quantum master equation (changes) in nLab

Context

Algebraic Quantum Field Theory

• Idea
• In causal perturbation theory
  • Background
  • Interacting quantum BV-differential
  • Renormalization and Master ward identity
• Related concepts
• References

Lemma

(global BV-differential is derivation on Wick algebra)

The global BV-differential

{−S′,−}:PolyObs(E BV-BRST) reg[[ℏ]]⟶PolyObs(E BV-BRST) reg[[ℏ]] \{-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 ⋆ H\star_H:

{−S′,A 1⋆ HA 2}={−S′,A 1}⋆ HA 2+A 1⋆ H{−S′,A 2}. \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

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

given by

𝒮(S int)=exp ⋆ F(1iℏS int))≔1+1iℏS int+121(iℏ) 2S int⋆ FS int+⋯. \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 intS_{int} here as an adiabatically switched non-point-interaction action functional.

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

𝒮(S int) −1⋆ H𝒮(S int)=1. \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 ⋆ F\star_F, which is 𝒮(−S int)\mathcal{S}(-S_{int}):

𝒮(−S int)⋆ F𝒮(S int)=1. \mathcal{S}(-S_{int}) \star_F \mathcal{S}(S_{int}) = 1 \,.

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∈PolyObs(E BV-BRST) regdeg=0[[ℏ]], 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)12{S′+S int,S′+S int} 𝒯+iℏΔ BV(S′+S int)=0. \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. 1) is BVBV-closed

   {−S′,𝒮(S int)}=0. \{-S', \mathcal{S}(S_{int})\} = 0 \,.

3. The quantum master Ward identity (MWI) on regular polynomial observables in terms of retarded products:

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

   (Dütsch 18, (4.2))

   expressing the interacting quantum BV-differential (def. 4) as the sum of the time-ordered antibracket (this def.) with the total action functional S′+S intS' + S_{int} and iℏi \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)𝒮(−S int)⋆ F{−S′,𝒮(S int)⋆ F(−)}=−({S′+S int,(−)} 𝒯+iℏΔ BV) \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)

   (Dütsch 18, (4.8))

Proof

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

(4){−S′,𝒮(S int)} ={−S′,exp 𝒯(1iℏS int)} ={−S′,exp 𝒯(1iℏS int)} 𝒯⏟−1iℏ{S′,S} 𝒯⋆ Fexp 𝒯(1iℏS int)−iℏΔ BV(exp 𝒯(1iℏS int))⏟(1iℏΔ BV(S int)+12(iℏ) 2{S int,S int} 𝒯)⋆ Fexp 𝒯(1iℏS int) =−1iℏ({S′,S int}+12{S int,S int}+iℏΔ BV(S int))⏟QME⋆ Fexp 𝒯(1iℏS int) \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

12{S′,S′} 𝒯⏟=0−iℏΔ BV(S′)⏟=0=0 \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′S' is independent of antifields.

But since the operation (−)⋆ Fexp 𝒯(1iℏS int)(-) \star_F \exp_{\mathcal{T}}\left( \tfrac{1}{i \hbar} S_{int} \right) has the inverse (−)⋆ Fexp 𝒯(−1iℏS int)(-) \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′,−}\{-S',-\} is a derivation of the Wick algebra product ⋆ H\star_H (lemma 1).

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

Thus we compute as follows:

{−S′,−}∘ℛ −1(A) ={−S′,ℛ −1(A)} ={−S′,𝒮(S int) −1⋆ H(𝒮(S int)⋆ Fa)} =+{−S′,𝒮(S int) −1}⏟=0⋆ H(𝒮(S int)⋆ FA) =+𝒮(S int) −1⋆ H{−S′,𝒮(S int)⋆ FA} =𝒮(S int) −1⋆ H(𝒮(+S int)⋆ F𝒮(−S int)⏟=1⋆ F{−S′,𝒮(S int)⋆ FA}) =𝒮(S int) −1⋆ H(𝒮(+S int)⋆ F𝒮(−S int)⋆ F{−S′,𝒮(S int)⋆ FA}⏟(*)) =ℛ −1(𝒮(−S int)⋆ F{−S′,𝒮(S int)⋆ FA}⏟(*)) \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 ℛ\mathcal{R} to both sides of this equation, this means first of all that the interacting quantum BV-differential is equivalently given by

ℛ∘{−S′,(−)}∘ℛ −1=𝒮(−S int)⋆ F{−S′,𝒮(S int)⋆ F(−)}, \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 𝒮\mathcal{S} (def. 1) and expressing {−S′,−}\{-S',-\} via the time-ordered antibracket (this def.) and the BV-operator Δ BV\Delta_{BV} (this prop.) as

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

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

(5) ℛ∘{−S′,(−)}∘ℛ −1(A) =exp 𝒯(−1iℏS int)⋆ F{−S′,exp 𝒯(1iℏS int)⋆ FA} =exp 𝒯(−1iℏS int)⋆ F({−S′,exp 𝒯(1iℏS int)⋆ FA} 𝒯−iℏΔ BV(exp 𝒯(1iℏS int)⋆ FA)) +=1iℏ{−S′,S int} 𝒯⋆ FA+{−S′,A} 𝒯 =−iℏexp 𝒯(−1iℏS int)⋆ F(Δ BV(exp 𝒯(1iℏS int))⏟(1iℏΔ BV(S int)+12(iℏ) 2{S int,S int})⋆ Fexp 𝒯(1iℏS int)⋆ FA+exp 𝒯(1iℏS int)⋆ FΔ BV(A)+{exp 𝒯(1iℏS int),A} 𝒯⏟exp 𝒯(1iℏS int)⋆ F1iℏ{S int,A}) =−({S′+S int,A} 𝒯+iℏΔ BV(A)) =−1iℏ(12{S′+S int,S′+S int} 𝒯+iℏΔ BV(S′+S int))⏟QME⋆ FA =−({S′+S int,A} 𝒯+iℏΔ BV(A)) \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} 𝒯\{(-),A\}_{\mathcal{T}} is a derivation of the time-ordered product. Finally we have collected terms, added 0={S′,S′}+iℏΔ BV(S′)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 ℛ\mathcal{R} is, so that with the above we have:

({−S′,−}) 2=0 ⇔ (ℛ∘{−S′,(−)}∘ℛ −1) 2=0 ⇔ ({S′+S int,(−)} 𝒯+iℏΔ BV) 2⏟{12{S′+S int,S′+S int} 𝒯+iℏΔ BV(S′+S int),(−)}=0 \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:

({S′+S int,(−)} 𝒯+iℏΔ BV) 2 =+{S′+S int,{S′+S int} 𝒯,(−)} 𝒯⏟12{{S′+S,S′+S} 𝒯,(−)} 𝒯 =+iℏ({S′+S int,(−)} 𝒯∘Δ BV+Δ BV∘{S′+S int,(−)} 𝒯)⏟{Δ BV(S′+S),(−)} 𝒯 =+(iℏ) 2Δ BV∘Δ BV⏟=0. \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 ℏ→0\hbar \to 0 of the quantum master Ward identity (2) is

{−S′,(−)}∘ℛ −1=−ℛ −1({S′+S int,(−)}). \{-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=∫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 becomes

0 ={−S′,ℛ −1(A)}+ℛ −1({S′+S int,A} 𝒯) =∫ΣδS′δΦ a(x)ℛ −1(A a(x))dvol Σ(x)+ℛ −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}

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).