Batalin-Vilkovisky formalism in the functional approach to classical field theory

Abstract

We develop the Batalin-Vilkovisky formalism for classical field theory on
generic globally hyperbolic spacetimes. A crucial aspect of our treatment is
the incorporation of the principle of local covariance which amounts to
formulate the theory without reference to a distinguished spacetime. In
particular, this allows a homological construction of the Poisson algebra of
observables in classical gravity. Our methods heavily rely on the differential
geometry of configuration spaces of classical fields.

Full text

Page 1

arXiv:1101.5112v4 [math-ph] 25 Feb 2011
Batalin-Vilkovisky formalism in the functional approach to
classical field theory
Klaus Fredenhagen, Katarzyna Rejzner
II Inst. f. Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany
klaus.fredenhagen@desy.de,katarzyna.rejzner@desy.de
February 28, 2011
Abstract
We develop the Batalin-Vilkovisky formalism for classical field theory on generic
globally hyperbolic spacetimes. A crucial aspect of our treatment is the incorpo-
ration of the principle of local covariance which amounts to formulate the theory
without reference to a distinguished spacetime. In particular, this allows a homo-
logical construction of the Poisson algebra of observables in classical gravity. Our
methods heavily rely on the differential geometry of configuration spaces of classical
fields.
Contents
1Introduction2
2The BV formalism for the scalar field3
3The BV formalism for the Yang-Mills theories
3.1 Geometrical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2Chevalley-Eilenberg complex. . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Koszul-Tate resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Classical master equation. . . . . . . . . . . . . . . . . . . . . . . . . . .
9
10
12
13
15
4Gauge fixing
4.1 Nonminimal sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2Gauge fixing for the Yang-Mills theory . . . . . . . . . . . . . . . . . . . .
17
17
18
5 Peierls bracket20
1

Page 2

6 General relativity
6.1 BV construction on a fixed background . . . . . . . . . . . . . . . . . . . .
6.2 BV construction for natural transformations . . . . . . . . . . . . . . . . .
24
24
25
7Conclusions 29
A Appendix
A.1 Calculus on locally convex vector spaces . . . . . . . . . . . . . . . . . . .
A.2 Topologies and completions . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Distributions with values in a locally convex vector space
30
30
32
36. . . . . . . . .
1Introduction
Classical field theory is an infinite dimensional generalization of classical mechanics,
Therefore the transfer of structures like Lagrangians, Hamiltonians, Poisson brackets etc.
involve problems of infinite dimensional analysis. To a certain extent these problems can
be circumvented by exploiting the locality principle of classical field theory and using as
variables finitely many derivatives of the field at a given point. A particular elegant way
of doing this is multisymplectic analysis [52, 39, 20].
A drawback of this approach is that the connection to quantum field theory is not
easy (for an attempt see [50, 51]). Other approaches use the path integral formalism of
quantum field theory. There it is extremely difficult to arrive at rigorous results.
A new way of formulating perturbative quantum field theory which combines aspects
of canonical quantization with aspects from the path integral formalism is perturbative
algebraic quantum field theory [17, 27, 29, 28]. It inherits from the canonical approach the
emphasis on algebraic structures and from the path integral formalism the insight that all
possible field configurations have to be considered, not only those which are solutions of
the field equations. This approach goes back to old ideas of Schwinger, Stückelberg and
Bogoliubov, was made mathematically rigorous by the method of causal perturbation
theory (Epstein-Glaser renormalization) and was mainly developed in order to treat
quantum field theory on generic curved spacetimes in agreement with the principle of
general covariance. The ? = 0 limit then is classical field theory [17] where the observables
are functionals of generic field configurations and the Poisson bracket is a functional bi-
differential operator. The general method was mainly developed on the example of a
scalar field.
A generalization to gauge theories was performed in several steps [26, 29], culmi-
nating in the work of Hollands [46] on quantization of Yang-Mills theories on curved
backgrounds. There it was observed that the Batalin-Vilkovisky formalism [6, 7, 8, 9] is
most suitable for the treatment of gauge theories.
The Batalin-Vilkovisky (BV) formalism (previously and independently introduced by
Zinn-Justin [83]) generalizes the BRST approach [10, 11]. In the literature it is also called
the antifield formalism. It is usually used as a covariant method to perform the gauge-
fixing in quantum field theory, but was also applied to other problems like analyzing
2

Page 3

possible deformations of the action and anomalies. For a review on BRST see [5], where
the cohomological interpretation of the antifield formalism is especially stressed. It was
already recognized in [41] (see also [42]) that the Koszul-Tate complex plays a very
important role in the antifield formalism and the methods of homological algebra can be
successfully applied. The book [41] shows also the geometrical interpretation of the BV
complex in both the Hamiltonian and the Lagrangian setting, but the mathematically
rigorous results are formulated only when the configuration space is finite dimensional. A
generalization to infinite dimensional spaces is usually achieved in the jet space formalism
[5], together with the variational bicomplex [70, 71, 72]. This formulation is sometimes
more convenient for calculation but it is hard to see the general structure behind it.
Furthermore it is usually applied only to the local functionals of the configuration fields
and in quantum field theory one needs more singular objects (see for example [17]).
Most of the mathematically rigorous versions of the BV formalism (see, e.g., [77,
35, 1]) typically use assumptions which are not satisfied by the examples of interest in
physics, e.g. that the underlying spacetime is compact (this is never true for a globally
hyperbolic spacetime), or that the field configurations of interest have compact support
(this excludes solutions of hyperbolic field equations), or spacetime is even replaced by a
finite set in order to make the path integral well defined. Moreover the formalism looks
like a formal recipe where a lot of additional structure is introduced (ghosts, antifields,...)
whose conceptual and mathematical status is often unclear. A more conscious approach,
with a due attention to the infinite dimensional nature of the problem was proposed in
the notes [23] but the formalism is still not complete.
In the present paper we attempt to provide a conceptually convincing and mathemat-
ically rigorous version of the BV formalism for classical field theory on globally hyperbolic
spacetimes. Our construction is based on the principle of local covariance [16]. According
to this principle one does no longer try to construct a theory on a given spacetime. In-
stead all constructions have to be performed simultaneously on all spacetimes of a given
class in a coherent way. This amounts to a functorial relation between spacetimes and
the associated Poisson algebras of observables. We hope that our formulation improves
the understanding of classical field theory (see also [18]), but our main goal is to use
it as a basis for quantum field theory, in particular for quantum gravity as proposed in
[33, 14].
2 The BV formalism for the scalar field
Before entering the more complicated realm of gauge theories, we want to illustrate the
basic structures on the example of a scalar field.
Let E(M) be the space of smooth functions on the smooth manifold M, equipped
with the locally convex topology of convergence of all derivatives, uniformly on compact
sets. Let χ : M → N be an embedding and let χ∗be the pullback, χ∗ϕ = ϕ ◦ χ for
ϕ ∈ E(M). Then E can be understood as a contravariant functor from the category Loc
of time-oriented globally hyperbolic spacetimes and with causal isometric embeddings as
morphisms to the category Vec of locally convex vector spaces (lcvs) with continuous
3

Page 4

injective linear maps as arrows. The embedding χ is associated to the pullback χ∗,
Eχ = χ∗. We interpret E(M) as the space of field configurations.1
Another functor between these categories is the functor which associates to a manifold
the space D(M) of compactly supported test functions with its standard topology. This
functor is covariant with respect to push forwards Dχ = χ∗,
?
0
χ∗f(x) =
f(χ−1(x)),
,
x ∈ χ(M),
else.
The observables of the theory are functions F : E(M) → R. They are supposed to
be smooth in the following sense: For each n ∈ N and each ϕ ∈ E(M) there exists a
distributional density F(n)(ϕ) with compact support on Mn. F(n)(ϕ) is symmetric under
permutations of the n arguments from M and satisfies
dn
dλnF(ϕ + λψ) ↾λ=0= ?F(n)(ϕ),ψ⊗n? .
Moreover, the map E(M)×E(M) ∋ (ϕ,ψ) ?→ ?F(n)(ϕ),ψ⊗n? is required to be continuous.
Since E(M) is a nuclear Fréchet space, this notion coincides with the standard definition
of smooth maps between locally convex vector spaces (see the Appendix A.1 for details).
The support of a function F on configuration space is defined as the set of points
x ∈ M such that F depends on the field configuration in any neighbourhood of x
suppF = {x ∈ M|∀ neighbourhoods U of x ∃ϕ,ψ ∈ E(M),suppψ ⊂ U
such that F(ϕ + ψ) ?= F(ϕ)} .
A very important property of functionals in the context of classical field theory is locality.
According to the standard definition it means that the functional F is of the form:
?
M
(1)
F(ϕ) =
f(jx(ϕ))dµ(x),
where f is a function on the jet space over M and jx(ϕ) = (x,ϕ(x),∂ϕ(x),... ) is the jet
of ϕ at the point x. It was already recognized in [28, 17, 18] in the context of perturbative
algebraic quantum field theory that the property of locality can be reformulated using the
notion of additivity of a functional. The concept itself dates back to the works of Chacón
and Friedman [19] and even to earlier ones as seen from the survey of Rao [66]. We say
that F is additive if for all fields ϕ1,ϕ2,ϕ3∈ E(M) such that supp(ϕ1) ∩ supp(ϕ3) = ∅
we have:
F(ϕ1+ ϕ2+ ϕ3) = F(ϕ1+ ϕ2) − F(ϕ2) + F(ϕ2+ ϕ3).
One can show that a smooth compactly supported functional is local if it is additive and
the wave front sets of its derivatives are orthogonal to the tangent bundles of the thin
(2)
1In view of the expected properties of the path integral one might conjecture that field configurations
which are not smooth will play an important role. Fortunately this is not the case in a perturbative
framework. Since our observables are functions of field configurations, an extension to more general
configurations would mean that we had to distinguish functions which coincide on smooth configurations.
4

Page 5

diagonals ∆k(M)
bundles of Mk:
.=
?(x,...,x) ∈ Mk: x ∈ M?, considered as subsets of the tangent
WF(F(k)(ϕ)) ⊥ T∆k(M).
In particular F(1)(ϕ) is a smooth section for each fixed ϕ. This property turns out to be
crucial in the context of the BV formalism. We shall come back to it later on.
The space of compactly supported smooth local functions F : E(M) → R is denoted
by Floc(M). The algebraic completion of Floc(M) with respect to the pointwise product
(3)
F · G(ϕ) = F(ϕ)G(ϕ),
(4)
is a commutative algebra F(M) consisting of finite sums of finite products of local func-
tionals. F becomes a (covariant) functor by setting Fχ(F) = F ◦ Eχ, i.e. Fχ(F)(ϕ) =
F(ϕ ◦ χ). Later we will enlarge these algebras by admitting more singular functionals
(Section 5 and Appendix A.2).
In the framework of infinite dimensional differential geometry one models manifolds
on general locally convex vector spaces.Many of the results known from the finite
dimensional case generalize to this setting. One can define vector fields, differential forms
and exterior derivative. We provide precise definitions in the Appendix A.1. For further
reference see for example [61, 56]. In particular a locally convex vector space E(M) is a
trivial manifold. General vector fields are derivations of the algebra of smooth functions
C∞(E(M)). Since we restricted our considerations to its subalgebra F(M), we have to
identify a submodule of vector fields that correspond to derivations of F(M). Moreover
we want to associate this module to M in a functorial way. These two requirements
already determine the class of vector fields we want to consider.
First we note that vector fields X on E(M) can be considered as smooth maps from
E(M) to E(M), since the tangent spaces of a vector space can naturally be identified
with the space itself. We restrict ourselves to smooth compactly supported maps X with
image in D(M). Moreover we require the wave front set condition similar to (3) but
formulated for distributions with values in D(M) (for a detailed discussion see [67] and
the Appendix A.3). Vector fields with these properties act on F(M) as derivations,
∂XF(ϕ) := ?F(1)(ϕ),X(ϕ)?.
(5)
The spacetime support of a vector field X is defined in the following way:
suppX = {x ∈ M|∀ neigh. U of x ∃F ∈ F(M),suppF ⊂ U such that ∂XF ?= 0
or ∃ ϕ,ψ ∈ E(M),suppψ ⊂ U such that X(ϕ + ψ) ?= X(ϕ)} .
The space of vector fields with compact support and properties mentioned above can
be completed to an F(M)-module (with respect to the pointwise product (4)) which is
denoted by V(M). V becomes a (covariant) functor by setting
(6)
Vχ(X) = Dχ ◦ X ◦ Eχ .
(7)
5