Presentation gauge field theory

Gauge Field Theory

August 20, 2013

Brief Outline

1. Gauge invariance in classical electrodynamics

2. Local gauge invariance in quantum mechanics

3. Yang-Mills theory

Gauge Invariance and Classical Electrodynamics

• In classical electrodynamics, the electric and magnetic fields canbe written in terms of the scalar and vector potentials

~B = ∇× ~A ~E = −∇φ− ∂~A∂t

• However, these potentials are not unique for a given physicalfield. There is a certain freedom in choosing the potentials.

• The potentials can be transformed as

Aµ(x)→ A′µ(x) = Aµ(x) + ∂µΛ(x)

without affecting the physical electric and magnetic fields.

Local Gauge Invariance in Quantum Mechanics

Charged Particle in the Electromagnetic Field

• Hamiltonian of a charged particle moving in the presence of theelectromagnetic field is given by

H =1

2m(~p− q~A)2 + qφ

• Quantum mechanically, the charged particle is described by theSchrodinger equation,

− 12m

(∇− iq~A

)2ψ(~x, t) = i

( ∂∂t

+ iqφ)ψ(~x, t)


Gauge Invariance and Quantum Mechanics

• Classically, the potentials φ and ~A are not unique for a givenphysical electromagnetic field.

• We can transform the potentials locally without affecting thephysical fields (and hence the behaviour of the charged particlemoving in the field).

• We want to investigate whether an analogous situation exists inquantum mechanics (i.e. whether quantum mechanics respectsthe gauge invariance property of electromagnetic fields)


• The gauge transformation of the potentials does not leave theSchrodinger equation invariant.

• However, it is possible to restore the form invariance of theSchrodinger equation, provided the transformation of thepotentials

Aµ → A′µ = Aµ + ∂µΛ(x)

is accompanied by a transformation of the wave function

ψ → ψ′ = e−iqΛ(x)ψ

• With these two transformations together, the form invariance ofthe Schrodinger equation is assured (i.e. A′µ and ψ′ satisfy thesame equation as Aµ and ψ.)


Summary

Quantum mechanics respects the gauge invariance property of theelectromagnetic field. It gives the freedom to change the

electromagnetic potentials but at the cost of a simultaneous change inthe phase of the wave function.


Reversing the Argument(Demanding Local Gauge Invariance)

• Instead of starting with the charged particle Schrodingerequation, we start with the free particle Schrodinger equation

− 12m∇2ψ(~x, t) = i

∂

∂tψ(~x, t)

• We demand that this equation remains invariant under the localphase transformation of the wave function

ψ(x)→ ψ′(x) = e−iqΛ(x)ψ(x)

• However, the new wave function ψ′(x) does not satisfy the freeparticle Schrodinger equation.


• We conclude that the local gauge invariance is not possible withthe free particle Schrodinger equation.

• However, the demand of local gauge invariance can be satisfiedby modifying the free particle Schrodinger equation.

• It turns out that by modifying the derivative operators in the freeparticle Schrodinger equation as

∂µ → Dµ = ∂µ + iqAµ

we can achieve the required goal, provided the vector field Aµalso transforms under the phase transformation of the wavefunction ψ.


Summary

• Local gauge freedom in the wave function in quantum mechanicsis not possible with the free particle Schrodinger equation.

• The insistence on the local gauge freedom forces us to introducein the equation a new field which interacts with the particle.

Yang-Mills Theory

• We now turn to extend the concept of local gauge invariance tofield theories.

• In field theory, the quantity of fundamental interest is theLagrangian density of the fields and accordingly, we demand thelocal gauge invariance of the Lagrangian density.

Yang-Mills Theory

Lagrangian Density

• We consider a Lagrangian density which depends upon the scalarfield φ and its first derivative ∂µφ

L ≡ L(φ(x), ∂µφ(x))

• We also assume that the Lagrangian density is constructed out ofthe inner products (φ, φ) and (∂µφ(x), ∂µφ(x)) (where bracketdenotes the inner product in field space), e.g.

L = (∂µφ)†(∂µφ)− m2φ†φ− λ(φ†φ)2

where the field φ, in general, is a multi component field.

Yang-Mills Theory(Infinitesimal group theory)

• We are mainly interested in the compact Lie groups such asSU(N) and SO(N).

• One basic property of the compact groups is that their finitedimensional representations are equivalent to the unitaryrepresentation.

• The advantage of unitary transformations is that they preservethe inner products

φ†φ→ (Uφ)†(Uφ) = φ†(U†U)φ = φ†φ

Yang-Mills Theory(Infinitesimal group theory)

• Associated with each Lie group is a Lie algebra. The elements ωof the group can be written as

T(ω) = eiλaT(ta)

where ta are the generators of the group and T is somerepresentation.

• One important representation is the adjoint representation forwhich the Lie algebra space coincides with the vector space onwhich the group elements act. The action is given by

Ad(ω)A = ωAω−1

where, A is an element of the Lie algebra space.

Yang-Mills Theory(Global symmetry transformation)

• We now assume that the Lagrangian density remains invariantunder a global symmetry transformation

φ(x)→ φ′(x) = T(ω)φ(x)

where ω is an element of the symmetry group and T(ω) is someunitary representation under which the fields φ transform.

• For example, the field φ may be a two component objecttransforming under the fundamental representation of the SU(2)group, i.e.

φ(x) ≡(φ1(x)φ2(x)

)→(φ′1(x)φ′2(x)

)= eiΛaσa/2

(φ1(x)φ2(x)

)

Yang-Mills Theory(Local symmetry transformation)

• We now generalize the global transformation to a localtransformation

φ(x)→ φ′(x) = T(ω(x))φ(x)

• Under a local transformation, the inner product φ†φ remainsinvariant. However, the inner product involving the derivative ofthe fields (∂µφ)†(∂µφ) does not remain invariant, since

∂µφ(x)→ ∂µφ′(x) = T(ω(x))∂µφ(x) + ∂µT(ω(x))φ(x)

(The second term in the right hand side prevents the invarianceof the inner product involving the derivatives)

Yang-Mills Theory(Introducing gauge fields)

• To ensure the invariance of the Lagrangian density, the sameprocedure, as in the case of quantum mechanics, is followed.

• We replace the ordinary derivative by a covariant derivative

∂µφ(x)→ Dµφ(x) = (∂µ − igT(Aµ))φ(x)

introducing a field Aµ known as the gauge field.

• The field Aµ is constructed in such a way that the covariantderivative transforms exactly as the field φ, namely

Dµφ(x)→ (Dµφ(x))′ = T(ω(x))Dµφ(x)

Yang-Mills Theory(Gauge fields belong to the lie algebra)

• The last demand leads to the following transformation propertyfor the gauge fields Aµ

T(A′µ) = T(ωAµω−1) +ig

T(ω∂µω−1)

• Both the terms in the right hand side belong to the lie algebra ofthe corresponding symmetry group.

• The first term is a result of the action of adjoint representation.For the second term, we look at the group elements near identity

ω(x) = 1 + iλa(x)ta + o(λ2)

This givesω∂µω

−1 = −i(∂µλa)ta

Yang-Mills Theory(Gauge fields belong to the Lie algebra)

• Since gauge fields Aµ belong to the Lie algebra space, it followsthat we can write them as a linear combination of the generatorsta

Aµ = Aµa ta

• From this, it also follows that the number of independent gaugefields is equal to the number of generators of the group. Thus,e.g., if the symmetry group is SU(N), the number of gauge fieldswill be (N2 − 1).

• Thus, the number of gauge fields depends only upon theunderlying symmetry group and is independent of the number ofmatter fields present in the system (of course, the number ofmatter fields should match with the dimension of somerepresentation of the symmetry group)

Yang-Mills Theory(Lagrangian density for the gauge fields)

• Since we have introduced the gauge fields Aµ in our system, weneed to have a term in the Lagrangian density which describestheir dynamical behavior.

• Moreover, this term should also be gauge invariant to preservethe gauge invariance of the Lagrangian density.

• We recall that the electromagnetic Lagrangian density is given by

L = −14

FµνFµν

where, Fµν = ∂µAν − ∂νAµ is the field strength tensor.


• To construct the field strength tensor for the gauge fields, we takeguidance from the following theorem (Rubakov, chapter 3)

“A Lie algebra is compact if and only if it has a(positive-definite) scalar product, which is invariant under theaction of the adjoint representation of the group ”

• Since our aim is also to have a gauge invariant term, we demandthat the field strength tensor for the gauge fields should alsotransform according to the adjoint representation, i.e.

Fµν → F′µν = Ad(ω)Fµν = ωFµνω−1

and we construct the gauge invariant Lagrangian density usingthis field tensor.


• This demand leads to the field strength tensor

Fµν = ∂µAν − ∂νAµ − ig[Aµ,Aν ]

• Since Fµν belongs to the Lie algebra, we can write it as a linearcombination of the generators

Fµν = Fµνa ta

• In terms of the components Aµa and Fµνa , we have

Fµνa = ∂µAνa − ∂νAµa + gfabcAµb Aνc

• This differs from the electromagnetic case by the presence of anon linear term.


• The Lagrangian density for the gauge fields is postulated to bethe inner product

Lgaugefield = −12

Tr(FµνFµν) = −14

Fµνa Faµν

• Since Fµν transforms as the adjoint representation, this innerproduct is invariant (basically due to cyclic property of trace).

Yang-Mills Theory(Full Lagrangian density)

• For the example given earlier, the complete Lagrangian densitythus becomes

L = (Dµφ)†(Dµφ)− m2(φ†φ)− λ(φ†φ)2 − 14

Fµνa Faµν

where,Dµφ = (∂µ − igT(Aµ))φ

Yang-Mills Theory(Energy-momentum tensor)

• The energy momentum tensor can be obtained using thedefinition

δS = −12

∫d4x√−g Tµν δgµν

• This gives

Tµν =14ηµνFλρa Faλρ − Fµλa Fνa λ + 2(Dµφ)†Dνφ− ηµνLφ

• Energy is given by integrating the (00)th component of thistensor over the spatial volume and is positive definite.

E =

∫d3x

((D0φ)†D0φ+ (Diφ)†Diφ+ m2(φ†φ) + λ(φ†φ)2

+12

F0ia F0i

a +14

Fija Fij

a

)

Yang-Mills Theory

Summary

The interaction between the scalar fields and the gauge fields can beobtained by invoking the local gauge invariance principle. This

principle also dictates the kind of terms which can be present in theLagrangian.

Why Yang-Mills Theory

• Every physical phenomenon is believed to be governed by fourinteractions. Two of these, namely, Gravity andElectromagnetism are felt in day to day life.

• Due to this, it is possible to formulate a classical version of theseinteractions.

• The formulation of the universal Gravitational force law by IsaacNewton from the observation of the motion of an apple and themoon is an excellent example of this.

• Similarly, the laws of Electrodynamics were discovered byobserving the behavior of magnets, current carrying wires and soon. James Clark Maxwell gave the exact mathematical form ofthese laws using these observation (and his excellent insight).

Discovering Gravity

Newton under the Apple tree

Discovering Electromagnetism

Why Yang-Mills Theory

• The quantum version of the Electrodynamics (QuantumElectrodynamics) was constructed with the help of its knownclassical version.

• However, there is no guidance in the form of classical laws forthe strong and weak interactions. We have to directly deal withthe quantum version of these interactions.

• The Gauge invariance principle comes to rescue. Themathematical form of the strong and weak interactions has beenconstructed by using this principle.

References

1. Valery Rubakov, Classical Theory of Gauge Field, PrincetonUniversity Press, Princeton, New Jersey (2002)

2. Aitchison and Hey, Gauge Theories in Particle Physics: Volume 1,3rd Ed., IOP (2004)

Thank You

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Presentation gauge field theory