Ward identity in nLab

Context

Algebraic Quantum Field Theory

• Idea
• Details
• Before renormalization
• Examples
• References

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.

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.