Stress Tensors, Goldstone Modes, and QED Ward Identities

The previous page derived Ward identities by localizing an internal symmetry. A conserved current is not just an operator whose divergence vanishes; it is the object that measures the response of the action when a constant symmetry parameter is allowed to vary in spacetime. This page applies that lesson in three directions.

First, Ward identities explain why spontaneous breaking of a continuous global symmetry produces a massless pole. Second, spacetime symmetries have their own current: the stress tensor $T_{\mu\nu}$. Translation invariance gives conservation, rotation invariance gives a symmetric improved stress tensor, and scale or conformal invariance is measured by the trace. Third, in QED the local $U(1)$ Ward identity becomes the Ward–Takahashi identity, relating the exact photon vertex to the exact charged-particle propagator and forcing the photon self-energy to be transverse.

The last section translates the stress-tensor story into two-dimensional complex coordinates. In two dimensions, conservation plus tracelessness implies that $T_{zz}$ is holomorphic and $T_{\bar z\bar z}$ is antiholomorphic. This is the doorway to the stress-tensor OPE and the Virasoro algebra.

Required background. Lesson 18 supplies the localized-symmetry derivation of current Ward identities and their contact terms. Helpful background. Lessons 13–16 develop scale and conformal transformations; only their basic geometric action is used here.

Goldstone pole from a Ward identity

Metric variation and QED momentum routing

Most stress-tensor formulas on this page are Euclidean. We define the metric stress tensor by

$$\delta S={1\over2}\int d^d x\sqrt g\,T^{\mu\nu}\delta g_{\mu\nu}.$$

With this convention, an infinitesimal diffeomorphism gives

$$\delta g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu.$$

Some references vary $g^{\mu\nu}$ instead of $g_{\mu\nu}$, or put a minus sign in the definition of $T^{\mu\nu}$. That changes some overall signs in Ward identities but not the conservation, symmetry, trace, or transversality statements.

For QED, $G(p)$ denotes the exact charged-field propagator and $\Gamma_\mu(p+q,p)$ denotes the exact one-particle-irreducible vertex with incoming charged momentum $p$, outgoing charged momentum $p+q$, and photon momentum $q$ flowing into the vertex. The charge is called $e$.

Let $J_\mu$ be the conserved current for a continuous global symmetry. Suppose a local operator $\Phi$ transforms nontrivially,

$$\delta\Phi=i\epsilon R\Phi.$$

The Ward identity for the correlator $\langle J_\mu(x)\Phi(0)\rangle$ has the form

$$\partial_\mu\langle J_\mu(x)\Phi(0)\rangle =-\delta^{(d)}(x)\langle\delta\Phi(0)\rangle,$$

up to the convention-dependent factor of $i$ in the generator. If the vacuum is symmetric, then $\langle\delta\Phi\rangle=0$ and no singularity is forced. If the symmetry is spontaneously broken, there is an order parameter with

$$\langle\delta\Phi\rangle\ne0.$$

Fourier transforming the Ward identity gives

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

A correlator regular at $q=0$ cannot satisfy this equation, because $q_\mu C_\mu(q)$ would vanish as $q\to0$. Rotational invariance fixes the leading singular longitudinal term:

$$\boxed{ C_\mu(q)\sim i{q_\mu\over q^2}\langle\delta\Phi\rangle. }$$

After continuation to Lorentzian signature, a pole at $q^2=0$ is the signature of a massless particle. Equivalently, there is a state $|\pi(q)\rangle$ with

$$\langle0|J_\mu(0)|\pi(q)\rangle=iFq_\mu.$$

The current creates the Goldstone mode, and the massless propagator supplies the $1/q^2$.

This argument assumes the usual relativistic vacuum, a well-defined conserved charge, and a phase in which a local order parameter can acquire a nonzero expectation value. In particular, the Coleman–Mermin–Wagner obstruction rules out this ordinary pattern of continuous internal-symmetry breaking in relativistic $1+1$-dimensional theories under its standard assumptions.

A broken continuous global symmetry forces a pole in the current–order-parameter correlator. Current conservation supplies the factor $q_\mu$, while the nonzero variation of the order parameter forces the $1/q^2$ singularity.

The word global is essential. If the broken symmetry is gauged, the pole is not a gauge-invariant massless particle in the physical spectrum. In the Higgs phase, the would-be Goldstone mode becomes the longitudinal polarization of the gauge field. The Ward identity remains true, but the interpretation of the pole changes.

Stress tensor as the current for translations

For an internal symmetry, the current is obtained by promoting a constant parameter to $\alpha(x)$. For translations, the corresponding operation is a local displacement

$$x^\mu\mapsto x^\mu+\xi^\mu(x).$$

In flat space, a local displacement changes the action by

$$\delta S=\int d^d x\,T_{\mu\nu}\partial_\mu\xi_\nu$$

up to terms proportional to equations of motion and boundary terms. Integrating by parts gives

$$\delta S=-\int d^d x\,\xi_\nu\partial_\mu T_{\mu\nu}.$$

Since $\xi_\nu(x)$ is arbitrary, translation invariance gives the local conservation law

$$\boxed{\partial_\mu T_{\mu\nu}=0.}$$

In Lorentzian signature, the conserved momentum is

$$P_\nu=\int d^{d-1}\mathbf x\,T_{0\nu}.$$

For scalar fields with Euclidean Lagrangian density $\mathcal L(\phi^a,\partial_\mu\phi^a)$, the canonical Noether tensor is

$$T_{\mu\nu}^{\mathrm{can}} ={\partial\mathcal L\over\partial(\partial_\mu\phi^a)}\partial_\nu\phi^a -\delta_{\mu\nu}\mathcal L.$$

It is conserved on the equations of motion when the theory has no explicit coordinate dependence. But it is not unique: one may add an identically conserved improvement term,

$$T_{\mu\nu}\mapsto T_{\mu\nu}+\partial_\rho B_{\rho\mu\nu}, \qquad B_{\rho\mu\nu}=-B_{\mu\rho\nu},$$

without changing the conserved momentum under standard boundary conditions. This freedom is crucial. It allows us to choose a tensor adapted to rotations and conformal transformations.

Metric variation and diffeomorphism Ward identities

The cleanest definition of the stress tensor is obtained by coupling the theory to a background metric and varying the action:

$$\delta S={1\over2}\int d^d x\sqrt g\,T^{\mu\nu}\delta g_{\mu\nu}.$$

Under an infinitesimal diffeomorphism generated by $\xi^\mu$,

$$\delta g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu.$$

Therefore, if $T^{\mu\nu}$ is symmetric,

$$\delta S=\int d^d x\sqrt g\,T^{\mu\nu}\nabla_\mu\xi_\nu.$$

Integrating by parts gives

$$\delta S=-\int d^d x\sqrt g\,\xi_\nu\nabla_\mu T^{\mu\nu}$$

up to a boundary term. Diffeomorphism invariance implies

$$\boxed{\nabla_\mu T^{\mu\nu}=0.}$$

In flat space this reduces to $\partial_\mu T_{\mu\nu}=0$.

A local displacement $\xi^\mu(x)$ changes the metric by $\delta g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$. The stress tensor is the operator conjugate to this metric strain. Diffeomorphism invariance gives stress-tensor conservation.

Inside a correlation function, conservation has contact terms. For scalar local operators $O_i(x_i)$,

$$\boxed{ \partial_\mu\left\langle T_{\mu\nu}(x)\prod_i O_i(x_i)\right\rangle =-\sum_i\delta^{(d)}(x-x_i){\partial\over\partial x_i^\nu} \left\langle\prod_i O_i(x_i)\right\rangle. }$$

The stress tensor generates translations of the insertions. If the operators carry spin, there are additional contact terms rotating their indices. If the operators are composite, there can also be scheme-dependent contact terms from operator mixing.

Rotations, dilatations, and conformal transformations

Special choices of $\xi_\mu(x)$ turn the stress-tensor Ward identity into familiar spacetime symmetries.

For rotations,

$$\xi_\nu=\omega_{\nu\rho}x_\rho, \qquad \omega_{\nu\rho}=-\omega_{\rho\nu},$$

so

$$\partial_\mu\xi_\nu=\omega_{\nu\mu}.$$

Only the antisymmetric part of $T_{\mu\nu}$ contributes. Rotation invariance therefore allows us to improve the canonical tensor to a symmetric one,

$$T_{\mu\nu}=T_{\nu\mu}.$$

For fields with spin, this improvement is the Belinfante construction. The angular-momentum current is schematically

$$M_{\lambda\mu\nu}=x_\mu T_{\lambda\nu}^{\mathrm{can}} -x_\nu T_{\lambda\mu}^{\mathrm{can}} +S_{\lambda\mu\nu},$$

where $S_{\lambda\mu\nu}$ is the spin current. Conservation of $M_{\lambda\mu\nu}$ says that the antisymmetric part of $T^{\mathrm{can}}$ is a total derivative, which can be absorbed into an improvement.

For a dilation,

$$\xi_\nu=\lambda x_\nu, \qquad \partial_\mu\xi_\nu=\lambda\delta_{\mu\nu}.$$

The variation becomes

$$\delta S=\lambda\int d^d x\,T^\mu{}_{\mu}.$$

Thus the trace is the local obstruction to scale invariance. A scale-invariant theory has

$$T^\mu{}_{\mu}=\partial_\mu V^\mu$$

for some virial current $V^\mu$. If the virial current can be removed by improvement, then one may choose

$$\boxed{T^\mu{}_{\mu}=0.}$$

Translation invariance gives conservation, rotation invariance gives a symmetric improved stress tensor, and scale invariance at a conformal fixed point gives a traceless improved stress tensor.

A conformal transformation is generated by a vector field satisfying the conformal Killing equation

$$\partial_\mu\xi_\nu+\partial_\nu\xi_\mu ={2\over d}(\partial\cdot\xi)\delta_{\mu\nu}.$$

If $T_{\mu\nu}$ is symmetric and traceless, then

$$T_{\mu\nu}\partial_\mu\xi_\nu ={1\over2}T_{\mu\nu}(\partial_\mu\xi_\nu+\partial_\nu\xi_\mu) ={1\over d}(\partial\cdot\xi)T^\mu{}_{\mu}=0.$$

Thus a symmetric, conserved, traceless stress tensor generates conformal Ward identities.

For $d>2$, the conformal Killing equation has finitely many independent solutions: translations, rotations, dilatations, and special conformal transformations. The infinitesimal special conformal transformation is

$$\xi^\mu(x)=2(b\cdot x)x^\mu-b^\mu x^2.$$

Equivalently, the finite map can be written as inversion, translation, inversion:

$${x'^{\mu}\over x'^2}={x^\mu\over x^2}+b^\mu,$$

with the sign of $b^\mu$ depending on convention.

Scale invariance versus conformal invariance

The implication from scale invariance to conformal invariance is not a purely algebraic identity. The safe local criterion is the existence of a conserved symmetric stress tensor whose trace can be improved to zero. Many unitary relativistic critical theories of interest satisfy this criterion, but it should not be used without checking its assumptions.

QED Ward–Takahashi identity

Let $G(p)$ be the exact propagator of a charged field and let $\Gamma_\mu(p+q,p)$ be the exact proper photon vertex. Gauge invariance implies

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

This is the Ward–Takahashi identity. A longitudinal photon insertion is equivalent to a gauge transformation on the charged line; therefore it measures the difference between the inverse propagator at the two ends.

Taking $q\to0$ gives

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

If the vertex is regular in this limit, then

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

This identity is exact. In perturbative QED it is the origin of the equality between vertex and wavefunction renormalization constants in compatible renormalization schemes.

The photon self-energy obeys the related transversality identity

$$\boxed{q_\mu\Pi_{\mu\nu}(q)=0.}$$

In a parity-even Lorentz- or rotation-invariant vacuum, this implies, for nonzero $q$,

$$\Pi_{\mu\nu}(q)=\left(q^2\delta_{\mu\nu}-q_\mu q_\nu\right)\Pi(q^2).$$

A local Proca contribution to the self-energy would be proportional to $m_\gamma^2\delta_{\mu\nu}$. Contracting with $q_\mu$ gives $m_\gamma^2q_\nu$, so such a term violates the Ward identity unless $m_\gamma^2=0$.

Transversality alone does not forbid every gauge-invariant mass gap. If the scalar form factor is nonanalytic, for example $\Pi(q^2)\sim m^2/q^2$, then the transverse combination becomes $m^2(\delta_{\mu\nu}-q_\mu q_\nu/q^2)$. The Schwinger model on the next page realizes precisely this possibility. The Ward identity excludes an elementary local Proca term; it does not exclude Higgs, Schwinger, or other dynamical mass-generation mechanisms.

Gauge invariance ties the longitudinal part of the exact matter–photon vertex to the inverse charged propagator. It also makes the photon self-energy transverse, excluding a local Proca self-energy while allowing nonlocal transverse mass generation.

Notice what the Ward–Takahashi identity does not say. It fixes the longitudinal part of the vertex. It does not determine all transverse structures. Those contain genuine dynamics, such as anomalous magnetic moments and form factors.

Two-dimensional stress tensor and holomorphy

In two Euclidean dimensions define

$$z=x^1+i x^2, \qquad \bar z=x^1-i x^2,$$

so

$$ds^2=(dx^1)^2+(dx^2)^2=dz\,d\bar z.$$

The stress tensor has complex components $T_{zz}$, $T_{\bar z\bar z}$, and $T_{z\bar z}$. Up to conventional normalization factors,

$$T_{zz}\sim T_{11}-T_{22}-2iT_{12},$$

and

$$T_{\bar z\bar z}\sim T_{11}-T_{22}+2iT_{12}.$$

The mixed component is proportional to the trace:

$$T_{z\bar z}\sim T_{11}+T_{22}.$$

At a conformal fixed point,

$$T_{z\bar z}=0.$$

Stress-tensor conservation then splits into

$$\bar\partial T_{zz}=0, \qquad \partial T_{\bar z\bar z}=0$$

away from insertions. Thus we write

$$T(z)=T_{zz}(z), \qquad \overline T(\bar z)=T_{\bar z\bar z}(\bar z).$$

In two dimensions, tracelessness removes the mixed component $T_{z\bar z}$, and conservation forces $T_{zz}$ to be holomorphic and $T_{\bar z\bar z}$ to be antiholomorphic away from insertions.

This is where two-dimensional conformal field theory becomes dramatically more powerful than its higher-dimensional cousin. Local conformal transformations

$$z\mapsto f(z), \qquad \bar z\mapsto \bar f(\bar z)$$

are generated by contour integrals of $T(z)$ and $\overline T(\bar z)$. The next step is to determine the singular terms in the OPE of $T$ with local fields and with itself.

Summary

A broken continuous global symmetry forces a massless pole in a current correlator. This is the Goldstone theorem in Ward-identity form. If the symmetry is gauged, the same local identity persists, but the Goldstone pole is reorganized into the longitudinal gauge-field degree of freedom.

The stress tensor is the current for spacetime symmetries. Conservation expresses translations; symmetry expresses rotations after improvement; tracelessness expresses scale or conformal invariance after improvement. Coupling the theory to a background metric gives the most invariant definition of $T_{\mu\nu}$ and directly produces diffeomorphism Ward identities.

In QED, local gauge invariance gives the Ward–Takahashi identity and transverse vacuum polarization. In two-dimensional CFT, conservation and tracelessness imply the holomorphic factorization of the stress tensor, preparing the ground for Virasoro symmetry.

Common pitfalls

A conserved current does not by itself imply a Goldstone boson. The pole appears only when the current acts nontrivially on the vacuum, equivalently when some operator has $\langle\delta\Phi\rangle\ne0$.

The stress tensor is not unique. The canonical Noether tensor, the Belinfante tensor, and the metric stress tensor can differ by improvement terms. The conserved charges agree under suitable boundary conditions, but local formulas may look different.

Tracelessness is a local operator statement modulo contact terms, improvements, and anomalies. In a quantum CFT on curved space, trace anomalies can appear even when the flat-space theory is conformal.

The Ward–Takahashi identity fixes only the longitudinal part of a QED vertex. Transverse form factors are not determined by current conservation alone.

Transverse vacuum polarization is not the same as a massless spectrum. A local term $m^2\delta_{\mu\nu}$ is forbidden, but a nonanalytic transverse structure can generate a gauge-invariant mass scale, as it does in QED$_2$.

Exercises

Exercise 1: Diffeomorphism Ward identity

Starting from

$$\delta S={1\over2}\int d^d x\sqrt g\,T^{\mu\nu}\delta g_{\mu\nu}$$

and

$$\delta g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu,$$

derive $\nabla_\mu T^{\mu\nu}=0$ from diffeomorphism invariance.

Solution

Substitute the metric variation:

$$\delta S={1\over2}\int d^d x\sqrt g\,T^{\mu\nu} (\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu).$$

Because $T^{\mu\nu}$ is symmetric in the metric definition,

$$\delta S=\int d^d x\sqrt g\,T^{\mu\nu}\nabla_\mu\xi_\nu.$$

Integrating by parts gives

$$\delta S=-\int d^d x\sqrt g\,\xi_\nu\nabla_\mu T^{\mu\nu}$$

up to a boundary term. If the action is invariant for arbitrary $\xi_\nu(x)$, then

$$\nabla_\mu T^{\mu\nu}=0.$$

Exercise 2: Belinfante symmetry from rotations

Let

$$\delta S=\int d^d x\,T_{\mu\nu}\partial_\mu\xi_\nu.$$

Show that a rotation $\xi_\nu=\omega_{\nu\rho}x_\rho$, with $\omega_{\nu\rho}=-\omega_{\rho\nu}$, couples only to the antisymmetric part of $T_{\mu\nu}$.

Solution

For the rotation,

$$\partial_\mu\xi_\nu=\omega_{\nu\mu}.$$

Thus

$$\delta S=\int d^d x\,T_{\mu\nu}\omega_{\nu\mu}.$$

Decompose

$$T_{\mu\nu}=T_{(\mu\nu)}+T_{[\mu\nu]},$$

where the first term is symmetric and the second is antisymmetric. Since $\omega_{\nu\mu}$ is antisymmetric,

$$T_{(\mu\nu)}\omega_{\nu\mu}=0.$$

Therefore

$$\delta S=\int d^d x\,T_{[\mu\nu]}\omega_{\nu\mu}.$$

Rotation invariance allows the antisymmetric part to be removed by a Belinfante improvement.

Exercise 3: Soft-photon Ward identity

Assume the Ward–Takahashi identity

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

Show that, if the vertex is regular as $q\to0$,

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

Solution

Expand

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

and

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

Then the Ward–Takahashi identity gives

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

Since this holds for arbitrary small $q$, the coefficients of $q_\mu$ agree:

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

Exercise 4: What transversality forbids

Show that transversality,

$$q_\mu\Pi_{\mu\nu}(q)=0,$$

forbids a local Proca contribution $m_\gamma^2\delta_{\mu\nu}$ to the exact inverse propagator. Then explain why the transverse but nonanalytic tensor $m^2(\delta_{\mu\nu}-q_\mu q_\nu/q^2)$ is not excluded.

Solution

A local Proca contribution to the self-energy would be

$$\Pi_{\mu\nu}^{\mathrm{mass}}(q)=m_\gamma^2\delta_{\mu\nu}.$$

Contracting with $q_\mu$ gives

$$q_\mu\Pi_{\mu\nu}^{\mathrm{mass}}(q)=m_\gamma^2q_\nu.$$

This vanishes for arbitrary $q$ only if $m_\gamma^2=0$. Thus gauge invariance excludes the local Proca tensor.

By contrast,

$$\Pi_{\mu\nu}^{\mathrm{gap}}(q) =m^2\left(\delta_{\mu\nu}-{q_\mu q_\nu\over q^2}\right)$$

is transverse. It is nonanalytic at $q=0$, because it corresponds to $\Pi(q^2)=m^2/q^2$ in the scalar decomposition. Transversality therefore permits a gauge-invariant dynamical mass gap even though it forbids a local Proca term.

Exercise 5: Holomorphy from conservation and tracelessness

In two Euclidean dimensions, show that conservation and tracelessness imply $\bar\partial T_{zz}=0$ away from insertions.

Solution

In complex coordinates, one component of stress-tensor conservation is

$$\bar\partial T_{zz}+\partial T_{\bar z z}=0.$$

The mixed component $T_{z\bar z}=T_{\bar z z}$ is proportional to the trace. At a conformal fixed point away from insertions,

$$T^\mu{}_{\mu}=0,$$

so

$$T_{z\bar z}=0.$$

The conservation equation reduces to

$$\bar\partial T_{zz}=0.$$

Thus $T_{zz}$ is holomorphic locally, and we write it as $T(z)$.

Exercise 6: Goldstone singularity

Suppose a current–order-parameter correlator satisfies

$$q_\mu C_\mu(q)=v, \qquad v\ne0.$$

Assuming rotational invariance, determine the leading singular part of $C_\mu(q)$ near $q=0$.

Solution

Write the vector correlator as a longitudinal part plus a transverse part:

$$C_\mu(q)=q_\mu f(q^2)+C_\mu^\perp(q), \qquad q_\mu C_\mu^\perp(q)=0.$$

Then

$$q_\mu C_\mu(q)=q^2 f(q^2)=v.$$

Therefore

$$f(q^2)={v\over q^2},$$

so the leading singular part is

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

This is the Goldstone pole.

References

• S. Coleman, Lectures of Sidney Coleman on Quantum Field Theory, Cambridge University Press (2019), lectures on spacetime symmetries, internal symmetries, and current algebra.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer (1997), chapters 4–6.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Westview Press (1995), chapters 2, 9, and 19.
• M. Srednicki, Quantum Field Theory, Cambridge University Press (2007), chapters 22, 32, 67, and 68.
• S. Weinberg, The Quantum Theory of Fields, volumes I–II, Cambridge University Press (1995–1996), volume I, chapters 7 and 10, and volume II, chapters 15 and 19.
• J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed., Oxford University Press (2021), chapters on symmetries, Ward identities, and critical correlation functions.