8. Forces, Connections and Gauge Fields 8.0. Preliminary 8.1. Electromagnetism 8.2. Non-Abelian...
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Transcript of 8. Forces, Connections and Gauge Fields 8.0. Preliminary 8.1. Electromagnetism 8.2. Non-Abelian...
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8. Forces, Connections and Gauge Fields
8.0. Preliminary
8.1. Electromagnetism
8.2. Non-Abelian Gauge Theories
8.3. Non-Abelian Theories and Electromagnetism
8.4. Relevance of Non-Abelian Theories to Physics
8.5. The Theory of Kaluza and Klein
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8.0 Preliminary
General relativity: gravitational forces due to geometry of spacetime.
Logical steps that lead to this conclusion:
1. Physical quantities (tensors) at different points in spacetime are related by an affine connection, which defines parallel transport.
2. Connection coefficients that cannot be set equal to zero everywhere by a suitable coordinate transformation indicate the presence of gra
vitational forces.
3. Such effects can be described by a principle of least action.
Gravitational forces arises from communication between points in spacetime.
Likewise for gauge theories.
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8.1. Electromagnetism
Internal Space
1 2x x i x i xx e ,x ct xComplex wavefunction:
Constant overall phase θ0 is not observable but θ(x) is.
E.g. 3 *p i d x x x
Consider (x) as a vector in the 2-D internal space of the spacetime point x.
→ Fibre bundle with spacetime as base manifold & internal space the typical fibre.
→ (x) is a vector field (cross section) of the bundle.
→ θ(x) gives the orientation of the vector at x.
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θ0 not observable → parallel transport to define parallelism.
Physically significant change is 2 1 2x x x
i i i j jx x x x x x x Γ = connection coefficients
“Flat” space : Directions of (x) can be referred to one global coordinate system.
→ (x1) and (x2) are parallel if 1 2 2x x n n = integer
→ Internal space is the same for all x.
→ Free particle.
“Curved” space : Electromagnetism.
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Connection Coefficients
* 2 21 2 = (measurable) probability amplitude
( x1 → x2 ) is physically equivalent to ( x1 ) → 2 2
1 2 1x x x
2 2 2
1 2 1 1 2 2 1 2x x x x x x
2
1 1 1 1 1 1 12 j jx x x x x
2 2
2 1 2 1 2 1 12 j jx x x x x O x
2 2
1 1 1 1 1 2 1 2 1 12 j j jx x x x x x x O x
→ 1 1 2 20 j j j i j i j
i j j i
i j i jx A x
→
Aμ= electromagnetic vector potential
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Group Manifold
Parallel transport preserves | | → it affects only phase θ.
Typical fibre is unit circle | | = 1 or θ [ 0 , 2π).
Phase transformation : i xx e x
→ e iθ is a symmetry transformation ~ ix e x
ie is a Lie group called U(1)
For θ = const:
with multiplication 1 21 2 ii ie e e
→ The typical fibre θ [ 0 , 2π) is also the (symmetry) group manifold.
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i xx e x Local gauge transformation:
ix e x Global gauge transformation:→ gauge tensors on fibre
= Gauge vector * = Gauge 1-form
Gauge tensor field of rank (nm) : *m n
mn x x x
i n m xmn mnx e x with
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Covariant Derivative
0
lim i ii
x
x x x x xD x
x
i i j jx x x i i j jx A x x
1 1 2D x x A x x 2 2 1D x x A x x
1 2D x D x i x x i A x x
→
Under gauge transformation
i xx x e x * * *i xx x e x
D i A ii A e i j i jx A x where
Note: D does not change the rank of gauge tensors.
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Dμ is a gauge vector : iD e D ie i A
ii A e
→ i i ie i A e i A e
i iiA A e e
A
In general, mn mn mnD i n m A
Same as EM gauge transformation
→ A μ(x) is called a gauge field.
Summary:
Phases of a complex wavefunction constitue a U(1) fibre bundle, whose geometry is determined by the gauge fields.
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Spin ½ Particles
Advantages of geometric point of view of interactions:• Easy generalization.• Provides classification of tensors.
4S d x i m 4S d x i A m
E.g., To include the effects of gauge fields, set D i A
→
i 0A m i 0m →
λ = charge
Minimal coupling : D i A promotes global to local gauge symmetry
In the absence of EM fields, there is a gauge such that 0i j x everywhere.
i j i jx A x i j A x x = 0 → A x x
Check: F A A 0
Indeed: A 0A A A
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Field Equations
,D D x R x x
. . .L H S i A i A i A i A
i A A A A i A A
R i A A → is gauge invariant
Simplest scalar under both Lorentz & gauge transformations is
R RL
with1
F Ri
A A = Maxwell field tensor
42
1
4S d x F F i A m
e
Action:
F scales with A , i.e., A A F F
λ ~ coupling strength
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42
1
1
4
n
j j j jj
S d x F F i A me
For system with n types of spin ½ particles :
Rescale: A eA F eF
4
1
1
4
n
j j j jj
S d x F F i e A m
Euler-Lagrange equations for A are just the Maxwell equations with
i ij x e x x (Prove it!)
e = elementary charge unit.
No restriction of λ derived → charge quantization not explained.
Remedy: grand unified theory
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8.2. Non-Abelian Gauge Theories
8.2.1. Isospin
8.2.2. Isospin Connection
8.2.3. Field Tensor
8.2.4. Gauge Transformation
8.2.5. Intermediate Vector Boson
8.2.6. Action
8.2.7. Conserved Currents
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8.2.1. IsospinProtons and neutrons are interchangeable w.r.t. strong interaction.
Conjecture: They are just different states of the nucleon.
pN
n
xx
x
Nucleon wavefunction :
Proton state: 0
pN
xx
Neutron state:
0N
n
xx
isotopic spin (isospin) state.
Complete set of independent operators in the isospin space: I, τ
Isospin operator = 1
2T τ
Any unitary operator that leaves * unchanged can be written as
1, exp
2U i I
α α τθ ~ gauge transformation
α ~ rotation in 3-D isospin space
Proton and neutron states are the isospin up and down states along z-axis.
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8.2.2. Isospin Connection
Fibre bundle with spacetime as base manifold & isospin space as typical fibre.
Reminder: Directions in isospin space have observable physical meanings.
Only meaningful change in isospin space is a rotation.
Parallel transport : i i i j jx x x x x x x
1
2N Nx x x I i x
α τ
1
2a a
NI i x
a aA x 1
2a a
i j i jx i A x
i, j = p,n
1st order in α:
→
There is no scale factor because the field tensor does not scale with the gauge fields.
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Typical fibre can be generated by rotations 2U α α α → SU(2)
Gauge covariant derivative :
i i i j jD x x x x , , 1, , 1,i j T T T T
a ai ji j
x i A x T x 1,2,3a
i ji jx i A x x a aA x A x T
D x x i A x x D i A x
Gauge transformation: x x U x α exp i x x α T
D i A i A U U U i A U →
D is a gauge scalar → D U D U iUA
iUA U i A U 1 1A UA U i U U →
EM case: U = e i θ(x)
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8.2.3. Field Tensor
,D D x R x x
. . .L H S iA iA iA iA
i A A A A A A A A
i A A A A A A
,R i A A A A
F iR ,A A i A A
Note: F is nonlinear in A. → F is not gauge invariant & doesn’t scale with A.
→ Different states of the same isospin must have the same isospin connection.
Only particles of different isospins can have different connections.
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Exact form of F depends on the representation of the gauge group used.
Generators of the gauge (Lie) group are T.
Corresponding Lie algebra is defined by ,a b abc cT T iC T abc ci T
Cabc = structure constants for SU(2) = εabc
,a a a a a b a bF A T A T i A A T T
a a a a a b abc cA T A T A A C T
a a b c bca aA A A A C T
a aF T
a a a abc b cF A A C A A
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8.2.4. Gauge Transformation
By definition, a gauge transformation is a rotation on given by
expU i α α T ( is a gauge vector )
Ta is a generator of the transformation → it is a gauge tensor of rank 2 :
1a aT UT U 1a a aU T UT U U T
1 1A UA U i U U
F F ,A A i A A 1F UF U
1a a a aF T UF T U 1 1a aUF U UT U
1a aUF U T
1a aF UF U
→ A is not a gauge tensor.
= gauge tensor of rank 2 ( proof ! )
→
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Alternatively, { Ta } is a basis for vector operators on the isospin space.
A gauge transformation is then a rotation operator defined by
a b aa a bT T T T αU U
1a aT U T U α αb a (α) is determined by comparison with
a aF F T expresses the vector F w.r.t. basis { Ta }
Gauge transformation: a aF F F F T U U a b baF T αU a aF T
→ aba bF F αU F F
αUor
There is an isomorphism between U and .
expU i α α T ~ exp i α αU T
The SU(2) representation formed by a is the adjoint representation,
aa bc
bciCTso called because
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8.2.5. Intermediate Vector Boson
42
1
4S d x Tr F F
g
a b a bTr F F F F Tr T T
a b abTr T T
42
1
4a aS d x F F
g
a aA gA
a a a abc b c aF g A A gC A A gF 41
4a aS d x F F
41
4a aS d x F F
a a a abc b cF A A gC A A
Task: Construct a gauge invariant action for the gauge fields.
where
To ensure that Tr( Fμν Fμν) is a gauge scalar, set
→
It is straightforward to show that the Pauli matrices satifsy a b abTr T T
Scaling:
Dropping ~ :
Quantized gauge fields → intermediate vector bosons (mediate weak interaction) S contains terms like g(A)AA & g2AAAA → IVBs are charged
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8.2.6. Action
42
1
1
4
na a
j j jj
S d x F F i A mg
4
1
1
4
nja a
j j jj
S d x F F i gA m
Rescaling by A → gA :
}4
}42 1 4
}4
jj T
jA A j aaA T
, 1, , 1,j j j ja T T T T
Each j is a 2T(j)+1 multiplet of 4-component Dirac spinors :
a a a abc b cF A A C A A where
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Euler-Lagrange equations for the field degrees of freedom :
4
1
1
4
nja a
j j jj
S d x F F i gA m
D F J
a abc b c aF gC A F J
1
n
j jj
J g
1
nj aa
j jj
J g T
3 1 01
0 12p
p nn
j g
1
2 p p n ng
3
,p n
T probability current deng sity
0jj ji gA m
or
whereor
For the nucleon doublet :
Euler-Lagrange eqautions for the spinor degrees of freedom:
(Dirac equations)
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8.2.7. Conserved Currents
Classical EM: gauge invariance → conservation of charges (μj μ = 0 ).
Gauge fields: conservation law is Dμj μ = 0 ( j is covariantly conserved).
Note: Dμj μ = 0 does not imply conservation of any physical scalar quantity.
Dirac particle: → conservation of charges.j e 0D j j
For the non-abelian SU(2) gauge group: 0a abc b cD J J gC A J
For the non-Abelian Maxwell equations
0a abc b cJ gC A F
a abc b c aF gC A F J
→0aF
a a abc b cJ J gC A F is the Noether current associated w
ith the non-Abelian symmetry.
= Fermion + vector bosons flows
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a a a abc b cF A A C A A Components of
can be thought of as ‘electric’ and ‘magnetic’ fields Ea and Ba.
i.e. 0 0ai ai a iE F F 1
2a i i j k a j kB F
ai abc b cii iB gC A B → ‘magnetic monopoles’ are allowed
Comment:
Bai here are not the usual magnetic fields.
However, the unified electroweak theories is a non-abelian gauge theory.
In that case, genuine magnetic monopoles are allowed.
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8.3. Non-Abelian Theories and Electromagnetism
1
2T τ , expU i I α α T
, 0i ie I e α T
Consider with
, 2 1U SU U α→
~ unification of EM & non-Abelian gauge fields (weak interaction)
Technical detail: The U(1) members should be EM gauge transformations so they can’t be eiθI .
0
0 1
ie
,U G α →
1 01
0 02I
α τ
Standard representations :
3 1 2
1 2 3
1 11 2 2
1 12
2 2
iI
i
α τ
→ 1 2 0 1
2 3
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For a general isospin T, 1
2
0
0
iQ
iQ
e
G e
, 1, , 1, 1j T T T T Qj = charge of the j-th isospin multiple.
In a representation where T 3 is diagonal :
1
2Y 3 1
2j jQ T Y Y = hypercharge
Largest charge of the multiplets is 3 1
2Q T Y Gell-Mann- Nishijima
relations
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8.3.a. Gell-Mann- Nishijima Law
The Gell-Mann- Nishijima law 3
1
2Q I Y
was proposed in 1953 to explain the “8-fold way” grouping of “stable” hadrons. “Stable” means no decay if electroweak interactions were absent.
0
0
0
0
n p
( Q, I, Y ) values
1 10, ,1 1, ,1
2 2
0,0,01, 1,0 1,1,0
0,0,0
1 11, , 1 0, , 1
2 2
Particles
Directions of increasing values are Q , ↗ I3→, and Y↑. Y = S for mesons
Y = S + 1 for baryons
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8.4. Relevance of Non-Abelian Theories to Physics
Pure geometrical consideration of the complex wavefunction
→ Abelian gauge fields
→ existence of electromagnetic forces
Application to isospin
→ non-abelian gauge fields (Yang-Mills theories)
→ nuclear weak interaction
Modern version:
Fundamental particles are quarks, leptons and quanta of fundamental interactions.
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8.5. The Theory of Kaluza and Klein
Classical (non-quantum mechanical) theory of Kaluza and Klein unifies gravity and electromagnetism by means of a 5-D spacetime.
5-D spacetime metric tensor ABg A, B 0, 1, 2, 3, 5
with 5 5 55g g g A 55g g g A A 0,1,2,3
g = metric tensor of the Einstein’s 4-D spacetime.
Action for “gravity” : 51
16S d x g R
G
Assumptions:
1. The 5th dimension is space-like, i.e.,
2. gμν and Aμ are independent of x5 and
55 0g → 0g
3. The 5th dimension rolls into a circle of radius r5
55 constg
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42
1 1
16 4S d x g R F F
G e
5 552
GG
r g
2
5
3/ 2
55
8Ge
gr
with
(a miracle!)
Objections:
• There is no physical justification to the required assumptions.
• The theory offers no new observable effects.
Update:
Supergravity and superstring theories also make use of spacetimes of more than 4 dimensions.