Commutators, Currents, and Ward Identities

The previous lesson explained why real-time correlators are boundary values of analytic functions. This page draws the first major consequence: a commutator is a discontinuity. In Euclidean signature a two-point function may be a smooth power law, but after analytic continuation its light cone becomes a branch cut. The difference between the two sides is the commutator expectation; multiplying it by the appropriate time-ordering step function gives the retarded or advanced response.

The second theme is local symmetry. A conserved current is not merely an operator whose divergence happens to vanish. It is the operator that records how the action changes when a global symmetry is promoted to a slowly varying local transformation. In a correlation function this produces contact terms at the charged insertions. These contact terms are the Ward identities.

The bridge between the two themes is locality. Commutators tell us where an operator can causally respond. Ward identities tell us how a local symmetry variation is distributed among operator insertions. Both are statements about singularities of correlation functions.

Commutators as discontinuities

Dimensions, boundary values, and charge generators

We write $D$ for spacetime dimension and $d=D-1$ for the number of spatial dimensions. Thus equal-time canonical relations contain $\delta^{(d)}(\mathbf x-\mathbf y)$, while Euclidean Ward identities contain $d^D x$ and $\delta^{(D)}(x-x_k)$.

For Lorentzian formulas we write

$$\rho=\mathbf x^2-t^2.$$

Thus spacelike separation has $\rho>0$, while timelike separation has $\rho<0$. Wightman boundary values are written as

$$W_+(t,\mathbf x)=\langle0|O(t,\mathbf x)O(0)|0\rangle, \qquad W_-(t,\mathbf x)=\langle0|O(0)O(t,\mathbf x)|0\rangle.$$

For Ward identities we mostly use Euclidean path-integral notation,

$$Z=\int D\phi\,e^{-S[\phi]}.$$

An infinitesimal internal symmetry acts as

$$\delta \Phi_k=\alpha(x)R_k\Phi_k.$$

For a $U(1)$ field of charge $e_k$, $R_k=i e_k$ in the convention $\Phi_k\mapsto e^{i e_k\alpha}\Phi_k$. Different authors absorb the factor of $i$ into the generator; the contact-term structure is independent of this bookkeeping.

Consider a scalar primary-like operator whose Euclidean two-point function has the short-distance form

$$\langle O(x_E)O(0)\rangle={C_O\over (x_E^2)^\Delta}.$$

After analytic continuation to Lorentzian time, the two Wightman orderings are boundary values on opposite sides of the light-cone singularity:

$$W_+(t,\mathbf x) ={C_O\over \left(\mathbf x^2-(t-i0)^2\right)^\Delta} ={C_O\over \left(\rho+i0\,\operatorname{sgn}t\right)^\Delta},$$

and

$$W_-(t,\mathbf x) ={C_O\over \left(\mathbf x^2-(t+i0)^2\right)^\Delta} ={C_O\over \left(\rho-i0\,\operatorname{sgn}t\right)^\Delta}.$$

Therefore

$$\langle0|[O(t,\mathbf x),O(0)]|0\rangle =W_+(t,\mathbf x)-W_-(t,\mathbf x).$$

For spacelike separation, $\rho>0$, the two boundary values are the same. The commutator vanishes:

$$\boxed{ \langle0|[O(t,\mathbf x),O(0)]|0\rangle=0 \qquad (\mathbf x^2>t^2). }$$

This is the two-point, vacuum-expectation version of causality. Microcausality is the stronger operator statement $[O(x),O(0)]=0$ at spacelike separation (or the graded version for fermionic fields).

The Wightman functions $W_+$ and $W_-$ are boundary values of the same Euclidean power law. Their difference is the vacuum expectation of the commutator and is a discontinuity across the timelike branch cut. In the spacelike region this expectation vanishes; microcausality is the corresponding operator identity.

For non-integer $\Delta$, the timelike discontinuity is easy to compute. When $\rho<0$, write $\rho=-|\rho|$. On the principal branch,

$$(\rho+i0)^{-\Delta}=|\rho|^{-\Delta}e^{-i\pi\Delta}, \qquad (\rho-i0)^{-\Delta}=|\rho|^{-\Delta}e^{+i\pi\Delta}.$$

Taking into account the factor $\operatorname{sgn}t$ in the boundary prescription gives

$$\boxed{ W_+(t,\mathbf x)-W_-(t,\mathbf x) =-2iC_O\sin(\pi\Delta)\,\operatorname{sgn}t\, {\theta(t^2-\mathbf x^2)\over (t^2-\mathbf x^2)^\Delta} }$$

for non-integer $\Delta$, away from the light cone.

The sign in front depends on which commutator and boundary-value convention one calls $W_+-W_-$. The important invariant statement is the support:

$$\operatorname{supp}\langle[O(x),O(0)]\rangle \subseteq \{x: t^2\ge \mathbf x^2\}.$$

For integer $\Delta$, the factor $\sin(\pi\Delta)$ seems to vanish. This does not mean that the commutator vanishes. Instead the discontinuity collapses to derivatives of delta functions on the light cone. For example,

$${1\over \rho+i0}-{1\over \rho-i0}=-2\pi i\,\delta(\rho).$$

This is why the free massless scalar in four dimensions has a commutator supported on the light cone rather than throughout the timelike interior.

Equal-time algebra and the ultraviolet cutoff

A continuum field theory is an infrared description of local degrees of freedom. If it came from a lattice with spacing $a$, then the ultraviolet cutoff is of order

$$\Lambda_{\mathrm{UV}}\sim {1\over a}.$$

Power-law correlators are trustworthy for separations much larger than $a$. Equal-time commutators, however, are deliberately local. Their singular support lies at coincident spatial points:

$$[\phi(t,\mathbf x),\phi(t,\mathbf y)]=0,$$

and

$$[\phi(t,\mathbf x),\pi(t,\mathbf y)]=i\delta^{(d)}(\mathbf x-\mathbf y), \qquad \pi=\dot\phi.$$

The delta function is the continuum limit of a sharply localized lattice kernel. It is not resolved by the long-distance CFT power law. This is the same warning in a different language: the normalization and possible local improvement of contact terms are ultraviolet data that survive as distributions.

The mechanical analogy is useful but must be handled carefully. For a single nonlinear oscillator,

$$\ddot x+\omega^2x+\lambda x^3=0,$$

the initial data $x(0),\dot x(0)$ determine $x(t)$. For a field, the equation of motion propagates operator data from one time slice to another. But products of fields at the same point are singular, so the equation of motion is an operator identity only after specifying renormalized composite operators and contact terms.

Canonical equal-time data determine the operator solution, just as $x(0),\dot x(0)$ determine a mechanical trajectory. In field theory, local equations of motion hold inside correlators away from coincident insertions; coincident points generate contact terms.

A precise path-integral statement is the Schwinger–Dyson identity. For a scalar field and any product of insertions $\mathcal O[\phi]$,

$$0=\int D\phi\,{\delta\over\delta\phi(x)} \left(e^{-S[\phi]}\mathcal O[\phi]\right).$$

Therefore

$$\boxed{ \left\langle {\delta S\over\delta\phi(x)}\mathcal O\right\rangle =\left\langle {\delta\mathcal O\over\delta\phi(x)}\right\rangle. }$$

For

$$\mathcal O=\phi(x_1)\cdots\phi(x_n),$$

this becomes

$$\left\langle {\delta S\over\delta\phi(x)}\phi(x_1)\cdots\phi(x_n)\right\rangle =\sum_{k=1}^n\delta^{(D)}(x-x_k) \left\langle\phi(x_1)\cdots\widehat{\phi(x_k)}\cdots\phi(x_n)\right\rangle,$$

where the hat means omission. Away from $x=x_k$, the classical-looking equation of motion is valid inside correlation functions. At $x=x_k$, the contact terms are part of the identity.

For example, if

$$S=\int d^D x\left[{1\over2}(\partial\phi)^2+{g\over3!}\phi^3\right],$$

then

$${\delta S\over\delta\phi(x)}=-\partial^2\phi(x)+{g\over2}\phi^2(x),$$

so the equation

$$\partial^2\phi={g\over2}\phi^2$$

is not a naive pointwise identity between unrenormalized products. It is a renormalized local-operator relation, valid inside correlators modulo contact terms.

Noether currents from localizing a symmetry

Suppose the action is invariant under a constant internal transformation

$$\delta\Phi_k=\alpha R_k\Phi_k, \qquad \alpha=\text{constant}.$$

To find the current, promote $\alpha$ to a function $\alpha(x)$. The action is no longer invariant, but locality implies that to first order in derivatives of $\alpha$,

$$\delta S=\int d^D x\,J_\mu(x)\partial_\mu\alpha(x).$$

Integrating by parts gives

$$\delta S=-\int d^D x\,\alpha(x)\partial_\mu J_\mu(x),$$

up to boundary terms. Since constant $\alpha$ is an exact symmetry, only derivatives of $\alpha$ appear. The coefficient of $\partial_\mu\alpha$ is the Noether current.

A global symmetry becomes a current when the parameter is made local. The variation of the action is proportional to $J^\mu\partial_\mu\alpha$, or after integration by parts to $-\alpha\partial_\mu J^\mu$.

For a complex scalar with Euclidean action

$$S=\int d^D x\left(\partial_\mu\phi^*\partial_\mu\phi+V(|\phi|^2)\right),$$

the global $U(1)$ transformation is

$$\phi\mapsto e^{i\alpha}\phi, \qquad \phi^*\mapsto e^{-i\alpha}\phi^*.$$

For local $\alpha(x)$,

$$\delta\phi=i\alpha\phi, \qquad \delta\phi^*=-i\alpha\phi^*.$$

The variation of the kinetic term is

$$\delta S =\int d^D x\,i\left(\phi\partial_\mu\phi^*-\phi^*\partial_\mu\phi\right)\partial_\mu\alpha.$$

Thus one convenient Euclidean convention is

$$J_\mu=i\left(\phi\partial_\mu\phi^*-\phi^*\partial_\mu\phi\right).$$

The conservation equation

$$\partial_\mu J_\mu=0$$

holds as an operator statement away from insertions and, equivalently, as a Ward identity inside correlators.

Ward identities in position space

Let

$$\mathcal O=\Phi_1(x_1)\Phi_2(x_2)\cdots\Phi_n(x_n).$$

Perform a change of variables in the path integral using a local symmetry transformation. The integral does not change:

$$0=\int D\Phi\,\delta\left(e^{-S}\mathcal O\right) =\left\langle\delta\mathcal O\right\rangle- \left\langle\mathcal O\,\delta S\right\rangle.$$

Using

$$\delta S=-\int d^D x\,\alpha(x)\partial_\mu J_\mu(x),$$

and

$$\delta\mathcal O=\sum_{k=1}^n\alpha(x_k) \Phi_1(x_1)\cdots R_k\Phi_k(x_k)\cdots\Phi_n(x_n),$$

we obtain

$$0=\int d^D x\,\alpha(x) \left[ \partial_\mu\langle J_\mu(x)\mathcal O\rangle +\sum_{k=1}^n\delta^{(D)}(x-x_k) \langle\Phi_1\cdots R_k\Phi_k\cdots\Phi_n\rangle \right].$$

Because $\alpha(x)$ is arbitrary,

$$\boxed{ \partial_\mu\langle J_\mu(x)\Phi_1(x_1)\cdots\Phi_n(x_n)\rangle =-\sum_{k=1}^n\delta^{(D)}(x-x_k) \langle\Phi_1\cdots R_k\Phi_k\cdots\Phi_n\rangle. }$$

For $U(1)$ charges $e_k$, with $R_k=i e_k$, this is

$$\boxed{ \partial_\mu\langle J_\mu(x)\Phi_1(x_1)\cdots\Phi_n(x_n)\rangle =-i\sum_{k=1}^n e_k\delta^{(D)}(x-x_k) \langle\Phi_1(x_1)\cdots\Phi_n(x_n)\rangle. }$$

The right-hand side is not a failure of current conservation. It says that the current insertion detects the charge carried by the local operators.

The divergence of a conserved current vanishes away from operator insertions. At the insertions it produces delta-function contact terms, with coefficients fixed by the symmetry action on the inserted operators.

As a useful check, integrate the Ward identity over a region $R$. Gauss’s theorem gives

$$\int_R d^D x\,\partial_\mu\langle J_\mu(x)\mathcal O\rangle =\int_{\partial R} d\Sigma_\mu\,\langle J_\mu(x)\mathcal O\rangle.$$

If the boundary term vanishes and $R$ contains all insertions, then

$$\sum_{k=1}^n\langle\Phi_1\cdots R_k\Phi_k\cdots\Phi_n\rangle=0.$$

For a $U(1)$ correlator of charged fields, this becomes the charge-selection rule

$$\left(\sum_{k=1}^n e_k\right) \langle\Phi_1(x_1)\cdots\Phi_n(x_n)\rangle=0.$$

Thus a correlator can be nonzero only if the total charge is zero, unless the vacuum or boundary conditions carry charge.

Momentum-space Ward identities

Ward identities are especially powerful in momentum space. Define a current-inserted correlator schematically by

$$G_\mu(q;p_1,\ldots,p_n) =\int d^D x\prod_k d^D x_k\, e^{-iq\cdot x-i\sum_k p_k\cdot x_k} \langle J_\mu(x)\Phi_1(x_1)\cdots\Phi_n(x_n)\rangle.$$

Fourier transforming the position-space Ward identity gives, with this convention,

$$\boxed{ q_\mu G_\mu(q;p_1,\ldots,p_n) =-\sum_{k=1}^n e_k\, G(p_1,\ldots,p_k+q,\ldots,p_n) }$$

for $U(1)$ fields, after the common factor of $i$ has been canceled. The precise sign changes if one reverses the Fourier convention or defines the generator without $i$; the invariant content is that contracting the current momentum is equivalent to inserting the symmetry generator on each charged leg.

In momentum space, the divergence of a current insertion gives a sum over contact terms. Each term shifts the momentum of one charged operator by the current momentum $q$ and weights it by the corresponding charge.

For a charged field two-point function, the most familiar form is the Ward–Takahashi identity for the proper current vertex. Let $G(p)$ be the full propagator and define the amputated vertex $\Gamma_\mu(p+q,p)$ by

$$G_\mu(p+q,p)=G(p+q)\Gamma_\mu(p+q,p)G(p).$$

Then the Ward identity becomes

$$\boxed{ q_\mu\Gamma_\mu(p+q,p) =e\left[G^{-1}(p+q)-G^{-1}(p)\right]. }$$

This formula is one of the cleanest expressions of symmetry in perturbative QFT. It says that the longitudinal part of the current vertex is not independent dynamical data; it is fixed by the inverse propagator.

Taking $q\to0$ gives

$$\Gamma_\mu(p,p)=e\,{\partial G^{-1}(p)\over\partial p_\mu},$$

again modulo the precise Euclidean/Lorentzian and Fourier conventions. Later, in QED, this identity becomes the reason the charge renormalization and wavefunction renormalization are tied together.

The first Goldstone pole

Ward identities also know about spontaneous symmetry breaking. Suppose a continuous symmetry acts nontrivially on an operator $\Phi$, and the vacuum expectation value is nonzero:

$$\langle\Phi\rangle=v\ne0.$$

Apply the Ward identity to the correlator $\langle J_\mu(x)\Phi(0)\rangle$. If

$$\delta\Phi=i e\alpha\Phi,$$

then

$$\partial_\mu\langle J_\mu(x)\Phi(0)\rangle =-i e v\,\delta^{(D)}(x).$$

Fourier transforming gives

$$q_\mu C_\mu(q)=\text{constant}, \qquad C_\mu(q)=\int d^D x\,e^{-iq\cdot x}\langle J_\mu(x)\Phi(0)\rangle.$$

If $C_\mu(q)$ were regular at $q=0$, the left side would vanish as $q\to0$. Therefore it must contain a singular term. For a scalar order parameter, rotational invariance fixes the singular longitudinal part to have the form

$$C_\mu(q)\sim {q_\mu\over q^2}.$$

Regular transverse terms may also be present, but they cannot saturate the nonzero divergence.

In Lorentzian signature, a pole at $q^2=0$ is the signature of a massless excitation. This is the seed of Goldstone’s theorem.

If the chosen vacuum is not invariant and $\langle\delta\Phi\rangle\ne0$, the Ward identity forces the current–order-parameter correlator to have a $1/q^2$ pole. The next page develops this into the Goldstone theorem and then compares it with gauge-theory Ward identities.

This argument is deliberately minimal. It does not require a Lagrangian, weak coupling, or a classical potential. It only uses a conserved current, a local operator transformed by the symmetry, and a vacuum expectation value that is not invariant.

Scope of the pole argument

The symmetry here is a genuine continuous global symmetry, not a gauge redundancy, and the non-invariant vacuum is an infinite-volume state. One also assumes a well-defined anomaly-free current and the usual relativistic spectral interpretation. In $1+1$ dimensions, the standard assumptions of the Coleman theorem prevent precisely this kind of continuous-symmetry-breaking vacuum; the conditional Ward-identity algebra remains correct, but its premise $v\ne0$ is then unavailable.

Summary

The Lorentzian commutator expectation of two local fields is the difference between two Wightman boundary values. For a conformal power law, this difference is a branch-cut discontinuity. It vanishes in the spacelike region because there the two boundary values coincide. On the light cone, and sometimes inside it, this discontinuity supplies the spectral function from which retarded and advanced response are built.

Current conservation has an equally local interpretation. A current is obtained by promoting a global symmetry parameter to $\alpha(x)$ and reading off the coefficient of $\partial_\mu\alpha$. Inside correlation functions, the divergence of this current vanishes away from operator insertions and produces contact terms at the insertions. These contact terms are Ward identities.

In momentum space, a Ward identity says that contracting a current insertion with its momentum is equivalent to acting with the symmetry generator on each external operator. If the chosen vacuum is not invariant, the same identity forces a massless pole: the first glimpse of Goldstone physics.

Common pitfalls

The Feynman propagator is not the commutator. The Feynman propagator is time ordered; the commutator is the difference of two Wightman orderings. The former can be nonzero at spacelike separation, while the latter must vanish for local observables. The retarded response includes the additional factor $i\theta(t)$, up to the chosen source-sign convention.

The formula with $\sin(\pi\Delta)$ is not valid as an ordinary function when $\Delta$ is an integer. In that case the discontinuity is a distribution supported on the light cone, such as $\delta(\rho)$ or derivatives of $\delta(\rho)$.

The equation $\partial_\mu J_\mu=0$ is incomplete inside a correlator. It is true away from insertions. At charged insertions it must be supplemented by contact terms.

A Ward identity is not a dynamical approximation. It follows from a change of variables in the path integral, assuming the measure is invariant. If the measure is not invariant, an anomaly term must be added.

Exercises

Exercise 1: The timelike discontinuity

Let

$$W_+(t,\mathbf x)={C\over(\rho+i0\operatorname{sgn}t)^\Delta}, \qquad W_-(t,\mathbf x)={C\over(\rho-i0\operatorname{sgn}t)^\Delta}, \qquad \rho=\mathbf x^2-t^2.$$

For non-integer $\Delta$, compute $W_+-W_-$ for $\rho>0$ and for $\rho<0$.

Solution

For $\rho>0$, the two boundary values approach the positive real axis from opposite sides. Since $\rho^{-\Delta}$ has no branch cut there,

$$(\rho+i0)^{-\Delta}=(\rho-i0)^{-\Delta}=\rho^{-\Delta}.$$

Therefore

$$W_+-W_-=0 \qquad (\rho>0).$$

For $\rho<0$, write $\rho=-|\rho|$. If $t>0$, then

$$\rho+i0=|\rho|e^{i\pi}, \qquad \rho-i0=|\rho|e^{-i\pi},$$

so

$$W_+-W_- =C|\rho|^{-\Delta}\left(e^{-i\pi\Delta}-e^{i\pi\Delta}\right) =-2iC\sin(\pi\Delta)|\rho|^{-\Delta}.$$

If $t<0$, the boundary prescriptions are reversed, so the result changes sign. Thus

$$W_+-W_- =-2iC\sin(\pi\Delta)\operatorname{sgn}t\,{\theta(-\rho)\over(-\rho)^\Delta}.$$

Since $-\rho=t^2-\mathbf x^2$, this is

$$W_+-W_- =-2iC\sin(\pi\Delta)\operatorname{sgn}t\, {\theta(t^2-\mathbf x^2)\over(t^2-\mathbf x^2)^\Delta}.$$

Exercise 2: Integer dimension and light-cone support

Use the distribution identity

$${1\over x+i0}-{1\over x-i0}=-2\pi i\delta(x)$$

to compute the commutator discontinuity for $\Delta=1$.

Solution

For $\Delta=1$,

$$W_+-W_-=C\left[{1\over\rho+i0\operatorname{sgn}t}-{1\over\rho-i0\operatorname{sgn}t}\right].$$

If $t>0$, this is

$$C\left({1\over\rho+i0}-{1\over\rho-i0}\right) =-2\pi iC\delta(\rho).$$

If $t<0$, the $i0$ prescriptions are reversed, giving the opposite sign. Hence

$$W_+-W_-=-2\pi iC\operatorname{sgn}t\,\delta(\rho).$$

Since $\rho=\mathbf x^2-t^2$, this distribution is supported on the light cone. This is the light-cone version of the massless free-field commutator.

Exercise 3: Deriving the current Ward identity

Derive the Ward identity

$$\partial_\mu\langle J_\mu(x)\Phi_1(x_1)\cdots\Phi_n(x_n)\rangle =-\sum_{k=1}^n\delta^{(D)}(x-x_k) \langle\Phi_1\cdots R_k\Phi_k\cdots\Phi_n\rangle$$

from a local change of variables in the Euclidean path integral.

Solution

Let

$$\mathcal O=\Phi_1(x_1)\cdots\Phi_n(x_n).$$

Perform the infinitesimal change of variables

$$\delta\Phi_k=\alpha(x)R_k\Phi_k.$$

Assume the measure is invariant. Then

$$0=\int D\Phi\,\delta(e^{-S}\mathcal O) =\langle\delta\mathcal O\rangle-\langle\mathcal O\delta S\rangle.$$

The localized symmetry variation of the action is

$$\delta S=\int d^D x\,J_\mu\partial_\mu\alpha =-\int d^D x\,\alpha\partial_\mu J_\mu.$$

The insertion varies as

$$\delta\mathcal O= \sum_{k=1}^n\alpha(x_k)\Phi_1\cdots R_k\Phi_k\cdots\Phi_n.$$

Write this as an integral over $x$:

$$\delta\mathcal O= \int d^D x\,\alpha(x) \sum_{k=1}^n\delta^{(D)}(x-x_k) \Phi_1\cdots R_k\Phi_k\cdots\Phi_n.$$

Substituting into $0=\langle\delta\mathcal O\rangle-\langle\mathcal O\delta S\rangle$ gives

$$0=\int d^D x\,\alpha(x)\left[ \sum_{k=1}^n\delta^{(D)}(x-x_k) \langle\Phi_1\cdots R_k\Phi_k\cdots\Phi_n\rangle +\partial_\mu\langle J_\mu(x)\mathcal O\rangle \right].$$

Since $\alpha(x)$ is arbitrary, the integrand must vanish. This gives the Ward identity.

Exercise 4: The charge-selection rule

Let $\phi$ have $U(1)$ charge $+1$ and $\phi^*$ charge $-1$. Use the Ward identity to show that

$$\langle\phi(x_1)\phi(x_2)\rangle=0$$

in a charge-neutral vacuum.

Solution

For the two-point function

$$\mathcal O=\phi(x_1)\phi(x_2),$$

both insertions have charge $+1$. The integrated Ward identity for a region containing both insertions and with no boundary flux gives

$$(e_1+e_2)\langle\phi(x_1)\phi(x_2)\rangle=0.$$

Here $e_1=e_2=1$, so

$$2\langle\phi(x_1)\phi(x_2)\rangle=0.$$

Thus

$$\langle\phi(x_1)\phi(x_2)\rangle=0.$$

By contrast, $\langle\phi(x_1)\phi^*(x_2)\rangle$ is not forbidden, because the total charge is $1-1=0$.

Exercise 5: The zero-momentum Ward–Takahashi vertex

Assume that the exact propagator of a charged scalar field is $G(p)$ and that the current three-point function is written as

$$G_\mu(p+q,p)=G(p+q)\Gamma_\mu(p+q,p)G(p).$$

Starting from the Ward–Takahashi identity

$$q_\mu\Gamma_\mu(p+q,p)=e\left[G^{-1}(p+q)-G^{-1}(p)\right],$$

show that the zero-momentum current vertex is

$$\Gamma_\mu(p,p)=e{\partial G^{-1}(p)\over\partial p_\mu}.$$

Solution

Expand the inverse propagator for small $q$:

$$G^{-1}(p+q)=G^{-1}(p)+q_\nu{\partial G^{-1}(p)\over\partial p_\nu}+O(q^2).$$

Then the Ward–Takahashi identity becomes

$$q_\mu\Gamma_\mu(p+q,p) =e q_\nu{\partial G^{-1}(p)\over\partial p_\nu}+O(q^2).$$

Assuming the vertex is regular as $q\to0$, set

$$\Gamma_\mu(p+q,p)=\Gamma_\mu(p,p)+O(q).$$

The leading terms give

$$q_\mu\Gamma_\mu(p,p) =e q_\mu{\partial G^{-1}(p)\over\partial p_\mu}.$$

Since this holds for arbitrary infinitesimal $q_\mu$,

$$\Gamma_\mu(p,p)=e{\partial G^{-1}(p)\over\partial p_\mu}.$$

Exercise 6: The Goldstone singularity

Suppose a continuous symmetry current satisfies

$$\partial_\mu\langle J_\mu(x)\Phi(0)\rangle=-v\delta^{(D)}(x), \qquad v\ne0.$$

Show that the Fourier transform $C_\mu(q)$ cannot be regular at $q=0$.

Solution

Define

$$C_\mu(q)=\int d^D x\,e^{-iq\cdot x}\langle J_\mu(x)\Phi(0)\rangle.$$

Fourier transforming the Ward identity gives, up to the conventional factor of $i$ from Fourier transforming the derivative,

$$q_\mu C_\mu(q)=\text{nonzero constant proportional to }v.$$

If $C_\mu(q)$ were regular at $q=0$, then $q_\mu C_\mu(q)$ would vanish as $q\to0$. This contradicts the nonzero right-hand side. Therefore $C_\mu(q)$ must be singular.

Rotational invariance implies that the singular longitudinal part has the form

$$C_\mu(q)\sim {q_\mu\over q^2}v.$$

Terms transverse to $q_\mu$ may be regular, but they cannot saturate the nonzero divergence. The displayed longitudinal singularity is the massless pole that becomes the Goldstone boson in Lorentzian signature.

References and further reading

• M. Srednicki, Quantum Field Theory, chapters 3, 22, 32, 67, and 68, for canonical commutators, Noether currents, spontaneous symmetry breaking, and Ward identities.
• S. Coleman, Lectures of Sidney Coleman on Quantum Field Theory, lectures on symmetries and conservation laws, scalar fields, and current algebra.
• S. Weinberg, The Quantum Theory of Fields, volume I, chapters 5 and 10, and volume II, chapter 19, for causal fields, Ward identities, and Goldstone bosons.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, chapters 4–6, for conformal correlators, analytic continuation, and current Ward identities in two dimensions.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, chapters on functional identities, symmetries, and critical correlation functions.
• A. M. Polyakov, Gauge Fields and Strings, especially the early chapters on statistical mechanics, local symmetries, and conserved currents.