ∂µ

(∂L

∂(∂µψR

)− ∂L∂ψR

= 0 ,

which giveiγµ∂µψL = 0 iγµ∂µψR = 0 Weyl equations

Exercise: verify that you obtain equivalent field equations when ψL and ψR are varied.

There is no Lorentz-invariant ‘mass’ term (i.e. no derivatives) that can be included in theaction so a single Weyl fermion is automatically massless.

Weyl spinors are complex, and the actions are invariant under ψL 7→ e−iαψL, ψR 7→ e−iαψR.It follows that we have conserved chiral currents

jµL(x) =∂L

∂(∂µψL)

∂ψL∂α

,

and similarly for jµR(x). These give

jµL = ψLγµψL and jµR = ψRγ

µψR

with conserved chiral charges (recalling ψL = ψ†Lγ0 and γ20 = 1)

QL =

∫d3x ψLγ0ψL =

∫d3xψ†LψL, and QR =

∫d3xψ†RψR.

For Dirac fermions, ψψ is a scalar, so the most general action is

LD = ψ (iγµ∂µ −m)ψ = ψLiγµ∂µψL + ψRiγ

µ∂µψR − m(ψRψL + ψLψR

)(exercise)

The field equations are simply

(iγµ∂µ −m)ψ = 0 Dirac equation

For plane wave solutions, since (γµpµ −m) (γµpµ +m) = p2 −m2 = 0, Dirac fermions aremassive.

In the chiral limit, m → 0, ψL and ψR decouple: LD = LL + LR, so we are left with twoindependent Weyl fermions.

21None of ψLψL, ψ†LψL, ψLψL, nor similar expressions involving γ5, are scalars.

37

Invariance under ψ 7→ e−iαψ gives the conserved vector current

jµ = ψ γµ ψ

with conserved charge

Q =

∫d3x ψγ0ψ =

∫d3xψ†ψ = QL+QR Fermion number: lepton or baryon number.

However, underψ 7→ eiβγ5 ψ and ψ 7→ ψ eiβγ5

the mass term in LD is not invariant. Consequently, the axial current jµ5 = ψγ5γµψ is not

conserved, but rather∂µj

µ5 = −2imψγ5ψ.

See tutorial question (5.1).

We say that axial symmetry is softly broken by the mass term; it is restored as m → 0,whereupon the axial charge Q5 ≡ QL − QR is conserved. (ψψ in the mass term mψψ is adimension 3 operator because each ψ has (mass) dimension 3/2.)

Fermions have no renormalisable self-interactions in 3 + 1 dimensions, however they caninteract with scalars: suitable interactions with a Dirac fermion are

gσψψ + ig′πψγ5ψ Yukawa interaction

where σ(x) and π(x) are scalar fields, and g and g′ are coupling constants. If g = g′, chiralsymmetry is preserved provided that the fields transform as (tutorial question (5.5)):

δψ = iβγ5ψ β infinitesimal

δσ = 2βπ

δπ = −2βσ

}SO(2) transformation on

(σπ

): σ2 + π2 is invariant

8 Compact Groups II

In the section Compact Groups I, we proved two theorems (5) & (6) and stated another (7)without proof:

Theorem 5: If G is compact, LG may be decomposed as

LG ∼= L0 ⊕ (L1 ⊕ · · · ⊕ Ln)

where L0 is Abelian (so L0∼= u(1)⊕ · · · ⊕ u(1)) and the Li are simple. So we need consider

only semisimple (or indeed simple) compact groups.

NB, this decomposition does not apply to non-compact groups because it relies on thegenerators being hermitian. A good example is the algebra of the Poincare group: thetranslation generators Pµ form an Abelian invariant subalgebra, but as we saw they don’tcommute with the generators Mµν of the Lorentz group, so the Poincare algebra isn’t adirect sum of translation and Lorentz generators.

38