nLab Ward identity

under construction

Idea

In perturbative quantum field theory the quantum master Ward identity (def. below) expresses the relation between the quantum (measured by Planck's constant $\hbar$) interacting (measured by the coupling constant $g$) equations of motion to the classical free field equations of motion at $\hbar, g\to 0$ (remark below). As such it generalizes the Schwinger-Dyson equation, to which it reduces for $g = 0$ (example below) as well as the classical master Ward identity, which is the case for $\hbar = 0$ (example below).

Applied to products of the equations of motion with any given observable, the master Ward identity becomes a particular Ward identity.

This is of interest notably in view of Noether's theorem, which says that every infinitesimal symmetry of the Lagrangian of, in particular, the given free field theory, corresponds to a conserved current, hence a horizontal differential form whose total spacetime derivative vanishes up to a term proportional to the equations of motion. Under transgression to local observables this is a relation of the form

$$div \mathbf{J} = 0 \phantom{AAA} \text{on-shell} \,,$$

where “on shell” means up to the ideal generated by the classical free equations of motion. Hence for the case of local observables of the form $div \mathbf{J}$, the quantum Ward identity expresses the possible failure of the original conserved current to actually be conserved, due to both quantum effects ($\hbar$) and interactions ($g$). This is the form in which Ward identities are usually understood (example below).

In terms of BV-BRST formalism, the master Ward identity is equivalent to the quantum master equation on regular polynomial observables (this prop.).

Neither of these equations is guaranteed to hold for any choice of ("re"-)normalization. If a Ward identity is violated by the ("re"-)normalized perturbative QFT, specifically if there is no possible choice of ("re"-)normalization that preserves it, the one speaks of a quantum anomaly. Specifically if the conserved current corresponding to a gauge symmetry is anomalous in this way, one speaks of a gauge anomaly.

Details

Before renormalization

Definition

Consider a free gauge fixed Lagrangian field theory $(E_{\text{BV-BRST}}, \mathbf{L}')$ (this def.) with global BV-differential on regular polynomial observables

$$\{-S',(-)\} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]$$

(this def.).

Let moreover

$$g S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar , g ] ]$$

be a regular polynomial observable (regarded as an adiabatically switched non-point-interaction action functional) such that the total action $S' + g S_{int}$ satisfies the quantum master equation (this prop.); and write

$$\mathcal{R}^{-1}(-) \;\coloneqq\; \mathcal{S}(g S_{int})^{-1} \star_H (\mathcal{S}(g S_{int}) \star_F (-))$$

for the corresponding quantum Møller operator (this def.).

Then by this prop. we have

(1)$$\{-S',(-)\} \circ \mathcal{R}^{-1} \;=\; \mathcal{R}^{-1} \left( \left\{ -(S' + g S_{int}) \,,\, (-) \right\}_{\mathcal{T}} - i \hbar \Delta_{BV} \right)$$

This is the quantum master Ward identity on regular polynomial observables, i.e. before renormalization.

(Rejzner 13, (37))

Remark

(quantum master Ward identity relates quantum interacting field EOMs to classical free field EOMs)

For $A \in PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g] ]$ the quantum master Ward identity on regular polynomial observables (1) reads

(2)$$\mathcal{R}^{-1} \left( \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}} - i \hbar \Delta_{BV}(A) \right) \;=\; \{-S', \mathcal{R}^{-1}(A) \}$$

The term on the right is manifestly in the image of the global BV-differential $\{-S',-\}$ of the free field theory (this def.) and hence vanishes when passing to on-shell observables along the isomorphism (this equation)

$$\underset{ \text{on-shell} }{ \underbrace{ PolyObs(E_{\text{BV-BRST}}, \mathbf{L}') }} \;\simeq\; \underset{ \text{off-shell} }{ \underbrace{ PolyObs(E_{\text{BV-BRST}})_{def(af = 0)} }}/im(\{-S',-\})$$

(by this example).

Hence

$$\mathcal{R}^{-1} \left( \left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}} - i \hbar \Delta_{BV}(A) \right) \;=\; 0 \phantom{AAA} \text{on-shell}$$

In contrast, the left hand side is the interacting field observable (via this def.) of the sum of the time-ordered antibracket with the action functional of the interacting field theory and a quantum correction given by the BV-operator. If we use the definition of the BV-operator $\Delta_{BV}$ (this def.) we may equivalently re-write this as

(3)$$\mathcal{R}^{-1} \left( \left\{ -S' \,,\, A \right\} + \left\{ -g S_{int} \,,\, A \right\}_{\mathcal{T}} \right) \;=\; 0 \phantom{AAA} \text{on-shell}$$

Hence the quantum master Ward identity expresses a relation between the ideal spanned by the classical free field equations of motion and the quantum interacting field equations of motion.

Example

(free field-limit of master Ward identity is Schwinger-Dyson equation)

In the free field-limit $g \to 0$ (noticing that in this limit $\mathcal{R}^{-1} = id$) the quantum master Ward identity (1) reduces to

$$\left\{ -S' \,,\, A \right\}_{\mathcal{T}} - i \hbar \Delta_{BV}(A) \;=\; \{-S', A \}$$

which is the defining equation for the BV-operator (this equation), hence is isomorphic (under $\mathcal{T}$) to the Schwinger-Dyson equation (this prop.)

Example

(classical limit of quantum master Ward identity)

In the classical limit $\hbar \to 0$ (noticing that the classical limit of $\{-,-\}_{\mathcal{T}}$ is $\{-,-\}$) the quantum master Ward identity (1) reduces to

$$\mathcal{R}^{1} \left( \left\{ -(S' + g S_{int}) \,,\, A \right\} \right) \;=\; \{-S', \mathcal{R}^{-1}(A) \}$$

This says that the interacting field observable corresponding to the global antibracket with the action functional of the interacting field theory vanishes on-shell, classically.

Applied to an observable which is linear in the antifields

$$A \;=\; \underset{\Sigma}{\int} A^a(x) \mathbf{\Phi}^\ddagger_a(x) \, dvol_\Sigma(x)$$

this yields

$$\begin{aligned} 0 & = \{-S', \mathcal{R}^{-1}(A)\} + \mathcal{R}^{-1} \left( \left\{ -(S' + S_{int}) \,,\, A \right\}_{\mathcal{T}} \right) \\ & = \underset{\Sigma}{\int} \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x)) \, dvol_\Sigma(x) + \mathcal{R}^{-1} \left( \underset{\Sigma}{\int} A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)} \, dvol_\Sigma(x) \right) \end{aligned}$$

This is the classical master Ward identity according to (Dütsch-Fredenhagen 02, Brennecke-Dütsch 07, (5.5)), following (Dütsch-Boas 02).

Example

(quantum correction to Noether current conservation)

Let $v \in \Gamma^{ev}_\Sigma(T_\Sigma(E_{\text{BRST}}))$ be an evolutionary vector field, which is an infinitesimal symmetry of the Lagrangian $\mathbf{L}'$, and let $J_{\hat v} \in \Omega^{p,0}_\Sigma(E_{\text{BV-BRST}})$ the corresponding conserved current, by Noether's theorem I (this prop.), so that

$$\begin{aligned} d J_{\hat v} & = \iota_{\hat v} \delta \mathbf{L}' \\ & = (v^a dvol_\Sigma) \frac{\delta_{EL} L'}{\delta \phi^a} \phantom{AAA} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}}) \end{aligned}$$

(by this equation), where in the second line we just rewrote the expression in components (using this equation)

$$v^a \,, \frac{\delta_{EL} L'}{\delta \phi^a} \;\in \Omega^{0,0}_\Sigma(E_{\text{BV-BRST}})$$

and re-arranged suggestively.

Then for $a_{sw} \in C^\infty_{cp}(\Sigma)$ any choice of bump function, we obtain the local observables

$$\begin{aligned} A_{sw} & \coloneqq \underset{\Sigma}{\int} \underset{ A^a(x) }{ \underbrace{ a_{sw}(x) v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots ) } } \mathbf{\Phi}^\ddagger_a(x) \, dvol_\Sigma(x) \\ & \coloneqq \tau_\Sigma( a_{sw} v^a \phi^{\ddagger}_a \, dvol_\Sigma) \end{aligned}$$

and

$$\begin{aligned} (div \mathbf{J})_{sw} & \coloneqq \underset{\Sigma}{\int} \underset{ A^a(x) }{ \underbrace{ a_{sw}(x) v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots ) } } \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \, dvol_\Sigma(x) \\ & \coloneqq \tau_\Sigma \left( a_{sw} v^a \frac{\delta_{EL} \mathbf{L}'}{\delta \phi^a} \, dvol_\Sigma \right) \end{aligned}$$

by transgression of variational differential forms.

This is such that

$$\left\{ -S' , A_{sw} \right\} = (div \mathbf{J})_{sw} \,.$$

Hence applied to this choice of local observable $A$, the quantum master Ward identity (3) now says that

$$\mathcal{R}^{-1} \left( {\, \atop \,} (div \mathbf{J})_{sw} \right) \;=\; \mathcal{R}^{-1} \left( \{g S_{int}, A_{sw} \}_{\mathcal{T}} {\, \atop \,} \right) \phantom{AAA} \text{on-shell}$$

Hence the interacting field observable-version $\mathcal{R}^{-1}(div\mathbf{J})$ of $div \mathbf{J}$ need not vanish itself on-shell, instead there may be a correction as shown on the right.

Examples

• chiral anomaly

• conformal blocks

References

Named after John Clive Ward.

Discussion of the master Ward identity in the rigorous context of relativistic perturbative quantum field theory formulated via causal perturbation theory/perturbative AQFT is in

• Michael Dütsch, F.-M. Boas, The Master Ward Identity, Rev. Math. Phys 14, (2002) 977-1049 (pdf)

• Michael Dütsch, Klaus Fredenhagen, equation (90) in The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory, Commun.Math.Phys. 243 (2003) 275-314 (arXiv:hep-th/0211242)

• Ferdinand Brennecke, Michael Dütsch, equation (5.5) in Removal of violations of the Master Ward Identity, in perturbative QFT, Rev.Math.Phys. 20 (2008) 119-172 (arXiv:https://arxiv.org/abs/0705.3160)

• Stefan Hollands, around (322) and (333) and (345) of Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys.20:1033-1172, 2008 (arXiv:0705.3340)

• Katarzyna Rejzner, section 5.3 of Batalin-Vilkovisky formalism in locally covariant field theory (arXiv:1111.5130)

• Katarzyna Rejzner, equation (37) of Remarks on local symmetry invariance in perturbative algebraic quantum field theory (arXiv:1301.7037)

• Michael Dütsch, equation (4.2) of From classical field theory to perturbative quantum field theory, 2018

See also

• Wikipedia, Ward-Takahashi identity