Ward identity


under construction


Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



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quantum mechanical system, quantum probability

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Perturbative QFT



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 gg) equations of motion to the classical free field equations of motion at ,g0\hbar, g\to 0 (remark below). As such it generalizes the Schwinger-Dyson equation, to which it reduces for g=0g = 0 (example below) as well as the classical master Ward identity, which is the case for =0\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

divJ=0AAAon-shell, 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 divJdiv \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 (gg). 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.


Before renormalization


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

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

(this def.).

Let moreover

gS intPolyObs(E BV-BRST) reg[[,g]] 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+gS intS' + g S_{int} satisfies the quantum master equation (this prop.); and write

1()𝒮(gS int) 1 H(𝒮(gS int) F()) \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,()} 1= 1({(S+gS int),()} 𝒯iΔ BV) \{-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))


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

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

(2) 1({(S+gS int),A} 𝒯iΔ BV(A))={S, 1(A)} \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,}\{-S',-\} of the free field theory (this def.) and hence vanishes when passing to on-shell observables along the isomorphism (this equation)

PolyObs(E BV-BRST,L)on-shellPolyObs(E BV-BRST) def(af=0)off-shell/im({S,}) \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).


1({(S+gS int),A} 𝒯iΔ BV(A))=0AAAon-shell \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 Δ BV\Delta_{BV} (this def.) we may equivalently re-write this as

(3) 1({S,A}+{gS int,A} 𝒯)=0AAAon-shell \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.


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

In the free field-limit g0g \to 0 (noticing that in this limit 1=id\mathcal{R}^{-1} = id) the quantum master Ward identity (1) reduces to

{S,A} 𝒯iΔ BV(A)={S,A} \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 𝒯\mathcal{T}) to the Schwinger-Dyson equation (this prop.)


(classical limit of quantum master Ward identity)

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

1({(S+gS int),A})={S, 1(A)} \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=ΣA a(x)Φ a (x)dvol Σ(x) A \;=\; \underset{\Sigma}{\int} A^a(x) \mathbf{\Phi}^\ddagger_a(x) \, dvol_\Sigma(x)

this yields

0 ={S, 1(A)}+ 1({(S+S int),A} 𝒯) =ΣδSδΦ a(x) 1(A a(x))dvol Σ(x)+ 1(ΣA a(x)δ(S+S int)δΦ a(x)dvol Σ(x)) \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).


(quantum correction to Noether current conservation)

Let vΓ Σ ev(T Σ(E BRST))v \in \Gamma^{ev}_\Sigma(T_\Sigma(E_{\text{BRST}})) be an evolutionary vector field, which is an infinitesimal symmetry of the Lagrangian L\mathbf{L}', and let J v^Ω Σ p,0(E BV-BRST)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

dJ v^ =ι v^δL =(v advol Σ)δ ELLδϕ aAAAΩ Σ p+1,0(E BV-BRST) \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,δ ELLδϕ aΩ Σ 0,0(E BV-BRST) 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 swC cp (Σ)a_{sw} \in C^\infty_{cp}(\Sigma) any choice of bump function, we obtain the local observables

A sw Σa sw(x)v a(Φ(x),DΦ(x),)A a(x)Φ a (x)dvol Σ(x) τ Σ(a swv aϕ a dvol Σ) \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}


(divJ) sw Σa sw(x)v a(Φ(x),DΦ(x),)A a(x)δSδΦ a(x)dvol Σ(x) τ Σ(a swv aδ ELLδϕ advol Σ) \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

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

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

1((divJ) sw)= 1({gS int,A sw} 𝒯)AAAon-shell \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 1(divJ)\mathcal{R}^{-1}(div\mathbf{J}) of divJdiv \mathbf{J} need not vanish itself on-shell, instead there may be a correction as shown on the right.



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

See also

Last revised on March 28, 2019 at 02:06:09. See the history of this page for a list of all contributions to it.