algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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 “”:
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 (Planck's constant) by the BV-operator :
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.
Throughout, let be a gauge fixed free Lagrangian field theory with global BV-differential (this def.)
Hence denotes the gauge fixed free action functional.
Moreover, let be a compatible choice of Wightman propagator with associated Feynman propagator .
(global BV-differential is derivation on Wick algebra)
The global BV-differential
is a derivation also with respect to the Wick algebra star product :
(Fredenhagen-Rejzner 11b, below (37), Rejzner 11, below (5.28)) For proof see this prop
(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
given by
We think of here as an adiabatically switched non-point-interaction action functional.
We write for the inverse with respect to the Wick product (which exists by this remark)
Notice that this is in general different form the inverse with respect to the time-ordered product , which is :
(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
then the corresponding quantum Møller operator on regular polynomial observables
is given by the derivative of Bogoliubov's formula
where is the perturbative S-matrix from def. .
This indeed lands in formal power series in Planck's constant (by this remark), instead of in more general Laurent series as the perturbative S-matrix does (def. ).
Hence the inverse map is
(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 in order to bring out the analogy to (the conventions for the) time-ordered product on regular polynomial observables.
notice the implicit dependencies
endomorphism of regular polynomial observables | meaning | depends on choice of |
---|---|---|
time-ordering | free Lagrangian density and Wightman propagator | |
S-matrix | free Lagrangian density and Wightman propagator | |
quantum Møller operator | free Lagrangian density and Wightman propagator and interaction |
Given an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable in degree 0
then the interacting field algebra structure on regular polynomial observables
is the conjugation of the Wick algebra-structure by the quantum Møller operator (def. ):
(e.g. Fredenhagen-Rejzner 11b, (19))
Recall how the global BV-differential
on regular polynomial observables (this def.) is conjugated into the time-ordered product via the time ordering operator (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:
(interacting quantum BV-differential)
Given an adiabatically switched non-point-interaction action functional in the form of a regular polynomial observable , then the interacting quantum BV-differential on the interacting field algebra (def. ) on regular polynomial observables is the conjugation of the plain BV-differential by the quantum Møller operator induced by (def. ):
(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
Then the following are equivalent:
The quantum master equation (QME)
The perturbative S-matrix (def. ) is -closed
The quantum master Ward identity (MWI) on regular polynomial observables in terms of retarded products:
expressing the interacting quantum BV-differential (def. ) as the sum of the time-ordered antibracket (this def.) with the total action functional and times the BV-operator (BV-operator).
The quantum master Ward identity (MWI) on regular polynomial observables in terms of time-ordered products:
(Rejzner 11, (5.35) - (5.38), following Hollands 07, (342)-(345))
To see that the first two conditions are equivalent, we compute as follows
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
which vanish because, by definition of gauge fixing (this def.), the free gauge-fixed action functional is independent of antifields.
But since the operation has the inverse , this implies the claim.
Next we show that the quantum master equation implies the quantum master Ward identities.
We use that the BV-differential is a derivation of the Wick algebra product (lemma ).
First of all this implies that with also .
Thus we compute as follows:
By applying to both sides of this equation, this means first of all that the interacting quantum BV-differential is equivalently given by
hence that if either version (2) or (3) of the master Ward identity holds, it implies the other.
Now expanding out the definition of (def. ) and expressing via the time-ordered antibracket (this def.) and the BV-operator (this prop.) as
(on regular polynomial observables), we continue computing as follows:
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 is a derivation of the time-ordered product. Finally we have collected terms, added 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 is, so that with the above we have:
Here under the brace we computed as follows:
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).
(classical master Ward identity)
The classical limit of the quantum master Ward identity (2) is
Applied to an observable which is linear in the antifields
this becomes
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).
The quantum master equation in the form of prop. is derived on regular polynomial observables, in particular hence for non-point-interaction action functionals . 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.
The concept originates with
Traditional review includes
Discussion in the rigorous context of relativistic perturbative QFT formulated in causal perturbation theory/perturbative AQFT is in:
Klaus Fredenhagen, Kasia Rejzner, Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun. Math. Phys. 314(1), 93–127 (2012) (arXiv:1101.5112)
Klaus Fredenhagen, Kasia Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys. 317(3), 697–725 (2012) (arXiv:1110.5232)
Katarzyna Rejzner, Batalin-Vilkovisky formalism in locally covariant field theory (arXiv:1111.5130)
Katarzyna Rejzner, Remarks on local symmetry invariance in perturbative algebraic quantum field theory (arXiv:1301.7037)
and surveyed in
Last revised on September 20, 2021 at 16:12:14. See the history of this page for a list of all contributions to it.