Contents

# Contents

## Idea

In perturbative quantum field theory formulated in terms of BV-BRST formalism, the classical master equation expresses the nilpotency of the BV-differential before quantization, with the latter regarded as a Hamiltonian vector field with respect to the antibracket $\{-,-\}$, for “Hamiltonian” the BV-BRST-extended action functional “S + S_{BRST}”:

$\left( s_{BV} \right)^2 = 0 \phantom{AA} \Leftrightarrow \phantom{AA} \left( \{S + S_{BRST},(-)\}\right)^2 = 0 \phantom{AA} \Leftrightarrow \phantom{AA} \{S + S_{BRST}, S + BRST\} = 0 \,.$

The quantum master equation (prop. below) is the version of this equation after quantization, in which case the the BV-differential picks up a quantum correction of order $\hbar$ (Planck's constant) by the BV-operator $\Delta_{BV}$:

$\left(\, \{S + S_{BRST},(-)\} + i \hbar \Delta_{BV} \, \right)^2 = 0 \,.$

## In causal perturbation theory

We discuss the quantum master equation in the rigorous formulation of relativistic perturbative quantum field theory via causal perturbation theory/perturbative AQFT (Fredenhagen-Rejzner 11b, Rejzner 11).

First we consider all structure just on regular polynomial observables, hence excluding non-linear local observables such as the usual point-interaction action functionals.

Then the extension of all structures from regular to local observables is the renormalization step, discussed furthter below.

### Background

Throughout, let $(E_{\text{BV-BST}},\mathbf{L}')$ be a gauge fixed free Lagrangian field theory with global BV-differential (this def.)

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

Hence $S'$ denotes the gauge fixed free action functional.

Moreover, let $\Delta_H$ be a compatible choice of Wightman propagator with associated Feynman propagator $\Delta_F$.

###### Lemma

(global BV-differential is derivation on Wick algebra)

The global BV-differential

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

is a derivation also with respect to the Wick algebra star product $\star_H$:

$\left\{ -S', A_1 \star_H A_2 \right\} \;=\; \left\{ -S', A_1 \right\} \star_H A_2 \;+\; A_1 \star_H \left\{ -S', A_2 \right\} \,.$
###### Definition

(perturbative S-matrix on regular polynomial observables)

The perturbative S-matrix on regular polynomial observables is the exponential with respect to the time-ordered product

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

given by

$\mathcal{S}(S_{int}) = \exp_{\star_F} \left( \tfrac{1}{i \hbar} S_{int}) \right) \coloneqq 1 + \tfrac{1}{\i \hbar} S_{int} + \tfrac{1}{2} \tfrac{1}{(i \hbar)^2} S_{int} \star_F S_{int} + \cdots \,.$

We think of $S_{int}$ here as an adiabatically switched non-point-interaction action functional.

We write $\mathcal{S}(S_{int})^{-1}$ for the inverse with respect to the Wick product (which exists by this remark)

$\mathcal{S}(S_{int})^{-1} \star_H \mathcal{S}(S_{int}) = 1 \,.$

Notice that this is in general different form the inverse with respect to the time-ordered product $\star_F$, which is $\mathcal{S}(-S_{int})$:

$\mathcal{S}(-S_{int}) \star_F \mathcal{S}(S_{int}) = 1 \,.$
###### Definition

(quantum Møller operator on regular polynomial observables)

Given an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable of degree 0

$S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ]$

then the corresponding quantum Møller operator on regular polynomial observables

$\mathcal{R}^{-1} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]$

is given by the derivative of Bogoliubov's formula

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

where $\mathcal{S}(S_{int}) = \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$ is the perturbative S-matrix from def. .

This indeed lands in formal power series in Planck's constant $\hbar$ (by this remark), instead of in more general Laurent series as the perturbative S-matrix does (def. ).

Hence the inverse map is

$\mathcal{R} \;=\; \mathcal{S}(-S_{int}) \star_F ( \mathcal{S}(S_{int}) \star(-) ) \,.$

(Bogoliubov-Shirkov 59; the above terminology follows Hawkins-Rejzner 16, below def. 5.1)

Notice that compared to Fredenhagen-Rejzner et. al. we have changed notation conventions $\mathcal{R} \leftrightarrow \mathcal{R}^{-1}$ in order to bring out the analogy to (the conventions for the) time-ordered product $A_1 \star_F A_2 = \mathcal{T}(\mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2))$ on regular polynomial observables.

notice the implicit dependencies

endomorphism of
regular polynomial observables
meaningdepends on choice of
$\phantom{AA}\mathcal{T}$time-orderingfree Lagrangian density and Wightman propagator
$\phantom{AA}\mathcal{S}$S-matrixfree Lagrangian density and Wightman propagator
$\phantom{AA}\mathcal{R}$quantum Møller operatorfree Lagrangian density and Wightman propagator and interaction
###### Definition

(interacting field algebra)

Given an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable in degree 0

$S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ] \,,$
$PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \otimes PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \overset{ \star_{int} }{\longrightarrow} PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]$

is the conjugation of the Wick algebra-structure by the quantum Møller operator (def. ):

$A_1 \star_{int} A_2 \;\coloneqq\; \mathcal{R} \left( \mathcal{R}^{-1}(A_1) \star_H \mathcal{R}^{-1}(A_2) \right)$

### Interacting quantum BV-differential

Recall how the global BV-differential

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

on regular polynomial observables (this def.) is conjugated into the time-ordered product via the time ordering operator $\mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-}$ (this prop.).

In the same way we may use the quantum Møller operators to conjugate the BV-differential into the regular part of the interacting field algebra of observables:

###### Definition

(interacting quantum BV-differential)

Given an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable $S_{int}$, then the interacting quantum BV-differential on the interacting field algebra (def. ) on regular polynomial observables is the conjugation of the plain BV-differential $\{-S',-\}$ by the quantum Møller operator induced by $S_{int}$ (def. ):

$\mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1} \;\colon\; PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ] \,.$
###### Proposition

(quantum master equation and quantum master Ward identity on regular polynomial observables)

Consider an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable in degree 0

$S_{int} \;\in\; PolyObs(E_{\text{BV-BRST}})_{{reg} \atop {deg = 0}}[ [\hbar] ] \,,$

Then the following are equivalent:

1. The quantum master equation (QME)

(1)$\tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}} + i \hbar \Delta_{BV}( S' + S_{int} ) \;=\; 0 \,.$
2. The perturbative S-matrix (def. ) is $BV$-closed

$\{-S', \mathcal{S}(S_{int})\} = 0 \,.$
3. The quantum master Ward identity (MWI) on regular polynomial observables in terms of retarded products:

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

expressing the interacting quantum BV-differential (def. ) as the sum of the time-ordered antibracket (this def.) with the total action functional $S' + S_{int}$ and $i \hbar$ times the BV-operator (BV-operator).

4. The quantum master Ward identity (MWI) on regular polynomial observables in terms of time-ordered products:

(3)$\mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F (-)\} \;=\; - \left( \left\{ S' + S_{int} \,,\, (-) \right\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)$

(Rejzner 11, (5.35) - (5.38), following Hollands 07, (342)-(345))

###### Proof

To see that the first two conditions are equivalent, we compute as follows

(4)\begin{aligned} \left\{ -S', \mathcal{S}(S_{int}) \right\} & = \left\{ -S' , \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right\} \\ & = \underset{ { \tfrac{-1}{i \hbar} \{S',S\}_{\mathcal{T}} } \atop { \star_F \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) } }{ \underbrace{ \left\{ -S' , \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right\}_{\mathcal{T}} } } - i \hbar \underset{ { \left( \tfrac{1}{i \hbar} \Delta_{BV}(S_{int}) + \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\}_{\mathcal{T}} \right) } \atop { \star_{F} \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) } }{ \underbrace{ \Delta_{BV} \left( \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \right) } } \\ & = \tfrac{-1}{i \hbar} \underset{ \text{QME} }{ \underbrace{ \left( \{S',S_{int}\} + \tfrac{1}{2}\{S_{int}, S_{int}\} + i \hbar \Delta_{BV}(S_{int}) \right) } } \star_F \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \end{aligned}

Here in the first step we used the definition of the BV-operator (this def.) to rewrite the plain antibracket in terms of the time-ordered antibracket (this def.), then under the second brace we used that the time-ordered antibracket is the failure of the BV-operator to be a derivation (this prop) and under the first brace the consequence of this statement for application to exponentials (this example). Finally we collected terms, and to “complete the square” we added the terms on the left of

$\frac{1}{2} \underset{= 0}{\underbrace{\{S', S'\}_{\mathcal{T}}}} - i \hbar \underset{ = 0}{\underbrace{ \Delta_{BV}(S')}} = 0$

which vanish because, by definition of gauge fixing (this def.), the free gauge-fixed action functional $S'$ is independent of antifields.

But since the operation $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right)$ has the inverse $(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int} \right)$, this implies the claim.

Next we show that the quantum master equation implies the quantum master Ward identities.

We use that the BV-differential $\{-S',-\}$ is a derivation of the Wick algebra product $\star_H$ (lemma ).

First of all this implies that with $\{-S', \mathcal{S}(S_{int})\} = 0$ also $\{-S', \mathcal{S}(S_{int})^{-1}\} = 0$.

Thus we compute as follows:

\begin{aligned} \{-S', -\} \circ \mathcal{R}^{-1}(A) & = \{-S', \mathcal{R}^{-1}(A)\} \\ & = \left\{ { \, \atop \, } -S', \mathcal{S}(S_{int})^{-1} \star_H \left( \mathcal{S}(S_{int}) \star_F a \right) {\, \atop \,} \right\} \\ & = \phantom{+} \underset{ = 0 }{ \underbrace{ \left\{ -S', \mathcal{S}(S_{int})^{-1} \right\} } } \star_H \left( \mathcal{S}(S_{int}) \star_F A \right) \\ & \phantom{=} + \mathcal{S}(S_{int})^{-1} \star_H \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} \\ & = \mathcal{S}(S_{int})^{-1} \star_H \left( \underset{ = 1 }{ \underbrace{ \mathcal{S}(+ S_{int}) \star_F \mathcal{S}(- S_{int}) } } \star_F \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} \right) \\ & = \mathcal{S}(S_{int})^{-1} \star_H \left( \mathcal{S}(+ S_{int}) \star_F \underset{ (\ast) }{ \underbrace{ \mathcal{S}(- S_{int}) \star_F \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} } } \right) \\ & = \mathcal{R}^{-1} \left( \underset{ (\ast) }{ \underbrace{ \phantom{\, \atop \,} \mathcal{S}(-S_{int}) \star_F \left\{ -S', \mathcal{S}(S_{int}) \star_F A \right\} } } \right) \end{aligned}

By applying $\mathcal{R}$ to both sides of this equation, this means first of all that the interacting quantum BV-differential is equivalently given by

$\mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1} \;=\; \mathcal{S}(-S_{int}) \star_F \{-S', \mathcal{S}(S_{int}) \star_F (-)\} \,,$

hence that if either version (2) or (3) of the master Ward identity holds, it implies the other.

Now expanding out the definition of $\mathcal{S}$ (def. ) and expressing $\{-S',-\}$ via the time-ordered antibracket (this def.) and the BV-operator $\Delta_{BV}$ (this prop.) as

$\{-S',-\} \;=\; \{-S',-\}_{\mathcal{T}} - i \hbar \Delta_{BV}$

(on regular polynomial observables), we continue computing as follows:

(5)\begin{aligned} & \mathcal{R} \circ \{-S', (-)\} \circ \mathcal{R}^{-1}( A ) \\ & = \exp_{\mathcal{T}} \left( \tfrac{-1}{i \hbar} S_{int} \right) \star_F \left\{ -S', \exp_{\mathcal{T}} \left( \tfrac{1}{i \hbar} S_{int} \right) \star_F A \right\} \\ & = \exp_{\mathcal{T}} \left( \tfrac{-1}{i \hbar} S_{int} \right) \star_F \left( \left\{ -S', \exp_{\mathcal{T}} \left( \tfrac{ 1 }{i \hbar} S_{int} \right) \star_F A \right\}_{\mathcal{T}} - i \hbar \Delta_{BV} \left( \exp_{\mathcal{T}} \left( \tfrac{ 1 }{i \hbar} S_{int} \right) \star_F A \right) \right) \\ & \phantom{+} = \tfrac{1}{i \hbar} \{ -S', S_{int} \}_{\mathcal{T}} \star_F A + \{-S', A\}_{\mathcal{T}} \\ & \phantom{=} - i \hbar \exp_{\mathcal{T}}\left( \tfrac{-1}{i \hbar} S_{int}\right) \star_F \left( \underset{ { \left( \tfrac{1}{i \hbar}\Delta_{BV}(S_{int}) + \tfrac{1}{2 (i \hbar)^2} \left\{ S_{int}, S_{int} \right\} \right) } \atop { \star_F \exp_{\mathcal{T}}\left( \tfrac{ 1 }{i \hbar} S_{int} \right) } }{ \underbrace{ \Delta_{BV} \left( \exp_{\mathcal{T}} \left( \tfrac{ 1}{i \hbar} S_{int} \right) \right) } } \star_F A \,+\, \exp_{\mathcal{T}} \left( \tfrac{ 1}{i \hbar} S_{int} \right) \star_F \Delta_{BV}(A) \,+\, \underset{ { \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right) } \atop { \star_F \tfrac{ 1}{i \hbar} \{S_{int}, A\} } }{ \underbrace{ \left\{ \exp_{\mathcal{T}} \left( \tfrac{ 1}{i \hbar} S_{int} \right) \,,\, A \right\}_{\mathcal{T}} } } \right) \\ & = - \left( \{ S' + S_{int}\,,\, A\}_{\mathcal{T}} + i \hbar \Delta_{BV}(A) \right) \\ & \phantom{=} - \tfrac{1}{i \hbar} \underset{ \text{QME} }{ \underbrace{ \left( \tfrac{1}{2} \{ S' + S_{int}, S' + S_{int} \}_{\mathcal{T}} + i \hbar \Delta_{BV}( S' + S_{int} ) \right) }} \star_F A \\ & = - \left( \{ S' + S_{int}\,,\, A\}_{\mathcal{T}} + i \hbar \Delta_{BV}(A) \right) \end{aligned}

Here in the line with the braces we used that the BV-operator is a derivation of the time-ordered product up to correction by the time-ordered antibracket (this prop.), and under the first brace we used the effect of that property on time-ordered exponentials (this example), while under the second brace we used that $\{(-),A\}_{\mathcal{T}}$ is a derivation of the time-ordered product. Finally we have collected terms, added $0 = \{S',S'\} + i \hbar \Delta_{BV}(S')$ as before, and then used the QME.

This shows that the quantum master Ward identities follow from the quantum master equation. To conclude, it is now sufficient to show that, conversely, the MWI in terms of, say, retarded products implies the QME.

To see this, observe that with the BV-differential being nilpotent, also its conjugation by $\mathcal{R}$ is, so that with the above we have:

\begin{aligned} & \left( \{-S',-\}\right)^2 = 0 \\ \Leftrightarrow \; & \left( \mathcal{R} \circ \{-S',(-)\} \circ \mathcal{R}^{-1} \right)^2 = 0 \\ \Leftrightarrow \; & \underset{ \left\{ {\, \atop \,} \tfrac{1}{2}\{S' + S_{int}, S' + S_{int}\}_{\mathcal{T}} + i \hbar \Delta_{BV}(S' + S_{int}) \,,\, (-) \right\} }{ \underbrace{ \left( \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)^2 } } = 0 \end{aligned}

Here under the brace we computed as follows:

\begin{aligned} \left( \{S' + S_{int}, (-)\}_{\mathcal{T}} + i \hbar \Delta_{BV} \right)^2 & = \phantom{+} \underset{ \tfrac{1}{2} \{ \{S' + S, S'+ S\}_{\mathcal{T}}, (-) \}_{\mathcal{T}} }{ \underbrace{ \{S' + S_{int}, \{S' + S_{int}\}_{\mathcal{T}}, (-) \}_{\mathcal{T}} }} \\ & \phantom{=} + i \hbar \underset{ \{ \Delta_{BV}(S'+ S)\,,\, (-) \}_{\mathcal{T}} }{ \underbrace{ \left( \{S' + S_{int}, (-)\}_{\mathcal{T}} \circ \Delta_{BV} + \Delta_{BV} \circ \{S' + S_{int}, (-)\}_{\mathcal{T}} \right) }} \\ & \phantom{=} + (i \hbar)^2 \underset{= 0} { \underbrace{ \Delta_{BV} \circ \Delta_{BV} } } \end{aligned} \,.

where, in turn, the term under the first brace follows by the graded Jacobi identity, the one under the second brace by Henneaux-Teitelboim (15.105c) and the one under the third brace by Henneaux-Teitelboim (15.105b).

###### Example

(classical master Ward identity)

The classical limit $\hbar \to 0$ of the quantum master Ward identity (2) is

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

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 becomes

\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}

In this form the classical Master Ward identity was originally identified in (Dütsch-Fredenhagen 02, (90), Brennecke-Dütsch 07, (5.5), following Dütsch-Boas 02).

### Renormalization and Master ward identity

The quantum master equation in the form of prop. is derived on regular polynomial observables, in particular hence for non-point-interaction action functionals $S_{int}$. But the interaction terms of interest are point-interactions, hence are local observables. The extension of the time-ordered product and hence of the perturbative S-matrix from regular to local onservables exsists but involves choices, these are the renormalization choices in the formulation of causal perturbation theory.

Since for gauged fixed gauge theories this physically relevant observables are not the plain (mcirocausal) polynomial observables, but the cochain cohomology of the BV-BRST differential on them, one needs to require for gauge theories that the quantum master equation still holds after renormalization. This is closely related to the renormalization condition called the master Ward identity (Rejzner 11 (prop. 5.3.1) and following paragraphs). If the quantum master equation cannot be retained in renormalization one says that the field theory suffers from a quantum anomaly.

## References

The concept originates with