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

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.

8.1. ElectromagnetismInternal 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.

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.

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 22 1 2 1 2 1 12 j jx x x x x O x

2 21 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 ji j j i

i j i jx A x

→

Aμ= electromagnetic vector potential

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.

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

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.

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.

Spin ½ ParticlesAdvantages 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

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

with1F Ri

A A = Maxwell field tensor

42

14

S d x F F i A me

Action:

F scales with A , i.e., A A F F

λ ~ coupling strength

42

1

14

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

14

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

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

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 = 12

T τ

Any unitary operator that leaves * unchanged can be written as

1, exp2

U i I α α τ θ ~ gauge transformation

α ~ rotation in 3-D isospin space

Proton and neutron states are the isospin up and down states along z-axis.

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

12N Nx x x I i x

α τ

12

a aNI 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.

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)

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.

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

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 ! )

→

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

8.2.5. Intermediate Vector Boson

42

14

S d x Tr F Fg

a b a bTr F F F F Tr T T

a b abTr T T

42

14

a aS d x F Fg

a aA gA

a a a abc b c aF g A A gC A A gF 414

a aS d x F F

414

a 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

8.2.6. Action

42

1

14

na a

j j jj

S d x F F i A mg

4

1

14

nja a

j j jj

S d x F F i gA m

Rescaling by A → gA :

}4}4

2 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

Euler-Lagrange equations for the field degrees of freedom :

4

1

14

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 1 010 12

pp n

n

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)

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

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 12

a 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.

8.3. Non-Abelian Theories and Electromagnetism

12

T 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 .

00 1

ie

,U G α →

1 010 02

I

α τ

Standard representations :

3 1 2

1 2 3

1 11 2 2

1 122 2

iI

i

α τ

→ 1 2 0 12

3

For a general isospin T, 1

2

00

iQ

iQ

eG e

, 1, , 1, 1j T T T T Qj = charge of the j-th isospin multiple.

In a representation where T 3 is diagonal :

12

Y 3 12j jQ T Y Y = hypercharge

Largest charge of the multiplets is 3 12

Q T Y Gell-Mann- Nishijima relations

8.3.a. Gell-Mann- Nishijima Law

The Gell-Mann- Nishijima law 312

Q 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, ,12 2

0,0,01, 1,0 1,1,0

0,0,0

1 11, , 1 0, , 12 2

Particles

Directions of increasing values are Q , ↗ I3→, and Y↑. Y = S for mesons

Y = S + 1 for baryons

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.

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” : 5116

S d x g RG

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

42

1 116 4

S d x g R F FG e

5 552GG

r g

2

53/ 2

55

8Gegr

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.