Noether’s Theorem (Quantum Part)

In section 3.3 we have seen the important connection between symmetries and conservation laws, known as Noether’s theorem, for the classical field theory.

Now we can describe the quantum field theory also by a variational principle, but with the quantum effective action Γ[ϕ] introduced in the last section, rather than with the classical action S[ϕ].

We start from the invariance of the classical action and ask for the consequences for the effective action of the corresponding quantised theory. For this we suppose, that the path integral measure of the fields Dφ is invariant under the symmetry transformation and that it can be written in the Lagrangian form.

Now we make the substitution φ = φ⁰ −δφ in the generating functional (4.152). Since S[φ] =S[φ⁰] and Dφ= Dφ⁰ by assumption we have

Z[J] =N Z

Dφ⁰exp{iS[φ⁰] + i J₁φ⁰₁

1}exp{iδhJ₁φ₁i}. (4.224) Now J is an auxiliary external current, which we can give arbitrary behaviour under the symmetry transformation. We shall assume that it transforms as a set of scalar fields under Poincar´e transformations and as a scalar under intrinsic transformations of the fieldsφ.

Now we subtract the original form (4.152) of the functional and expand the expression to linear order in the infinitesimal transformation:

Z

DφδhJ₁φ₁i1exp[iS[φ] + ihJ₂φ₂i] = 0. (4.225) Now we have definedJ to be a set of scalar fields, leading to

δhJ₁φ₁i₁ = Z

d⁴xJ(x)[δφ(x)−δx^µ∂_µφ], (4.226) as it was shown when proving Noether’s theorem for classical fields in section 3.3.

2Since in the perturbative diagrams a line stands for iG0 any diagram with this properties contributes to

−iΣ.

4.6 · Generating Functionals

To understand the meaning of (4.225) better, we write the most general infinitesimal trans-formation we want to take into account in the following form (also used in section 3.3):

δφ(x) =τ_a(x, φ)δη^a, δx^µ=−T_a^µ(x)δη^a, (4.227) where theδη^a are the infinitesimal independent parameters of the Lie group.

According to the general Feynman-Kac formula (4.151) and using the effective action formal-ism we obtain from (4.216)

Z

d⁴xδΓ[ϕJ]

δϕ_J(x)δη^ah0|T_c[τ_a(x,φ)−T_a^µ∂_µφ]|0i= 0, (4.228) which is exactly of the same form, we deduced for the classical action to prove Noether’s theorem in the classical case,if τ is linear in φ, because then we have hτ(φ, x)i=τ(hφi, x) andhφi=ϕ_J, whereh· · ·iis the vacuum expectation value for the operators under influence of the external currentJ.

Together we have shown, that, if the classical action functional is invariant under a infinites-imal transformation, which is linearly realized on the fields and leaves the path integral measure invariant, then the effective action of the corresponding quantum field theory also is invariant under the same transformation. We can immediately conclude, that then Noether’s theorem holds for the quantum case, i.e., there exists a conserved Noether current for each independent symmetry transformation, which operates linearly on the fields.

Warning! All this is purely formal, because we have not taken into account that one has to regularise and renormalise the field theory. All this formal manipulations with path integrals have only sense if we are able to regularise the theory in some way. This regularisation has to be done such that the symmetries hold for the regularised theory too. One nice regularisation technique is the dimensional regularisation invented by Veltman and van ’t Hooft especially for gauge theories (explained in chapter 6 of this notes). Now it may happen that there is a transformation which leaves the classical action invariant in the physical 1 + 3-dimensional space-time, but that there is no symmetry in other space-time dimensions. Then the regularised theory has no symmetry and thus it happens that the effective action fails also to be symmetric under the symmetry transformation, although the formal assumptions may be fulfilled.

It can also be that the path integral measure is not invariant under the symmetry trans-formation, which again breaks the symmetry for the effective action. All these cases, when a symmetry of the classical action is not a symmetry of the effective action, are called an anomaly. Thus we should keep in mind that we have to check carefully if the theory is free of anomalies before we use any symmetry argument in the quantum case.

In the next section we show in which sense Noether’s theorem holds for the Dyson-Wick expansion of the quantum field theory, if all the conditions for the transformation to be a symmetry of the exact effective action is fulfilled.

4.6.6 ~-Expansion

In this section we reintroduce ~ explicitly in the path integral formulation of quantum field theory. We expect that it should be possible to expand around the classical solutions for

the fields which we denote by ϕ₀. In order to discuss the generating functionals for discon-nected and condiscon-nected Green’s functions and the effective action which generates proper vertex functions we include an external source J:

δS

The generating functional for disconnected Green’s functions is defined according to 4.187.

As explained above we reintroduce ~explicitely:

Z[J] =N motion (4.229). The classical field is of course of order O(~⁰) since it is completely classical while the quantum part should be scaled with √

~ according to its dimension. With this substitutions we find

This shows that we can evaluate an~-expansion forZ1if we apply the perturbative expansion given forZ itself by (4.188). There are only two differences: The first is that the propagator is given by D₁₂ which depends on the classical field ϕ₀ and thus implicitly on the external sourceJ. The second is that we are not allowed to cancel the additional factor for Z10[J, K]

(where K denotes a new independent external source), because it depends on ϕ₀ and thus also onJ. Here we have abbreviated the interaction parts of the Lagrangian in (4.232) by

V˜⁽³⁾(ϕ) = λ

6ϕ₀φ⁰³, V˜⁽⁴⁾(ϕ) = λ

24φ⁰⁴, (4.234)

while Z₁⁽⁰⁾(J, K) is given by the quadratic part of the Lagrangian in (4.233). We apply to it the same trick as for obtaining (4.186) but here we have to keep the overall factor, because it depends inJ via its dependence onϕ₀:

4.6 · Generating Functionals

To calculate the path integral factor we think of the path integral in terms of its discretised form. Then after a Wick rotation of the integrand we can use (1.73) to obtain immediately

Z

where we have introduced other indefinite factorsN⁰ andN⁰⁰ which are independent of both J andK. In the second term we have used the free propagator, defined by

D⁻¹ = (−21−m²)δ⁽⁴⁾(x₁−x₂), (4.237) to normalise the whole expression and to avoid dimensionful quantities within logarithms. As we shall see below in the Tr ln-form we can calculate the functional determinant with help of a perturbation series at any given order of ~.

Now it is our goal to calculate the effective action, defined by (4.212), up to order ~². Ac-cording to (4.230) for that purpose we have to expand the functional Z₁ (4.232) up to order

~. Thus we expand the functional differential operator (4.233) up to order~ leading to Z₁⁽²⁾[J] =N

Straight-forward calculation leads after some algebra³ to the result W[J] =~lnZ[J] =S[ϕ₀] +{J₁ϕ₀₁}₁+ i

The next task is to invert the definition

ϕ₁ = δW[J]

δJ1

(4.240) and substitute ϕ instead ofJ in order to perform the Legendre transformation according to (4.212). This is of course impossible to obtain exactly. But it is not too difficult to provide the systematic ~-expansion. Here it is important to note that by definition (4.229) ϕ₀ is the stationary point of the functional S[ϕ] +{J₁ϕ₁}1. Thus we have

S[ϕ] +{J₁ϕ₁}₁ =S[ϕ₀] +{J₁ϕ₀₁}₁+1 2

D₁₂ϕ⁰₁ϕ⁰_{2 12}+O(~³) withϕ⁰ =ϕ−ϕ₀. (4.241) Here we have used the fact that ϕ⁰ = O(~) which we have to prove now. To that end we derive the expansion (4.239) with respect to J. Since (4.239) depends only implicitly on J we do first the derivative with respect toϕ0. Using (4.229) we find

δW

δϕ₀₁ =−iλ

2~ϕ₀₁D₁₁+O(~²). (4.242)

3A simple computer algebra system like FORMis recommended for such cases!

Again using (4.239) we get

Combining (4.242) and (4.243) leads to ϕ₁ = δW Now we are ready to expand (4.239) systematically up to orderO(~²).

Making use of (4.239), (4.244) and (4.245) we find S[ϕ₀] +{J₁ϕ₀₁}1 =S[ϕ] +{J₁ϕ₁}1+λ²

In the remaining terms of (4.239) which are explicitely of order O(~²) we can simply set ϕ instead ofϕ₀ which gives only differences of orderO(~³).

The final result for the action up to order~² is thus Γ[ϕ] =S[ϕ] + i For sake of completeness we derive the rule how to get derivatives ofD(ϕ) with respect toϕ.

These are needed if we like to calculate the proper vertex functions using (4.214). We have δ₁₂=

Using the explicit form of D⁻¹ given by (4.245) we find δD₁₂(ϕ)

δϕ₃ =λD₁₃(ϕ)ϕ₃D₃₂(ϕ). (4.250) Now we shall give a diagrammatical interpretation in terms of the propagator D. To obtain an interpretation in terms of the perturbative propagator D = D(ϕ = 0) we have to use