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Covariant phase space, constraints, gauge and the Peierls formula

It is well known that both the symplectic structure and the Poisson brackets of classical field theory can be constructed directly from the Lagrangian in a covariant way, without passing through the non-covariant canonical Hamiltonian formalism. This is true even in the presence of constraints and gauge symmetries. These constructions go under the names of the covariant phase space formalism and the Peierls bracket. We review both of them, paying more careful attention, than usual, to the precise mathematical hypotheses that they require, illustrating them in examples. Also an extensive historical overview of the development of these constructions is provided. The novel aspect of our presentation is a significant expansion and generalization of an elegant and quite recent argument by Forger&Romero showing the equivalence between the resulting symplectic and Poisson structures without passing through the canonical Hamiltonian formalism as an intermediary. We generalize it to cover theories with constraints and gauge symmetries and formulate precise sufficient conditions under which the argument holds. These conditions include a local condition on the equations of motion that we call hyperbolizability, and some global conditions of cohomological nature. The details of our presentation may shed some light on subtle questions related to the Poisson structure of gauge theories and their quantization.


Introduction
A classical field theory is essentially defined by a local variational principle for a given set of dynamical fields on a given spacetime manifold. The variational principle determines a set of partial differential equations (PDEs), the equations of motion, to be emposed on the dynamical fields. The equations of motion are typically hyperbolic (or can be made so with gauge fixing if gauge invariance is present). What distinguishes variational PDEs among the more general class of hyperbolic PDEs is that their solution spaces can be naturally endowed with symplectic and hence Poisson structure, making it into a phase space. The algebra of smooth functions on the phase space with the corresponding Poisson bracket then constitutes the Poisson algebra of observables. The existence of this algebraic formulation is what allows for quantization. It is clear that the symplectic and Poisson structure on its phase space is a crucial ingredient in the description of a classical field theory. The most common way of building these structures is via the canonical formalism 1, 2 (sometimes known as the 3+1 Hamiltonian formalism, when spacetime is 4-dimensional), which requires an explicit choice of a time function, even when no one such choice is natural, and the application of a Legendre transform, which may be only ambiguously defined, for instance, in the presence of gauge invariance. However, it is also well known that both can be defined completely covariantly (that is, without choosing an explicit time function or applying a Legendre transform) directly from the Lagrangian, without going through the canonical formalism. These methods are known, respectively, as the covariant phase space formalism [3][4][5] and the Peierls bracket. 6,7 They are clearly preferable when the canonical formalism explicitly breaks some of the natural symmetries of the theory (any relativistic theory is an example). The symplectic 2-form and the Poisson bivector constructed in this way are equivalent (they are mutual inverses). Despite the covariance of both constructions, until recently, their equivalence was only known via the intermediary of the non-covariant canonical formalism. 8,9 That is, until Forger & Romero 10 provided an elegant, completely covariant proof of the equivalence in the case of a scalar field.
In this review, we describe in depth the constructions of symplectic and Poisson structures of classical field theory, as well as their equivalence, all in a covariant way. A novel contribution of our exposition is an extension of the Forger-Romero argument to field theories where constraints and gauge invariance are present. Another is that we do not restrict ourselves to the class of wave-like equations defined on Lorentzian manifolds (though that will be the main source of our examples). We also pay special attention to several aspects that are often omitted or left implicit in the existing literature. Neither the covariant phase space nor the Peierls constructions are automatic. That is, besides a given Lagrangian density, several conditions must be fulfilled for the corresponding formulas to make sense. We make these conditions explicit in both cases: existence of a Cauchy surface and spacelike compact support for solutions, in the construction of the symplectic structure, and existence of hyperbolic PDE system, with retarded and advanced Green functions, closely related to the equations of motions, in the construction of the Poisson structure. Furthermore, the statement of equivalence also requires a certain sufficient condition on the cohomological properties of the constraint and gauge generator differential operators.
Note that we restrict our attention to the geometric and algebraic aspects of the constructions and systematically avoid analytical details. In particular, the construction of the space of solutions of a PDE system on a spacetime of dimension greater than one (which corresponds to an ordinary mechanical system) requires a theory of infinite dimensional differential geometry. We treat the minimal amount of the needed infinite dimensional geometry in a formal way. On the other hand, an honest attempt to present the relevant functional analytical details can be found in Refs. 11-14. Our results could then form the core identities of a future investigation along similar lines that could extend them beyond the formal level.
In Sec. 2, we present some background material on Green functions and hyperbolic PDEs. There we define the notion of Green hyperbolicity that will be crucial in the later construction of the Peierls formula and the study of its properties. The main body of this review is contained in Sec. 3. There, we review the formal differential geometry of the (possibly infinite dimensional) solution spaces (Sec. 3.2). Also, we review the covariant phase space formalism (Secs. 3 5. In addition, some further needed background information is given in the appendices, which includes the jet bundle approach to PDEs, conservation laws and variational forms, as well as a generalization of causal structure on smooth manifolds beyond Lorentzian geometry. In particular, Appendix B sets up some notation that is used throughout the paper to describe PDE systems. Finally, we conclude this introductory section with an extensive (though still incomplete) historical overview of the literature on the covariant phase space formalism and the Peierls formula. This historical material may be safely skipped at first reading.

Historical overview
The canonical formalism in mechanics [15][16][17][18][19] (what we would now call the construction of the symplectic and Poisson structures on the phase space of a mechanical system) has a long history and is most closely associated with the names of Hamilton and Jacobi. Though, undoubtedly, its roots go back even to Lagrange. Its main features are (a) the identification of the phase space with the space of initial data and (b) the use of the Legendre transform to determine special coordinates on the phase space in which the symplectic form takes a certain canonical form (hence naming the formalism). Because of the relation of Poisson brackets to quantization, the problem of the quantization of fields in the early 20th century required the translation (see for instance Refs. 1,20) of the canonical formalism from mechanics to field theories (multiple independent variables instead of just one).
As already mentioned earlier, the main features of the canonical formalism quickly began to clash with the relativistic nature and spacetime covariance of field theories. That was already clear in the works of Rosenfeld 20 and Dirac. 1 The possibility that both of these unpleasant features could be avoided became realized very slowly and is still not in mainstream use by theoretical physicists. It appears that parts of it were rediscovered multiple times and it is hard to trace them to any one source. Below, we discuss some key references that made it clear that it is possible to construct the phase space itself, as well as its symplectic and Poisson structures in a fully covariant way, avoiding both features (a) and (b) of the canonical formalism.
Covariant phase space, constraints, gauge and the Peierls formula 5 An important figure in some of the developments described below is that of Souriau, despite lack of many explicit references to his work. Perhaps his importance is not surprising, since he was one of the people responsible for abstracting (in the 1960s) the modern notion of a symplectic manifold as the appropriate arena for mechanics. 19 In particular, the name of Souriau is closely associated to identifying the classical phase space with the set of solutions of the equations of motion, rather than the set of initial data. Also, in Souriau's book 19 can be found a construction of the symplectic structure on the phase space directly from the Lagrangian, which he attributes to Lagrange himself. 21 Even though his book treated only mechanical systems and not field theories, these ideas seem to have been rather influential.

Peierls formula
The covariant construction of Poisson brackets in field theory can in fact be traced to a single source: the seminal 1952 paper of Peierls. 6 In that paper, he introduced what is now known as the Peierls bracket, which we prefer to call the Peierls formula, as reviewed in Sec. 3.3.4. The formula for the causal Green function as the difference of the retarded and advanced Green functions, G = G + − G − , appeared there, though in a somewhat implicit form. It is likely that Peierls was guided by experience. Having seen the unequal time Poisson bracket (or rather the quantum commutator) of point fields in many examples, computed using the canonical method, but expressed in relativistically invariant form, he was probably lead to a guess for its general formula. He showed that this formula is in fact antisymmetric and is equivalent to the canonical bracket for equal time fields, in the non-singular case (without gauge invariance). He, however, did not give an independent proof of its non-degeneracy or the Jacobi identity. Peierls showed that gauge invariance was not an obstacle to defining the Poisson bracket by his method, as long as one restricted oneself to gauge invariant observables. He also showed how the formula extends to fermionic fields: each fermionic field can be reduced to a bosonic one after multiplication by a formal anticommuting parameter.
A somewhat later 1957 paper of Glaser, Lehmann & Zimmermann 22 treated the perturbative expansion of interacting fields in terms of retarded products of incoming free fields, in contrast to the usual expansion in terms of time ordered products of asymptotic fields. As their name suggests, retarded fields are defined using retarded Green functions. They did not explicitly discuss Poisson structures, but their work folds into the thread of ideas we are discussing in a slightly different way. Their formula for the commutator of interacting fields involved differences of of retarded and advanced Green functions, what we would now call causal Green functions, which also occur in the Peierls formula. This is not so surprising given the intimate relationship between quantum commutators and Poisson brackets.
In 1960 came a paper of Segal 23 where he discussed the canonical quantization of field theories with non-linear hyperbolic equations of motion by identifying their phase space with the space of solutions and endowing it with a Poisson structure in a covariant way. His formula for the Poisson bracket also involved the causal Green function G. His construction appears to have been independent of Peierls, but motivated much in the same way. In particular, he constructs G not as the difference of retarded and advanced Green functions, but as a distributional solution of the linearized equations with specific initial conditions designed to reproduce the equal time commutation relations. Further, though minor, developments of these ideas appeared in a monograph 24 and in some conference proceedings, including Ref. 25 Unfortunately, not many people paid attention to Peierls' paper. Notable exceptions were Bergmann and DeWitt. In fact, DeWitt 7, 26 quickly became an early adopter and proponent. He can be said to be responsible for clarifying the role of the causal Green function of the Jacobi equation (the linearized Euler-Lagrange equation) in Peierls' construction and showing that it can be consistently used with gauge fixing. Incidentally, he also clarified the role of classical fermi fields in terms of anticommuting Grassmann variables and thus seeded the germs of supergeometry.
In 1971, Steinmann 27 published a monograph where he adapted the retarded products of Glaser, Lehmann & Zimmermann as a way of formalizing renormalized perturbation theory within the context of Axiomatic Quantum Field Theory.
DeWitt's formulation saw relatively few improvements until the early '90s when Marolf [28][29][30] (DeWitt's PhD student at the time) realized that the Peierls formula can be taken off-shell and define a Poisson bracket for off-shell observables. He essentially showed that, by using the Peierls formula directly, the (off-shell) field configuration space can be given the structure of a degenerate (though regular) Poisson manifold. The symplectic leaves of this Poisson structure are copies of the (on-shell) solution space with the standard canonical symplectic structure. The following interpretation is present though implicit in Marolf's papers: given a Lagrangian density L[φ] + φ · J, where φ denotes the dynamical fields and J the corresponding external sources, the solution space corresponding to a fixed external source profile J is one leaf of the Poisson structure on the field configuration space constructed from the source-less Lagrangian density L[φ]. Unfortunately, most of Marolf's discussion is non-covariant, as for simplicity it introduces a fixed time coordinate.
More recently, the Peierls formula became an important ingredient in the construction of perturbative Algebraic Quantum Field Theory (pAQFT). Its use came to prominence in the 2000s with Refs. 31, 32. In these papers, cues were taken partially from the perturbative QFT tradition of Refs. 22, 27, with retarded, advanced and time-ordered products adapted to an off-shell setting. At the same time, they realized that very similar formulas come about at the classical level from the use of the off-shell Peierls formula (as defined by Marolf) in perturbative classical field theory. It was there that the use of the off-shell Peierls formula was formulated in a systematic and covariant way. The first direct demonstration of the Jacobi identity for the Peierls formula was given in Ref. 32. More recently, this formulation of the off-shell Peierls formula has been extended to classical fermi fields, 33 which paved the way for its inclusion in a BV-BRST treatment of gauge theories in pAQFT. 12,13 Another recent development is the incorporation of the off-shell Peierls formula Covariant phase space, constraints, gauge and the Peierls formula 7 in a serious functional analytical effort to describe the infinite dimensional spaces of field configurations and algebras of observables using infinite dimensional differential geometry. [12][13][14] Finally, we should mention another recent paper 34 that has a non-trivial overlap with this review. Its goals also include clarifying the specific conditions needed to be satisfied by a linear gauge theory that guarantee that the Peierls construction works. On the other hand, their geometric set up is somewhat less general than ours and they do not consider the relation with the covariant phase space formalism in detail. Though, unlike here, they also treat fermi fields and further consider quantization.

Covariant phase space formalism
Much of the impetus for the development of a covariant Poisson bracket came from covariant quantization, or more precisely the need for a solid classical analog of the unequal time commutation relations in QFT. On the other hand, the development of the covariant phase space formalism was spurred by the desire to understand conservation laws in field theory, as well as trying to improve upon the canonical quantization program of Dirac and Bergmann. In the physics literature, the roots of this formalism, though in a rather obscure form, can be found in the 1953 paper of Bergmann & Schiller. 35 This paper was part of Bergmann's program to study the implications of general covariance in General Relativity (or any other second order covariant theory) for the structure of its conservation laws, its stress energy tensor, and its canonical quantization. There already appear formulas for what we call the presymplectic current density and its potential.
This possibility of obtaining the symplectic structure of a theory directly from the Lagrangian by the methods of Bergmann & Schiller remained somewhat unknown, except possibly to a small group of experts. For example, in 1962, Komar 36 (a student of Bergmann) used these methods to define the symplectic structure on the space of initial data on a null surface. Few other papers using this formalism appeared until its apparently independent resurgence in the '80s.
The 1975 article by Ashtekar & Magnon 37 used the integrated symplectic potential current density as a symplectic form for the Klein-Gordon field on curved spacetime, citing the work of Segal 25 as a similar previous treatment in Minkowski space, which goes back to the aforementioned Ref. 23. However, it appears that the formalism of Ashtekar & Magnon was privately inspired 38 by the ideas of Souriau, which they generalized to field theories. Later Ashtekar also applied the same formalism to general relativity. 39 Starting around this time, perhaps again due to the influence of the ideas of Souriau, the identification of the phase space with the space of solutions rather than with that of initial data starts to become more prevalent.
Another independent appearance of the (pre)symplectic form as an integral of the covariant (pre)symplectic current density is found in the 1978 article of Friedman 40 and a later article of Friedman & Schutz. 41 The examples considered there consisted of a scalar field and of the combined system of linearized gravitational and hydrodynamic modes describing a relativistic star. The latter presymplectic form is degenerate due to general covariance (gravitational gauge symmetry) and, in fact, its kernel was used in the analysis of linear stability to discard possibly unstable unphysical modes. Friedman did not cite any preceding sources, but apparently was inspired 42 by some lectures on the covariant treatment of conservation laws in variational theories (obviously connected to symplectic structure by Noether's theorem) that were delivered by Trautman at Chicago in 1971. The relevant content from these lectures later appeared as Ref. 43. Trautman's main influence seems to have been the gradual refinement of the original ideas of Noether 44 in connecting symmetries (including gauge symmetries) with conservation laws in the calculus of variations with multiple independent variables. [45][46][47] In the latter formalism, the geometric structure that is closely related (but not identical to) the covariant presymplectic (potential) current density is the Poincaré-Cartan form.
Another independent instance of the covariant symplectic formalism, though only for mechanical systems and not field theories, appeared in the 1982 article of Henneaux 48 on the inverse problem of the calculus of variations. No sources were cited in the article, but apparently the main inspiration 49 were the ideas of Souriau. 19 The breakthrough point for a more widespread appreciation of the covariant phase space formalism was the 1987 paper of Crnković & Witten, 4 which gave explicit covariant constructions for the symplectic forms of scalar fields, Yang-Mills theory and General Relativity. This paper was in fact an expansion of one section, where this formalism was laid out in some generality, of an earlier paper of Witten 50 on open string field theory. Witten  to be singled out for its exceptional clarity of presentation. The generality of the presentation was lacking only the treatment of classical fermi fields, which appeared even in a cursory way only in Ref. 52. That was rectified only much later in a paper of Hollands & Marolf. 55 In the mean time, some of the formal manipulations involved in deriving the covariant presymplectic current density were formalized in the framework of the variational bicomplex, as can be seen for instance in Sec. 8.3 of Ref. 56. It is essentially this presentation that we review later in Sec. 3.1.

Equivalence
While symplectic and Poisson structures are obviously related, as can be seen from the above references, the two covariant formalisms that we have described naturally appear in somewhat different problems. Thus, it is not surprising that most researchers would prefer to use just one or the other, without attempting to relate the two. Some have actually done so, though, until recently only using the fact that they are both equivalent (under appropriate assumptions of course) to the corresponding canonical constructions.
Recall that in his original paper Peierls 6 showed, by explicit calculation, the equivalence of the Peierls formula with the canonical Poisson bracket for nonsingular field theories. DeWitt 26 extended that to the case of gauge fixed singular theories, provided the canonical formalism was applied after gauge fixing.
Even before that, having taken note of Peierls' paper, Bergmann's group showed 57 that both the Bergmann-Schiller and Peierls formulations of Poisson brackets are essentially equivalent for non-singular field theories (those without gauge invariance). They also compare the Peierls bracket with the Dirac bracket in singular theories. Unfortunately, by modern standards their discussion is rather obscure.
Much later, Barnich,Henneaux & Schomblond 8 showed that the Peierls bracket coincides with the Dirac bracket in the canonical formalism even if both first and second class constraints are present (hence including gauge theories). Their presentation is very clear. It is also very insightful in the way that they included the covariant phase space formalism. They showed that the canonical formalism is in fact a special case of the covariant one, provided one uses Hamilton's least action principle as the variational functional. Further, they noted that the canonical variables can be introduced by adjoining some auxiliary fields 9 (the canonical momenta) to the Lagrangian formulation, while showing that the symplectic structure is invariant under the adjunction or elimination of auxiliary fields.
Finally, rather recently, a breakthrough appeared in a paper of Forger & Romero. 10 They managed to prove the equivalence of the covariant constructions of the symplectic and Poisson structures in an elegant and fully covariant way, thus without using the canonical formalism as an intermediary. Their proof was restricted to the case of a non-singular scalar field theory. It is their argument whose generalization we present in expanded detail in the bulk of this review, Sec. 3. It should be mentioned that Forger & Romero also very clearly compared the geometric structure of the covariant phase space formalism to the related but non-identical geometric structure of the multisymplectic formalism.

Linear PDE theory
In this section, we describe a number of important technical results about linear hyperbolic equations that will be crucial for the later discussion of the Peierls bracket in Sec. 3. Here we are mostly concerned with the linear algebra of the inhomogeneous system of partial differential equation (PDE system) on an n-dimensional spacetime manifold M , where f : Γ(F ) → Γ(F * ) is a linear partial differential operator acting on smooth sections φ ∈ Γ(F ) of a field (vector) bundle F → M and taking values in the densitized dual bundleF * → M , wherẽ F * ∼ = F * ⊗ M Λ n M is the linear dual bundle F * tensored with the bundle of volume forms on M . Typically,α * ∈ Γ(F ) is a compactly supported dual density. We choose to always have the PDE valued in dual densities out of convenience, as will be evidenced later in Sec. 2.5. Keeping with the terminology of Appendix B, we use such an (f,F * ) as our preferred equation form for linear PDE systems.
We will consider the case where f is hyperbolic and hence possesses Green functions (Sec. 2.1). An important idea that is often necessary to relate physical equations of motion to hyperbolic equations is that of compatible constraints (Sec. 2.2). Solution spaces can be conveniently parametrized using special, causal Green functions (Sec. 2.3 and 2.4). Both the differential operator f and its Green functions have adjoints, which are important in the definitions of various natural bilinear pairings (Sec. 2.5).
Sometimes we will refer to basic background information on jet bundles, the interpretation of differential operators as maps between jet bundles and the interpretation of PDEs as submanifolds of jet bundles. The relevant information is summarized in Appendix A and Appendix B. Also, hyperbolic PDEs naturally define a generalized kind of causal structure on M . The necessary ideas and definitions are summarized in Appendix C, with the notation similar to the standard one used in Lorentzian geometry. This causal structure can be used to restrict the supports of field and dual density sections.

Green hyperbolicity
Below, we define the notion of a Green hyperbolic PDE system as one that possesses unique advanced and retarded Green functions. We will rely heavily on the existence and properties of these Green functions in later sections. Before proceeding, we need the notion of a causal structure (a priori independent of any Lorentzian metric), with respect to which the notions of advanced and retarded support will be defined. In the literature on relativity, the two are usually introduced together. However, a deeper investigation of hyperbolic PDE systems shows that the notion of causality can be defined independently and intrinsically from a given PDE. It so happens that, for equations with a d'Alambert-like principal symbol, the causal relations deduced directly from the PDE, on the one hand, and from the background Lorentzian metric, on the other, actually coincide. The basic relevant notions and definitions are summarized in Appendix C. See Ref. [60,Secs.3,4] for a more in depth discussion.
The PDE system f [φ] = 0 is said to be Green hyperbolic if there exists a globally hyperbolic conal structure on M such that (a) the inhomogeneous equation f [φ ± ] =α * is solvable forα * ∈ Γ ± (F * ) and (b) a solution φ ± ∈ Γ ± (F ) exists, is unique and satisfies the support condition supp φ ± ⊆ I ± (suppα * ). We denote the unique two-sided inverses by G ± : Γ ± (F * ) → Γ ± (F ) and refer to them as the retarded (+) and advanced (−) Green functions. In adapted local coordinates (x i , u a ) on F and (x i , u b ) onF * , where u a (φ(x)) = φ a (x), u b (α * (x)) = α b (x) and dx = dx 1 ∧ · · · ∧ dx n , the Green functions can be represented as integral kernels It is sufficient that G ± be defined on Γ 0 (F * ). It can then be extended to Γ ± (F * ) by an exhaustion argument (Cor.5 in Ref. 61). Ideally, the Green functions should be well-defined distributions (be continuous in the appropriate function space topology), but we will not discuss this issue here and instead concentrate on their algebraic and geometric properties.

Compatible constraints
Consider a linear PDE system on the field bundle F → M that consists of where f : Γ(F ) → Γ(F * ) is a hyperbolic partial differential operator and c : Γ(F ) → Γ(E) is another partial differential operator valued in a vector bundle E → M . We refer to f [φ] = 0 as the hyperbolic subsystem and to c[φ] = 0 as the constraints subsystem, with E → M the constraints bundle. The equation form of the total system is (f ⊕ c,F * ⊕ E).
The constraints are said to be hyperbolically integrable if there exists a pair of linear differential operators h : Γ(E) → Γ(Ẽ * ) and q : Γ(F * ) → Γ(Ẽ * ) that satisfy the identity h = 0 is said to be hyperbolic with constraints. Whenever we refer to a causal structure induced by a hyperbolic system with constraints, we actually mean the one induced by the corresponding hyperbolic compound system.

Causal Green function (without constraints)
Now that we are sure to have access to the retarded/advanced green functions G ± for the Green hyperbolic system f [φ] = 0, we can define the so-called causal Green function This new Green function helps to conveniently parametrize the space of solutions S SC (F ) ∼ = ker f ⊂ Γ SC (F ) by featuring in the following is exact (in the sense of linear algebra).
That is, the image of each map coincides with the kernel of the next map. The proof for wave-like equations, which is given in [ 61. A complete proof actually follows from the identities given in Lem. 2.1 below and the fact that f is invertible on Γ ± (F ), and hence a fortiori injective on Γ 0 (F ). We can interpret the above proposition in the following way. Since S SC (F ) ∼ = im G, we can express any solution to the homogeneous problem as φ = G[α * ], where α ∈ Γ 0 (F * ) is some smooth dual density of compact support. Equivalently, by exactness, S SC (F ) ∼ = coker f = Γ 0 (F * )/ im f . Also, since Γ SC (F ) ∼ = im f , for any dual densityα * with spacelike compact support, there exists a solution φ with spacelike compact support of the inhomogeneous problem f [φ] =α * . Definition 2.3. Consider one Cauchy surface Σ ⊂ M and two more Cauchy surfaces Σ ± ⊂ M to the past and future of Σ, where Σ ± ⊂ I ± (Σ), and let S ± = I ± (Σ ∓ ). Let {χ + , χ − } be a partition of unity adapted to the open cover {S + , S − } of M , that is, χ + + χ − = 1 and supp χ ± ⊂ S ± . We call {χ + , χ − } a partition of unity adapted to the Cauchy surface Σ.
Given a partition of unity {χ + , χ − } adapted to a Cauchy surface Σ, there exist (noncanonical) splitting maps Proof. Note that these splitting maps are not canonical, as they depend on the choice of a Cauchy surface and a partition of unity adapted to it.
does in fact have compact support, as supp φ is spacelike compact while supp dχ ± ⊂ S + ∩ S − is timelike compact, a and f [χ ± φ] = 0 only on supp φ ∩ supp dχ ± , which is by definition compact. Also, since d(χ + + χ − ) = 0, we have f + which means that the map f χ = ±f ± χ is well defined. On the other hand, we have G ± [f [χ ± φ]] = χ ± φ from the uniqueness of solutions to the inhomogeneous problem with retarded/advanced support. The definition of the causal Green function then immediately implies that G • f χ = ±id on S SC (F ). Also, a direct calculation shows that f • G χ = id on Γ SC (F * ): This concludes the proof.

Causal Green function (with constraints)
We will not discuss the most general kind of constraints and restrict our attention only to parametrizable ones. By the term parametrizable, we mean that there exist an additional vector bundle E ′ → M and additional differential operators h ′ , c ′ and q ′ , which fit into the following commutative diagram Note that the causally restricted supports in the above diagram should be defined with respect to a causal structure that is defined by the total compound system with equation Lemma 2.2. The retarded/advanced inhomogeneous problem We obviously have f [φ] =β * . It remains to check This concludes the proof.
It is convenient to state here a lemma concerning formally exact complexes of differential operators, which shall be referred to in later sections. Lemma 2.3. Suppose that linear differential operators c ′ : Γ(E ′ ) → Γ(F ) and c : Γ(F ) → Γ(E) form a formally exact complex. Then any linear differential operator l : Γ(F ) → Γ(L) such that l • c ′ = 0 factors as l = l c • c, for some linear differential operator l c : Γ(E) → Γ(L). Similarly, any linear differential operator r : Γ(R) → Γ(F ) such that c • r = 0 factors as r = c ′ • r c , for some linear differential operator r c : Γ(R) → Γ(E ′ ).
Proof. We represent all differential operators as maps from appropriate jet bundles (see Sec. Appendix B). The fact that the differential operators c ′ : J ∞ E ′ → J ∞ F and c : J ∞ F → E form a formally exact complex shows that the prolongations p ∞ c ′ : J ∞ E ′ → J ∞ F and p ∞ c : J ∞ F → J ∞ E compose into an exact sequence of vector bundle maps. Hence, the image of p ∞ c ′ coincides with the kernel of p ∞ c. By hypothesis, the kernel of the linear bundle map l : J ∞ F → L contains im p ∞ c ′ , while the image of the linear bundle map p ∞ r : J ∞ R → J ∞ F is contained in the image of p ∞ c ′ . Therefore, desired factorization formulas follow straight forwardly from linear algebra.
Such arguments are common in the formal theory of PDEs and can even be generalized to the nonlinear setting. 66-68

Pairings and adjoints
We conclude this section by remarking the identities where on the left hand side (G ± ) * denotes the adjoint of the retarded/advanced Green function G ± of the equation f [φ] = 0, and on the right hand side G * ∓ denotes the advanced/retarded Green function of the adjoint equation f * [φ] = 0. Note that taking the adjoint flips the support between retarded and advanced.
Definition 2.4. Given two differential operators f, f * : Γ(F ) → Γ(F * ) are said to be mutually adjoint if there exists a bilinear differential operator G : Γ(F )×Γ(F ) → Ω n−1 (M ) such that Note that we may introduce a natural pairing between elements φ ∈ Γ(F ) and α ∈ Γ(F * ) given by The pairing is only partially defined. That is, there exist arguments for which the integral does not converge. For simplicity, we will consider it only for those those pairs of sections for which the integrand φ ·α * has compact support. This pairing is non-degenerate in either argument, as follows from the standard argument of the fundamental lemma of the calculus of variations [ It is straight forward that that the natural pairing −, − is non-degenerate on the spaces Γ ± (F ) × Γ ∓ (F * ). And since, the identities f • G ± = G ± • f = id hold on Γ ± (F ), it is now easy to verify the adjoint identities (21) for any φ ± ∈ Γ ± (F ) andα * ± ∈ Γ ± (F ). The causal Green functions then satisfy (G) * = −G * , where G * is the causal Green function for f * .
Lemma 2.4. Following the notation of Def. 2.4, the Green pairing −, − G depends only on the equivalence class [G] of G and is independent of Σ.
Proof. Any two representatives G 1 and G 2 of [G] will differ by an exact term dH, with H[−, −] a bilinear bidifferential operator. Therefore, the integrands ι * G i [ϕ 1 , ϕ 2 ] will differ by the exact term dι * H[ϕ 1 , ϕ 2 ], with necessarily compact support. Therefore, since Σ has no boundary, we can use any representative of [G] to evaluate the pairing. Now, let ι ′ : Σ ′ ⊂ M be another Cauchy surface and let S ⊆ M be such that ∂S = Σ ′ − Σ. Then an application of Stokes' theorem shows the following: (27) where we used the solution properties f [φ] = 0 and f * [ψ] = 0. This shows the independence of the Green pairing from the choice of a Cauchy surface.
Lemma 2.5. The Green pairing has the following alternative forms: Note that the following proof was partly inspired by Sec Proof. Consider a future oriented Cauchy surface ι : Σ ⊂ M (see Appendix C) and a partition of unity {χ ± } adapted to it. Then, recalling the notation for the splitting maps in Lem. 2.1, direct calculation gives Appealing to the exact sequence of Prp. 2.1, the dual densitiesα * andβ * are only defined up to an element of im f and im f * respectively. However, the formulas show that this freedom does not affect the result. Now, based on the formula ψ, ξ G = − M ψ ·β * , the fact thatβ * could be arbitrary and the non-degeneracy of the natural pairing −, − we can see that −, − G must be non-degenerate in its second argument. The same reasoning establishes non-degeneracy in the first argument as well.

Covariant phase space formalism, Peierls formula
This section constitutes the main body of this review. At this point it is helpful to at least skim the contents of Appendix A and Appendix B, as they summarize relevant concepts and notation. Its culmination is a precise set of conditions (Secs Below, we study their argument in depth and generalize it to include field theories with constraints and gauge invariance. Sec. 3.1 defines variational PDE systems and shows how the covariant symplectic formalism arises from their geometry. Sec. 3.2 discusses the differential geometry of the space of solutions of the PDE system. Since the focus of this work is more geometrical than analytical, we avoid a detailed discussion of the subtleties of infinite dimensional manifolds. Instead, we define so-called formal tangent and cotangent spaces to the solution space and then fix a particular background solution, so that we need only consider a single formal tangent and cotangent fiber at that point of the phase space. This is sufficient for defining the formal symplectic form and the Poisson bivector and proving their equivalence. An in-depth discussion of the functional analytical details that go into defining the necessary infinite dimensional differential geometry can be found in Refs. 11-14.

Consider a field vector bundle
where L, the Lagrangian density, is a section of the bundle (Λ n M ) k → J k F densities, which could depend on jet coordinates of order up to k (see Appendix A). The Lagrangian density is called local because, given a section φ and local coordinates (x i , u a I ) on J k F , the pullback at x ∈ M can be written as which depends only on x and on the derivatives of φ at x up to order k. For the most part, the integral over M can be considered formal, since all the necessary properties will be derived from L. On the other hand, the finiteness of S[φ] or related quantities may be important while discussing boundary conditions in spacetimes with non-compact spatial extent. However, we will not discuss these issues below.
Recall that Appendix A introduces the variational bicomplex Ω h,v (F ) of vertically and horizontally graded differential forms on J ∞ F . Below, we use the notation introduced there. A Lagrangian density is then an element L ∈ Ω n,0 (F ) that can be projected to J k F . Incidentally the usual variational derivative of variational calculus can be put into direct correspondence with the vertical differential d v on this complex, which is how the name variational bicomplex was established. 71,73 Let (x i , u a I ) be a set of adapted coordinates on the ∞-jet bundle J ∞ F , where all the following calculations can be lifted. Any result that depends only on jets of finite order can then be projected onto the appropriate finite dimensional jet bundle. Using the integration by parts identity (A. 19) if necessary, we can always write the first vertical variation of the Lagrangian density as All terms proportional to d v u a I , |I| > 0, have been absorbed into d h θ. In the course of the performing the integrations by parts, EL a can acquire dependence on jets up to order 2k (see Appendix B), and θ on jets up to order 2k − 1. Note that EL a = 0 are the Euler-Lagrange equations associated with the action functional S[φ] or the Lagrangian density L. We can identify the form EL a ∧ d v u a with a possibly nonlinear differential operator EL : Γ(F ) → Γ(F * ), or equivalently a bundle morphism EL : J 2k F →F * . Therefore, (EL,F * ) is an equation form of a PDE system E EL ⊂ J 2k F on F of order 2k. A PDE system with an equation form given by Euler-Lagrange equations of a Lagrangian density is said to be variational. Also, the form θ is an element of Ω n−1,1 (F ), projectable to J 2k−1 F . It is referred to as the presymplectic potential current density. Applying the vertical exterior differential to θ we obtain the presymplectic current density (or the presymplectic current density defined by L if the extra precision is necessary): with ω ∈ Ω n−1,2 (F ). This terminology implies that ω can be integrated over a codim-1 spacetime surface to construct a presymplectic form (Sec. 3.3.3). Such a form on the solution space is referred to as local. This method of constructing a symplectic form on the phase space of classical field theory is sometimes referred to as the covariant phase space method. [3][4][5]54 The following lemma is an easy consequence of the definition of ω.
is both horizontally and vertically closed when pulled back to ι ∞ : Proof. The horizontal and vertical differentials on E ∞ EL are defined by pullback Since ω = d v θ is already vertically closed on J ∞ F , it is a fortiori vertically closed on E ∞ EL . The rest is a consequence of the nilpotence and anti-commutativity of d h and d v : since EL a and d v EL a generate the differential ideal in Ω * (J ∞ F ) that is annihilated by the pullback ι * ∞ .
In fact, we will promote the name presymplectic current density to any form satisfying these properties.
Definition 3.1. It is interesting to note that the existence of a presymplectic current density compatible with a PDE E is almost equivalent to E being variational. 74 Given a PDE system ι : E ⊂ J k F we call a form ω a presymplectic current density compatible with E if ω ∈ Ω n−1,2 (F ) and it is both horizontally and vertically closed on solutions: The particular form ω defined by Eq. (39) will be referred to as the presymplectic current density associated to or obtained from the Lagrangian density L, if there is any potential confusion.

Formal differential geometry of solution spaces
Before describing the symplectic and Poisson structures on the space of solutions, we should say something about the differential geometry of the manifold of solutions of a PDE system as well as its tangent and cotangent spaces. As usual for infinite dimensional manifolds, there are some subtleties.
The main goal of this section is to describe the formal tangent and cotangent spaces of the manifold of arbitrary field sections and the manifold of solution sections. The adjective formal, in the last sentence, alludes to the fact that we avoid most technical issues of infinite dimensional analysis and concentrate on what would be dense subspaces of the true tangent and cotangent spaces with a reasonable choice for their topologies. Results are algebraic and (finite dimensional) geometric identities that would form the core of an earnest functional analytical formulation of their non-formal versions. The formal tangent and cotangent spaces have a natural dual pairing, which we prove to be non-degenerate, as a substitute for the absence of true topological duality between them. In the presence of constraints, the proof is carried out under some additional sufficient conditions.
We start with Sec

Non-linear equations and linearization
In the preceding section (Sec. 3.1) we have discussed general variational systems, without regard for either linearity or hyperbolicity. Note that the notion of Green hyperbolicity that we discussed earlier in Sec. 2 is only applicable to linear systems. The way that we shall restrict our discussion to linear systems is by linearization, which is justified below.
Let us denote by S(F ) the set of solutions of the equations of motion of a given non-linear classical field theory on a field bundle F → M . For instance, for General Relativity S(F ) would include metrics of all possible signatures, not just Lorentzian ones. So, obviously, we shall not be interested in all possible solutions, but those that have good causal behavior. We shall not delve here into the question of what constitutes good causal behavior in a non-linear field theory, but refer the reader to Sec. 4.2 of Ref. 60. We shall simply postulate that there is a subset S H (F ) ⊆ S(F ) that consists of all solutions with good causal behavior, with the subscript H nominally standing for globally hyperbolic. For us, the most important property of any background solution φ ∈ S H (F ) is that the linearized equations of motion about φ are Green hyperbolic (Sec. 2). Sometimes, we shall also refer to a background solution φ as a dynamical linearization point.
We shall also assume the hypothesis that S H (F ) can be seen as a possibly infinite dimensional manifold (see Refs. 11-14 for attempts to make that precise). When dealing with either symplectic or Poisson structure, we need also the notions of the tangent T S H (F ) and cotangent T * S H (F ) bundles, since these structures are needed to define 2-form or bivector tensors on S H (F ). The de Rham closedness and Jacobi identities that respectively identify symplectic and Poisson structures require a notion of differentiation, that is, a differential structure on T S H (F ) and T * S H (F ) as well. However, if we concentrate on the mutual inverse relationship between a symplectic form Ω and a Poisson bivector Π, we are allowed to work with a single tangent space T φ S H (F ) and a single cotangent space T * φ S H (F ) at a time, with φ ∈ S H (F ), verifying this property for each pair Ω φ and Π φ individually. This is precisely what we do below.
From now on, we fix φ ∈ S H (F ) to be a particular dynamical linearization point. Given any, possibly non-linear, differential operator, we denote its linearization by the same symbol but with a dot, e.g.,ḟ : Γ(F ) → Γ(F * ) is the linearization of f : Γ(F ) → Γ(F * ) about φ. Since solution space S H (F ) is embedded in the total field configuration space Γ(F ), the tangent space at φ is defined by the space of linearized solutions, that is, solutions of the linearized equations. The following sections, Secs. 3.2.2 and 3.2.3, introduce the linear differential operators that we expect to obtain after linearizing the equations of motion with constraints and gauge invariance. Later, in Sec. 3.3.1, we consider the Euler-Lagrange equations of a possibly non-linear classical field theory and linearize them, together with the corresponding hyperbolic, constraint and gauge generator differential operators.

Constraints
Earlier, in Sec. 2.2, we discussed linear constrained hyperbolic systems. This notion can actually be extended to non-linear systems, with a very similar structure of identities satisfied by the differential operators involved. See Refs. 75 and 60 for details. At this point, w will presume that we are dealing with a linearization of a possibly non-linear constrained hyperbolic system, whose linearization is itself a linear constrained hyperbolic system of the kind described in Sec. 2.2. To keep the linearization in mind, we put a dot on all the differential operators. Thus, we have a linear constrained hyperbolic system defined by the operatorsḟ : An important thing to note is that their formal adjoints will satisfy the related identityċ * •ḣ * =ḟ * •q * , which is exploited below.
When dealing with constrained systems, some results covered later will require the further sufficient condition that the constraints be parametrizable (see Sec. 2.4) so that we can extend both the linearized system and its adjoint to the following commutative diagrams: and The rows form exact sequences, while the columns form formally exact complexes, as described in Sec. 2.4. Note that the adjoint diagram also describes a hyperbolic system with hyperbolically integrable constraints, except that the role of the constraint subsystem is now played by (q ′ * , E ′ ) and the consistency subsystem is (ḣ ′ * ,Ẽ ′ * ), which satisfies the consistency identityḣ ′ * •q ′ * =ċ ′ * •ḟ * . It is convenient to introduce here a cohomological condition, to be applied in later sections, on the columns of the above commutative diagrams. First, let us introduce some notation for the respective cohomologies. Because the columns are so short, the cohomologies can only be defined at the middle nodes. Each cohomology can be identified by the vector bundle where it is defined, F orF * , the support restriction, 0 or SC, and the diagram used to define it, (47) or (48). Thus, we denote the cohomologies defined by the columns of diagram (47) , with appropriately restricted support, represents a cohomology class denoted by [ψ] c or [ψ] c * , and similarly for cocycle sections in Γ(F * ) (which are annihilated byq orċ ′ * ).
Second, recall that, given ψ ∈ Γ(F ) andα * ∈ Γ(F * ), there is a natural paring ψ,α * = M ψ ·α * , provided the integral is finite. This pairing is indeed well defined between the corresponding nodes of the diagrams (47) and (48). Moreover, when restricted to cocycle sections, this pairing descends to cohomology classes, . Third, we must recall that the commutativity of the diagrams (50) and (51) allows us to consider the respective cohomologies at Γ 0 (F * ) modulo imḟ and at Γ SC (F ) restricted to im G * . In both cases, by the exactness of the rows of these diagrams, we are simply describing the vertical cohomologies in the space of solutions ofḟ andḟ * , respectively. We shall denote them by H c SC (F,ḟ ) and H c * SC (F,ḟ * ), respectively. The natural pairing also descends to the cohomologies in solutions as follows, with say ψ = G[α * ] and ξ = G * [β * ], where on the right-hand-side −, − G is the Green pairing from Def. 2.5.
Definition 3.2. The constraints are said to be globally parametrizable if the natural pairing between the vertical cohomologies in solutions, H c SC (F,ḟ ) and H c * SC (F,ḟ * ), defined using the commutative diagrams (47) and (48), is non-degenerate.

Gauge transformations
Many important classical field theories exhibit gauge invariance, like Maxwell theory, Yang-Mills theory, and General Relativity. A gauge transformation is a family of maps g ε : Γ(F ) → Γ(F ), parametrized by sections ε ∈ Γ(P ) of the gauge parameter bundle P → M , that take solutions to solutions, while not modifying a field section outside the support of δ ∈ Γ(P ), which may be compact and arbitrarily small. If we linearize about some pair of background section δ → δ + ε, we obtain a linearized gauge transformation g δ It is another requirement on gauge transformations that the generator of linearized gauge transformationsġ : Γ(P ) → Γ(F ) is a differential operator, which may depend on the background sections δ and φ.
Equivalence classes of sections under gauge transformations are considered physically equivalent. Therefore, physical observables will consist only of those functions on phase space that are gauge invariant (constant on orbits of gauge transformations). Equivalently, observables are annihilated by the action of linearized gauge transformations. Another way to look at it, is to consider observables as functions on the space of gauge orbits. We denote the space of solutions of the possibly nonlinear equations of motion (with good causal behavior, cf. Sec. 3.2.1) modulo gauge transformations,S H (F ) = S H (F )/∼ and call it the physical phase space.
Often it is convenient to impose subsidiary conditions on field sections, called gauge fixing, that restrict the choice of representatives of gauge equivalence classes. The gauge fixing is called full if they only allow a unique representative from each equivalence class, and otherwise called partial. The gauge transformations that are compatible with a partial gauge fixing are called residual.
Unfortunately, PDE systems with gauge invariance cannot have a well-posed initial value problem, and hence cannot be hyperbolic. In particular, their linearizations cannot be Green hyperbolic. However, the addition of subsidiary conditions on field sections can make the new PDE system equivalent to a hyperbolic one, usually with constraints. In practice, many hyperbolic systems with constraints arise after adding such gauge fixing conditions to a non-hyperbolic system with gauge invariance. Interestingly, after many convenient gauge fixings, there may remain nontrivial residual gauge freedom. For later convenience, as we did with constraints, we restrict our attention to what we call recognizable gauge transformations. That is, given linearized gauge transformations of the formġ[ε] and a partially gauge fixed hyperbolic system with equation form (ḟ ,F * ), we can fit them into the following commutative diagram, whose columns form formally exact complexes: Their adjoints fit into the adjoint diagram whose columns are also formally exact complexes: The systems with equation forms (k,P * ) and (k ′ * ,P ′ * ) are required to be hyperbolic and P ′ → M is called the gauge invariant field bundle, whileġ ′ is called the operator of gauge invariant field combinations.
The above commutative diagrams are very similar to those used to define parametrizable constraints in Sec. 3.2.2. Thus, we can define all the same cohomologies: (50), (51). Also, in exactly the same way, there are bilinear pairings −, − and −, − G defined on respective pairs of these cohomologies. On the other hand, the following definition is not quite analogous, reflecting how this hypothesis is used in later sections.
Definition 3.3. The gauge transformations are said to be globally recognizable if the natural pairing between the cohomologies H g SC (F ) and H g * 0 (F * ), defined using the commutative diagrams (50) and (51), is non-degenerate.

Formal T and T * for configurations
Here we consider a section φ ∈ S H (F ) ⊂ Γ(F ) and examine the formal tangent and cotangent spaces at T φ Γ = T φ Γ(F ) and T * φ Γ = T * φ Γ(F ) at φ.
Definition 3.4. We define the formal full tangent space at φ as the set of spacelike compactly supported sections and we define the formal full cotangent space at φ as the set The natural pairing −, − : Since gauge transformations act on field configurations and not just solutions, it makes sense to consider all field configurations related by gauge transformations as physically equivalent. The tangent space T φ Γ will be reduced to the quotient (or physical) tangent space T φΓ and the cotangent space T * Γ to the subset T * Γ of gauge invariant elements. The natural pairing between them is shown to be non-degenerate under the condition of global recognizability, that is, the vertical formally exact complexes in diagrams (50) and (51) are exact. We deal with field configurations first and delay the discussion of solutions to the next section.
The exactness of the compositionġ ′ •ġ = 0 ensures that we can recognize pure gauge field configurations, which are of the form ψ =ġ[ε] for some spacelike compactly supported section ε : M → P , precisely as those spacelike compact field sections ψ : M → F that give vanishing gauge invariant field combinationsġ ′ [ψ] = 0. On the other hand, the exactness of the dual compositionġ * •ġ ′ * = 0 ensures that we can parametrize gauge invariant, compactly supported dual densitiesα * : M →F * , those satisfyingġ * [α] = 0, precisely as the image of the differential operatorġ ′ * acting on compactly supported sections ofP ′ * → M .
The formal gauge invariant full cotangent space at φ is the set of compactly supported gauge invariant dual densities, The natural pairing −, − : Proof. Non-degeneracy in the second argument follows once again from the fundamental lemma of the calculus of variations: [ψ],α * = ψ,α * = 0 for all ψ ∈ T φ Γ implies thatα * = 0. Non-degeneracy in the first argument is more complicated, since we can no longer use arbitraryα * in the second argument. It now requires an appeal to the global recognizability of the gauge transformations. Suppose that [ψ],α * = 0 for allα * ∈ T * φΓ . We need to show that this implies ψ =ġ[ε] is pure gauge, for some spacelike compactly supported ε ∈ Γ SC (P ).

Formal T and T * for solutions
The formal tangent space T φ S will consist of linearized solutions, that is solutions of the linearized constrained hyperbolic systemḟ [ψ] = 0 andċ[ψ] = 0. The formal cotangent space will naturally consist of equivalence classes of dual densities up to the images of the adjoints ofḟ andċ. After giving the precise definitions below, we prove that that the natural pairing between these formal tangent and cotangent spaces is non-degenerate.
Definition 3.6. We define the formal solutions tangent space at φ as the set of spacelike compactly supported linearized solution sections, We define the formal solutions cotangent space at φ as the set of equivalence classes of compactly supported dual densities, The natural pairing −, − : As a warm-up before the main result of this section, we fist handle the case where the constraints and gauge transformations are trivial. Proof. Non-degeneracy in the first argument follows again from the fundamental lemma of the calculus of variations: ψ, [α * ] = 0 for allα * ∈ Γ 0 (F * ) implies that ψ = 0. Non-degeneracy in the second argument is more tricky, since now ψ can no longer be arbitrary. Suppose we have ψ, [α * ] = 0 for all spacelike compactly supported linearized solutions ψ ∈ T φ S. From this, we need to deduce that [α * ] = [0], which meansα * =ḟ * [ξ] for some compactly supported ξ ∈ Γ 0 (F ).
Remark 3.2. At this point, it is worth mentioning that the natural pairing between T φ S and T * φ S (again, in the absence of constraints or gauge transformations) is essentially equivalent, via Lem. 2.5, to the Green pairing (Def. 2.5), which is also non-degenerate.
In the presence of gauge symmetries, the formal tangent space consists of equivalence classes of linearized solutions up to gauge transformations. On the other hand, the formal cotangent space is restricted to equivalence represented by gauge invariant dual densities. After giving the precise definitions below, we prove that the natural pairing between these formal tangent and cotangent spaces is nondegenerate, provided the constraintsċ[φ] = 0 are globally parametrizable and the gauge transformation are globally recognizable.
The following technical definition is motivated by following steps: we first construct the solution space T φ S and then quotient by its purge gauge subspace. Definition 3.7. We define the formal gauge invariant solutions tangent space at φ as the set of gauge equivalence classes of φ-spacelike compactly supported linearized solution sections, The formal gauge invariant solutions cotangent space at φ is the set of equivalence classes of compactly supported gauge invariant dual densities, The natural pairing −, − : We now prove the main result of this section. Proof. Unfortunately, we now cannot directly rely on the fundamental lemma of the calculus of variations to prove non-degeneracy in either argument. Instead, we proceed roughly as in the proof of Lem. 3.3.

Symplectic and Poisson structure
In this section, we endow the space of solutions S H (F ) of a variational PDE system with the structures of both a symplectic and a Poisson manifold (or rather formal versions of these structures), really turning it into the phase space of classical field theory.
In general, a variational system may have gauge symmetries. These must be gauge fixed. The resulting system should then be put into the form of a constrained hyperbolic system. Or, rather, what is most important for us is that these steps can be carried out for the linearization of our variational system (Sec For the remainder of this section, let us fix a Lagrangian density L ∈ Ω n,0 (F ). Following Sec. 3.1, it defines a presymplectic current density ω ∈ Ω n−1,2 (F ) and its Euler-Lagrange equations define a PDE system with equation form (EL,F * ).

Variational systems, gauge fixing and constraints
There are many reasons why the equation form (EL,F * ) of the equations of motion of the classical field theory is not optimal for our analysis. As we shall see later on, the Peierls formula calls for a Green function of the linearized equations of motion. However, the particular form of the differential operator EL may not be one that directly falls into one of the classes of differential operators that are easily recognized as hyperbolic, so that its linearizations possess Green functions. For instance, in the presence of gauge invariance, we must first adjoin a gauge fixing condition, say c g [φ] = 0 valued in a bundle E g → M . Also, in many cases, either due to the extra gauge fixing equations or due to internal integrability conditions (Sec. Appendix B.1), the equations can only be cast in hyperbolic form with constraints. Of course, let us not forget that, a priori, we haven't yet restricted the choice of L in any way that would guarantee that its Euler-Lagrange system is not elliptic or of some other hyperbolic type. So, we call the Euler-Lagrange system hyperbolizable if, after a possible gauge fixing, it can be shown to be equivalent to a constrained hyperbolic system in a way that we make precise below. From now on, we require that for a classical field theory the Lagrangian L is chosen such that its Euler-Lagrange equations are hyperbolizable. We shall see later in Sec. 4, that many relativistic field theories of physical interest are in fact hyperbolizable.
Consider the gauge fixed Euler-Lagrange system, whose equation form is (EL ⊕ c g ,F * ⊕ M E g ). It is hyperbolizable if it equivalent (in the sense of Sec. Appendix B.1) to a constrained hyperbolic system (f ⊕ c,F * ⊕ M E). Again, we are not going into the details of what constitutes a non-linear constrained hyperbolic system but defer instead to Refs. 75 and 60. The equivalence must have the following form: where the R,R, R g andR g are possibly non-linear differential operators. As discussed earlier, in Sec. 3.2.1, for the purposes of our discussion, it is sufficient to pick a single dynamical linearization point φ ∈ S H (F ) and linearize the above PDE systems about it. In particular, the linearized equations will be sufficient to define the formal tangent and cotangent spaces T φ S, T * φ S and their gauge invariant analogs T φS , T * φS . In other words, we need to work with the linearized versions of each of the hyperbolic system, the constraints, the Euler-Lagrange system, the gauge fixing conditions, the gauge transformations, as well as the hyperbolization. As before, we denote the equation form of the linearized hyperbolic system (ḟ ,F * ). The linearized constraints are presumed to be globally parametrizable and fit into the commutative diagrams (47) and (48). The linearized gauge transformations are presumed to be globally recognizable and fit into the commutative diagrams (50) and (51). The linearized EL equations are denoted (J,F * ) and are also called the Jacobi system, with J the Jacobi operator, 7 while the linearized gauge fixing conditions are denoted by the equation form (ċ g , E g ). In local coordinates (x i , u a ) on F , the components of the Jacobi operator satisfy the identity The equivalence of the linearized systems takes the following form: If the operatorr J is non-vanishing, it means that part of the constraints consist of integrability conditions of the Jacobi system. Note that, strictly speaking, the r-andr-differential operators effecting the equivalence are not inverses of each other. Their compositions may differ from the identity by some differential operator that factors through a differential identity, that is,q •ḟ −ḣ •ċ = 0 orġ * • J = 0. In other words, we must have for some differential operators p J , p f , p g and p c . Also, the identityq •ḟ −ḣ •ċ = 0, when expressed in terms of the J andċ g operators, is identically satisfied wheṅ It is worth noting that the above relations involving the r-andr-operators follow from the equivalence (76) only when J andċ g satisfy no additional differential identities. However, we will simply presume that they hold as needed sufficient conditions for the derivation of the Peierls formulas later in Sec. 3.3.4. Finally, to make sure that the conditionċ g [ψ] = 0 in fact constitutes a gauge fixing condition, we require the following compatibility between the gauge transformation operator and the constraints that we shall refer to as the gauge fixing compatibility condition:ṡ This condition connects constraints (represented byċ g ) and gauge transformations (represented byṡ ′ ). Roughly speaking, this condition says that the part of the constraintsċ[ψ] = 0 that comes fromċ g [ψ] = 0 is sufficient, when adjoined to J[ψ] = 0 to make the gauge fixed system hyperbolizable. This compatibility condition becomes important later on, in Lem. 3.9, to show that the gauge invariant formal cotangent space T * φS can be equivalently defined in two ways, involving either the J operator or theḟ ,ċ operators. Also, it helps prove that the hyperbolic differential operatork, that acts on gauge invariant field combinations in the presence of recognizable gauge transformations, is actually independent of the choice of gauge fixing operatorċ g as long as the compatibility condition is satisfied (see Cor. 3.2).
For future reference, it is convenient to state here the following Lemma 3.6. The gauge fixing compatibility condition Eq. (83) is equivalent to the existence of a differential operatorr s : Γ(F ) → Γ(P * ) such that That is,r c •ċ g factors throughṡ.
Proof. This follows directly from the gauge compatibility condition (83) and Lem. 2.3.
We summarize the conditions listed in this section in the following The consequences of hyperbolizability are explored in the following section. We stress that these conditions are sufficient for our purposes and can in fact be satisfied by many relativistic field theories of physical interest, but some of the same results could also hold under weaker conditions.

Causal Green functions
The goal of this section is to use the gauge fixed equivalence (76) with a constrained hyperbolic system to construct a causal Green function for the Jacobi system. First, we show that the residual gauge transformations (those that are still allowed by the gauge fixing conditionċ g [ψ] = 0) essentially come from gauge parameters that satisfy the symmetric hyperbolic equationk[ε] = 0. The main purpose of this lemma is to serve as a reference argument for one of the sub-results of Thm. 3.1. Proof. Recall that the Jacobi system, due to its variational character is easily shown to be self-adjoint: Also, gauge invariance and Noether's second theorem imply the identities The equivalence of the gauge fixed Jacobi system with the constrained hyperbolic system postulated in (76) then giveṡ Suppose that ψ ∈ Γ SC (F ) such thatċ g [ψ] = 0 and ψ =ġ[ε ′ ] for some ε ′ ∈ Γ SC (P ). Letβ * =k[ε ′ ] ∈ Γ SC (P * ) and note thaṫ Let {χ ± } be a partition of unity adapted to a Cauchy surface (Def. 2.3) and recall the associated splitting map (Lem. 2.1) K χ : Γ SC (P * ) → Γ SC (P ) that invertṡ k from the right. We can make two observations: (a) the difference * ] has compact support sinceṡ[β * ] = 0. Hence η and ψ define the same cohomology class, [η χ ] g = [ψ] g . Moreover, the choice of the adapted partition of unity {χ ± } doesn't matter, since for any other choice {χ ′ ± } the differences (χ ′ ± − χ ± )β * have compact support, so that [η χ ′ ] = [η χ ].
Thus, the composition of maps is independent of the choice of the adapted partition of unity and, as desired, its image coincides with the image of the subset of S SC (F ) consisting of elements of the form ψ =ġ[ε ′ ] with ε ′ ∈ Γ SC (P ). Thus, each such ψ =ġ [ε] wherek[ε] = 0 precisely when the image of g K is trivial.
It was remarked in the above proof that J •ġ = 0 andġ * • J = 0. This is actually enough information to prove that the Jacobi operator must factor throughġ ′ on the right and throughġ ′ * on the left. Lemma 3.8. There exists a differential operator J g such that J = J g •ġ ′ =ġ ′ * • J * g .
Finally, we motivate the Peierls formula and then state and prove the main theorem of this section. Equivalence with a constrained hyperbolic system now allows us to solve the inhomogeneous problem where the source must necessarily satisfy the gauge invariance conditionġ * [α * ] = 0. The equivalent inhomogeneous problem in symmetric hyperbolic form iṡ Recall from Lem. 2.2 that this system is solvable iff the sources satisfy the consistency identity:q which is obviously satisfied, after using identity (81) Motivated by this formula, we introduce the following retarded, advanced and causal Green functions for the gauge fixed Jacobi system.
We also call E the Peierls or Jacobi causal Green function.
Finally, a Cauchy surface Σ ⊂ M and a partition of unity {χ ± } adapted to it define the following splittings at Γ 0 (F * ) and Γ SC (F ): The conclusion of the theorem is rather dense with information, so its proof is somewhat lengthy. However it simply consists of checking the properties of the horizontal sequence in the above diagram at each of its objects.
Proof. The fact that successive maps compose to zero, after taking the vertical cohomologies, is established in items (2) and (3) below, which also prove exactness of the resulting complex at Γ 0 (F * ) and Γ SC (F ). On the other hand, cohomologies at Γ 0 (F ) and Γ SC (F * ) are computed in items (1) and (4).  (3) and (4).
Note that below we make liberal use of various maps defined using the adapted partition of unity {χ ± } introduced in the hypothesis of the theorem (cf. Lem. 2.1).
where we have used the identity (81) and exactness of the sequence in Prp. 2.1. For the second part, letα * = J χ [ψ] = ±J[χ ± ψ], so thatα * ∈ Γ 0 (F * ). We claim that ψ ′ = E[α * ] differs from ψ only by a pure gauge termġ[ε], with ε ∈ Γ SC (P ). We examine closely the following expression, which appears in the definition of ψ ′ ,r where we have used the identities (77) and (78). Note that, while the above expression has compact support, the two individual terms do not. It is important to be able to decompose this expression in two different ways: into terms having retarded support (+) or advanced support (−). Direct calculation then shows Therefore, the desired conclusion holds, where we have used the formal adjoint versions of Eqs. (76) and the gauge fixing compatibility condition (83). Therefore, the desired conclusion holds, with Therefore, we have established that, after taking vertical cohomologies, the cohomology of (100) at Γ SC (F * ) is isomorphic to H g * SC (F * ). As mentioned before, it is easy to see from its variational nature that the Jacobi operator is self-adjoint J * = J. If it were directly invertible, the Green functions E ± would satisfy the same relation with their adjoints as shown in Sec. (2.5), making the causal Green function anti-self-adjoint, (E) * = −E. However, due to gauge invariance the relation of the gauge fixed Green functions to their adjoints is more complicated.
Lemma 3.10. When restricted to act on gauge invariant dual densities, the causal Green function of the gauge fixed Jacobi system is anti-self-adjoint up to gauge: Proof. First, note that from identities (76) and (77) we have It then follows that and Given that E = E + − E − , we then have which gives the desired conclusion.
We conclude this section by drawing attention to the fact that the kind of gauge fixing that features in a hyperbolization, as discussed in Sec. 3.3.1 is a special kind of partial gauge fixing. We refer to it as purely hyperbolic. Any further gauge fixing conditions are then called residual. We leave the consideration of residual gauge fixing to future work. A principal difficulty in dealing with residual gauge fixing conditions is that the resulting constraints are no longer parametrizable (such as operators that are elliptic on a family of spatial slices). Thus, the kernel of the gauge fixing conditions may contain very few, if any solutions with spacelike compact support, which would be difficult to fit into the current formal framework for tangent and cotangent spaces to the space of solutions.

Formal symplectic structure
Below, we construct a formal symplectic formΩ using the covariant phase space formalism. That is, we will integrate the presymplectic current density ω, derived in Sec. 3.1, over a Cauchy surface. Any Cauchy surface would do, giving the same result. The resulting form can in general be degenerate, though, and only becomes symplectic once projected to the gauge invariant formal tangent space T φS . Definition 3.10. Consider a variational system (Sec. 3.1) with presymplectic form ω ∈ Ω n−1,2 (F ) (Appendix A). Suppose that it is hyperbolizable (Sec. 3.3.1) and φ ∈ S H (F ) is a background solution with good causal behavior (Sec. 3.2.1), so that the linearized equations of motion endow M with a globally hyperbolic causal structure (Appendix C). Then, given a Cauchy surface Σ ⊂ M , we define the formal presymplectic 2-form Ω on the formal tangent space T φS by the formula Recall that for any section ψ, ξ ∈ Γ(F ) we can define the prolonged evolutionary vector fieldsψ,ξ on J ∞ F ) (Appendix A), which can be then be contracted with ω ∈ Ω * (J ∞ F ). Ideally, we would now show that Ω defines a smooth, closed differential form on the possibly infinite dimensional space of solutions S H (F ). However, we would then need to make explicit use of the infinite dimensional differential structure on S H (F ) and T S H (F ), which we have consistently avoided doing in this review, preferring a formal approach, with minimal analytical details. So instead, we will settle for showing that it is formally smooth and closed. These names are simply place holders for the identities demonstrated in the proof of the following Proof. First, we note that if χ, ξ ∈ T φ S then the integral defining Ω is necessarily finite, since the integrand ω[χ, ξ] = (j ∞ φ) * [ιξιχω] has spacelike compact support as both χ and ξ do. Independence of the choice of Σ follows if we can show that ω[χ, ξ] is de Rham closed on M . This follows directly from the horizontal, on-shell closedness of ω in the variational bicomplex (Lem. 3.1): For a fixed background solution φ ∈ S H (F ), the formal 2-form Ω(χ, ξ) is defined as a Cauchy surface integral of a bidifferential operator ω[χ, ξ], which is defined by a form ω ∈ Ω * (J ∞ F ). Hence we are happy to declare Ω to be formally smooth in its dependence on φ, as long as ω itself is smooth, which it is by construction. Also, in this simple case, we are justified in declaring the formal de Rham differential δ on S H (F ) to act on Ω as the vertical differential d v under the integral sign. Therefore, in this context, it is straight forward to check that Ω is formally closed since ω is on-shell, vertically closed (Lem. 3.1): This concludes the proof.
Though this was not attempted in Refs. 12-14, their rigorous setting for infinite dimensional geometry can be used to remove the formal character of the above lemma. Also, it is quite clear from the proof that the integration surface Σ in the definition of Ω need not actually be a Cauchy surface. It need only be in the same homology b class as a Cauchy surface. In particular, it is enough that Σ coincides with some Cauchy surface outside a compact set. A bilinear form defines a linear map from a vector space to its algebraic dual. A similar statement holds for a continuous bilinear form and the topological dual space. However, our formal cotangent spaces T * φ S and T * φS are neither the algebraic b The appropriate homology theory here should correspond to a variant of locally finite Borel-Moore homology, where one considers only chains whose intersection with every spacelike compact set is compact. This variant does not appear to have gotten any attention in the literature and thus deserves further study.
nor the topological duals of the formal tangent spaces T φ S and T φS . Thus we have to check this property for Ω by hand. This will be accomplished using one of the splitting maps for the Jacobi system from Thm. 3 where J : Γ(F ) → Γ(F * ) the Jacobi differential operator and {χ ± } is a partition of unity adapted to a Cauchy surface Σ.
Proof. Using the adapted partition of unity, we can write any spacelike compactly supported solution ψ ofḟ [ψ] = 0 as ψ = ψ + + ψ − , with ψ ± = χ ± ψ now being of retarded and advanced supports. If ψ also satisfies the constraintsċ . Note that the support of J[ψ ± ] is compact, since ψ ± satisfy the Jacobi equation away from the intersection S + ∩ S − ∩ supp ψ, which is by hypothesis compact. Next, we want to find a compactly supported dual densityα * that satisfies Ω(ξ, ψ) = ξ,α * for any ξ ∈ T φ S, which in particular satisfies J[ξ] = 0. Recall that an adapted partition of unity also depends on two additional Cauchy surfaces Σ ± ⊂ I ± (Σ) and the supports of the partition are contained in supp χ ± ⊆ S ± = I ± (Σ ∓ ).
The following direct calculation helps us identifyα * .
Note that after the integration by parts, the boundary integrals over Σ ± were dropped since they did not intersect the support of their integrands. Then, since supp ψ ± ⊆ S ± , the integration over S ± ∩ I ∓ (Σ) was extended to all of I ∓ (Σ). Finally, the term ψ ± · J[ξ] was dropped since ξ is a linearized solution.
To complete the proof, we use the non-degeneracy of the natural pairing between T φ S and T * φ S (Lem. 3.5, which we can invoke because of the global parametrizability and recognizability hypotheses) to define the operator Ω by the formula Moreover, this map is independent of the choice of adapted partition of unity {χ ± }.
Proof. Notice that in the presence of gauge symmetries (residual gauge freedom is present after a purely hyperbolic gauge fixing) the form Ω is degenerate, since every pure gauge solution lies in its kernel: for any ψ, since Noether's second theorem implies 3 thatġ * • J = 0. So, the first part is established. For the second part, recall that we are not interested in the dual densitỹ α * = J χ [ψ] specifically, which explicitly depends on the adapted partition of unity, but rather the equivalence class [α * ] ∈ T * φ S, which is defined modulo imḟ * and imċ * . Equivalently, following a conclusion of Thm. 3.1, since we actually want [α * ] ∈ T * φS ∼ = T * φ S/ imġ, it is enough to considerα * modulo im J. Consider another adapted partition of unity {χ ′ ± }. Because each partition of unity provides a splitting map (Thm. 3.1), if we consider equivalence classes of solutions modulo imġ, So, by exactness of the sequence in Thm. 3.1, J χ [ψ] and J χ ′ [ψ] must differ by an element of im J; in other words, they represent the same equivalence class in T * φS . Finally, the projected mapΩ : withα * = J χ [ψ], which is sufficient because the natural pairing between T φS and T * φS is non-degenerate (Lem. 3.5, again which we can invoke by the global parametrizability and recognizability hypotheses).
So, formally, the quotient projection to the physical phase space effects a presymplectic reduction (S H (F ), Ω) → (S H (F ),Ω). We shall see later on thatΩ is nondegenerate and hence symplectic.
Remark 3.4. Note that the relation betweenΩ as a bilinear form on the formal tangent space T φS and the linear map J χ : T φS → T * φS relies on the non-degeneracy of the natural pairing between the formal tangent and cotangent spaces. This nondegeneracy, as proven in Sec. 3.2.5, relies on the cohomological conditions that we call global parametrizability and global recognizability (Secs. 3.2.2 and 3.2.3). It is clear that, if these conditions fail and hence the natural pairing is degenerate, the formΩ(ψ, ξ) = ψ, J χ [ξ] may be degenerate, even if the operator J χ is not. This is bound to happen, because, as one of the conclusions of Thm. 3.1, J χ is invertible under the weaker hypotheses of local parametrizability and local recognizability. Such a degeneracy has already been noted, for instance, in Refs. 34 and 76.

Formal Poisson bivector, Peierls formula
Below, we construct a formal Poisson bivector Π, using the Peierls formula where E is again the causal Green function of the Jacobi operator J as defined in Sec. 3.3.1. To show that Π is indeed a Poisson bivector, it suffices to show that (a) it is an antisymmetric bilinear form on the formal cotangent space, (b) it defines a map from the formal cotangent space to the formal tangent space and (c) it is a two-sided inverse ofΩ defined in Cor. 3.1. We actually postpone part (c) to Sec. 3.3. The fact that Π defines a Poisson bracket, with its Leibniz and Jacobi identities, then formally follows from standard arguments.
Proof. Recall that the representatives always satisfyġ * [α * ] =ġ * [β * ] = 0. Appealing directly to the anti-self-adjointness identity (140) we have First of all, the result itself is not completely new. On the one hand, Peierls' original paper 6 already outlined an argument for the equivalence of his proposed bracket and the standard Poisson bracket of the Hamiltonian formalism, defined with respect to a preferred time foliation. On the other hand, when the covariant phase space formalism was introduced, the use of the symplectic current density 3-5, 54 was justified by its agreement with the standard symplectic structure of the Hamiltonian formalism. These two observations were joined into a detailed argument by Barnich, Henneaux and Schomblond, 8 which covered the case when the Hamiltonian formalism includes first class (gauge) and second class constraints.
Note that both the covariant phase space and Peierls bracket formalisms are fully covariant, but their equivalence had only been demonstrated using a non-covariant Hamiltonian formalism as an intermediate step. So, one reason to look for improvements is the desire to make the argument covariant throughout and bypass the Hamiltonian formalism all together. Another reason is to make clear all mathematical assumptions necessary to make the intermediate constructions well defined. In particular, the existence of advanced and retarded Green functions, needed by the Peierls formula, is guaranteed by standard mathematical results in PDE theory only if the field theory equations of motion (the Euler-Lagrange equations) satisfy some local and global hyperbolicity c requirements. We have exhibited these assumptions bundled within the notions of hyperbolizability (Def. 3.8), a global causal condition generalizing global hyperbolicity (Appendix C), as well as global parametrizability and recognizability (Secs. 3. 2.2 and 3.2.3). Also, the formal presymplectic form (Def. 3.10) is defined only when the integral over the presymplectic current converges. Again, a sufficient condition for this integral to converge is to restrict the support of linearized solutions plugged into the presymplectic form to be spacelike c In the spirit of being inclusive, we have equated our basic notion of hyperbolicity precisely with the existence of retarded and advanced Green functions (Green hyperbolicity). However, as pointed out earlier, there are large classes of PDEs easily identifiable by their principal symbols (including wave-like and symmetric hyperbolic systems) that are well known to be Green hyperbolic.
compact. This restriction is the main reason for defining the formal tangent spaces to consist of field sections of spacelike compact support (Secs. 3.2.4, 3.2.5).
The main technical result leading up to the theorem below is of course Thm. 3.1, which reduces to Prp. 2.1 when the Euler-Lagrange equations are directly in hyperbolic form (gauge invariance and constraints are absent). The exactness of parts of the horizontal sequence (100) (after taking the vertical cohomologies) can be seen as a precise characterization of the kernel and cokernel of the causal Green function E, defined in Eq. (99). It is this characterization that is the main motivation behind defining the formal cotangent spaces to consist of dual densities of compact support (Secs. 3.2.4, 3.2.5). If these support restrictions were relaxed, for instance, to timelike compact support for dual densities, then the causal Green function E need not be invertible due to global Aharonov-Bohm type effects. 76 In that case, the relation of the Peierls formula to the presymplectic form must be more subtle. The final technical result that is used in the proof below is the non-degeneracy of the natural pairing between the formal tangent and cotangent spaces (Sec. 3.2.5), which rely on rather technical sufficient conditions that we have dubbed global parametrizability of constraints (Sec = ψ +ġ[ε],α * (for some ε ∈ Γ SC (P )) (178) Therefore, from the non-degeneracy of the natural pairing between T φS and T * φS (Lem. 3.5), we concluded that ΠΩ = id. Similarly, we have But then for some ε ∈ Γ SC (P ). But, by the exactness (after taking vertical cohomologies) of the horizontal sequence (100) Therefore, from the non-degeneracy of the natural pairing between T φS and T * φS , we concluded thatΩΠ = id.
It is interesting to note that the construction of the Poisson bivector Π via the Peierls formula requires gauge fixing the equations of motion. On the other hand, the construction of the symplectic formΩ does not. Since, after gauge reduction, the two are mutual inverses, the Poisson bivector on the gauge invariant solutions space ultimately does not depend on gauge fixing. There is another way to see that result. For recognizable gauge transformations, the gauge invariant field combina- On the other hand, the equivalence formulas (76) and the gauge fixing compatibility condition (83) imply that the same is true even if only J[ψ] = 0. In other words, the systemk ′ [ξ] = 0 depends on J andġ ′ but not on the choice of gauge fixing. This is the case, for example, for Maxwell electrodynamics, where Maxwell's equations for the gauge invariant field strength F = F [A], where A is the gauge variant vector potential, by themselves constitute a (constrained) hyperbolic system. It is then not surprising that we can express the Poisson bivector acting on gauge invariant observables formed with respect to the gauge invariant field combinations directly in terms of the causal Green for thek ′ PDE system. This observation was known already to Peierls and this example of Maxwell electrodynamics appeared in his original paper. 6 Of course, if appropriate cohomologies in diagram (51) do not vanish, there may be gauge invariant observables not of that form, for which the gauge fixed Peierls Green function E would be necessary. Corollary 3.2. Given two gauge invariant dual densities of the formα * =ġ ′ * [α ′ * ] andβ * =ġ ′ * [β ′ * ], withα * ,β * ∈ Γ 0 (F * ) andα ′ * ,β ′ * ∈ Γ 0 (P ′ * ), we have the following identity Proof. Direct calculation shows which concludes the proof.
We conclude with a simple corollary that is sometimes known as classical microcausality.

Examples
In this section, we give a few examples of common relativistic field theories and show how they fit into the framework presented in this review. In particular, we make explicit the various identities needed to show that they are hyperbolizable according to Def. 3.8. We freely use the notation introduced in Sec. 3.3.1. In all the examples, we will concentrate on linear theories or the linearizations of non-linear ones, as discussed in Sec. 3.2.1.
g is a globally hyperbolic Lorentzian metric on M , and |g| = det g ij . The Jacobi equations have a wave-like principal symbol and, given the global hyperbolicity of the metric, are well known to be Green hyperbolic, 58, 63 soḟ = J. The constraints and the gauge transformations are trivial,ċ = 0 andġ = 0. Note that, for the existence of Green functions, no constraints need to be imposed on V ′ (φ) beyond smoothness, so tachyonic theories and theories with variable mass are hyperbolizable as well.
A more detailed treatment can be found for instance in Ref. 10.
Before proceeding, we introduce some key linear differential operators and their adjoints. To start, the trace reversal operator ρ : Γ(S 2 T * M ) → Γ(S 2 T * M ) does not actually involve any derivatives and in components is given by ρ ij [ψ] = ψ ij − 1 2 φ ij ψ. With our conventions, it is self-adjoint, ρ * = ρ, and also idempotent, with the usual notation (−) ;i = ∇ i (−), is the linearized Riemann curvature operator Γ(S 2 T * M ) → Γ(RM ) and RM → M is the sub-bundle of (T * ) ⊗4 M that satisfies the algebraic symmetries of the Riemann tensor. It's adjoint operatoṙ Finally, we define the following self-adjoint hyperbolic differential operator with W : Γ(S 2 T * M ) → Γ(S 2 T * M ). Note that W has a wave-like principal symbol so it is known to be Green hyperbolic. 58,63,79 The Jacobi equations are invariant under gauge transformations (linearized diffeomorphisms) with generatorġ[v] = K[v], the Killing operator, where the gauge parameter bundle is P = T * M . The de Donder gauge plays the role of a purely hy- The equivalence is effected by the operatorsr = −2ρ,r c = 2K,r J = 0 andr g = id. The parametrizability (diagram 47) and recognizability (diagram 50) identities are generated by the following commutative diagrams: We do not give explicit general expressions for the operators K ′ and W ′ , simply because they do not seem to be available in the literature. On the other hand, they must exist for abstract reasons. Namely, if we define K ′ as differential operator extending K to a formally exact sequence, then it always exists, as mentioned in Appendix B.2. Further, the composition of operators clearly annihilates the image of the Killing operator K, due to the gauge invariance of L. Therefore, by Lem. 2.3, there must exist a factorization K ′ • W = W ′ • K ′ , which we will conjecture to have a wave-like principal symbol (like W and + Λ do) and hence be Green hyperbolic.
If we assume that the background metric tensor φ to have constant curvature, we can say much more. In particular we know that K ′ =Ṙ and that the compositioṅ R ′ • K = 0 forms part of a larger elliptic complex, 80 in many ways analogous to the de Rham complex. In the even more special case of zero curvature (and hence also Λ = 0), all operators can be expressed with constant coefficients in local inertial coordinate systems formal, which makes it easier to check formal exactness directly. Also, in that case all the hyperbolic operators become equal to , the wave operator.
A more detailed treatment of the quantization of the graviton field on arbitrary cosmological vacuum backgrounds can be found in Ref. 79, though without introducing operators analogous to K ′ and W ′ .

Discussion
We have reviewed in detail the covariant phase space formalism and the Peierls formula, which endow the space of solutions of a classical field theory, respectively, with symplectic and Poisson structures, thus giving it the structure of a phase space, well known to be equivalent to the canonical phase space. Each of these constructions is covariant and does not require the non-covariant, canonical Hamiltonian formalism as an intermediary. In distinction with much of the existing literature, where the following aspects have often been left implicit, we have spelled out precise conditions under which these constructions succeed without mathematical ambiguities or difficulties. While it has long been known that the resulting symplectic and Poisson structures are equivalent (the symplectic form and the Poisson bivector are mutual inverses), despite the covariant construction, existing proofs still required the canonical Hamiltonian formalism as an intermediary. The main result in our presentation, which also happens to be novel, is a detailed and completely covariant proof of the equivalence under a precise set of sufficient conditions. The proof follows the ideas of the previous work of Forger & Romero, 10 but is generalized to field theories more general than scalar fields. Our argument holds for theories that also include constraints and that may have gauge symmetries. The list of examples to which the argument is applicable includes essentially all relativistic field theories of physical interest.
Despite the fact that the phase space of a field theory in more than one spacetime dimension (which corresponds to ordinary mechanical systems) is infinite dimensional, we have systematically avoided a discussion of functional analytical details needed in a theory of infinite dimensional geometry. Instead, we have treated formally the minimal geometric details needed in our presentation. Essentially, we have restricted our discussion to linear PDEs (or rather, linearizations of non-linear ones) and their solution spaces by appealing to the fact that the inversion of a symplectic form or a Poisson bivector requires only the tangent or cotangent space at a single point of the phase space (a background solution). However, the precise algebraic and differential geometric identities given here can be used as a core in a future investigation that would fill in the missing functional analytic details. In fact, some attempts along these lines have already been made elsewhere. For instance, Ref. 14 has done precisely that but only for the more restrictive class of scalar field theories. On the other hand, Refs. 12, 13 have considered more general theories, including those with gauge theories. Incidentally, these references have concentrated on the so-called off-shell formalism and, while heavily relying on the Peierls formula, did not consider its relation to the corresponding covariant symplectic structure, which requires restriction to solutions to be well defined.
The sufficient conditions we have introduced for the Peierls inversion formula to hold, the (global) parametrizability of constraints and the (global) recognizability or gauge transformations, have two aspects. See Remarks 3.3 and 3.4 regarding the subtle interplay between these conditions and the hypotheses that are sufficient to establish non-degeneracy of symplectic and Poisson structures described in this review. The local version is expected to hold generically for relativistic field theories of physical interest, as illustrated by the examples of Sec. 4. The global version, on the other hand has a cohomological character and it is actually known to fail in spacetimes with certain topological properties. 34,76 The main examples of these problematic cases have come from studying Maxwell electrodynamics on spacetimes with non-trivial spatial topology. 34,76 It would be nice to identify more key examples and study their properties. This would require the computation of cohomologies of the de Rham and other formally exact complexes with causally restricted supports (e.g., advanced, retarded, spacelike compact, timelike compact). The techniques needed for such computations go a bit beyond the standard treatments of de Rham cohomology with unrestricted or compact supports, as presented in standard differential geometry and differential topology texts. They will be addressed elsewhere. 81 More generally, compact or spacelike compact supports, featuring in the sufficient conditions discussed above, may be too restrictive for physical purposes, for example when dealing with infrared issues on spatially non-compact spacetimes. In those cases, the solution, of course, is to introduce boundary conditions at infinity. However, as is well known, there may not always be a uniquely preferred set of boundary conditions. In fact, boundary conditions are expected to be dictated by detailed physical considerations, which may vary from problem to problem. The main difficulty in relaxing the spacelike compact support condition on linearized solutions is the divergence of the integral in the definition of the covariant symplectic form, Def. 3.10. This situation is reminiscent of the problem of extending unbounded, symmetric operators on a Hilbert space to larger domains, while maintaining their self-adjointness. 82 Perhaps a similar approach can be applied to the symplectic form, where its anti-symmetry would replace the self-adjointness condition, can be used to study the space of possible boundary conditions at infinity. Notably, an attempt in a direction implicitly similar to this suggestion can be found in Sec. 5.1 of Ref. 56. These ideas will be explored further in future work.

Acknowledgments
The author would like to thank Claudio Dappiaggi, Thomas-Paul Hack, Alexander Schenkel and Urs Schreiber for fruitful discussions, also Béatrice Bonga for feedback on the manuscript, and acknowledges support from the Netherlands Organisation for Scientific Research (NWO) (Project No. 680.47.413).

Appendix A. Jet bundles and the variational bicomplex
In this appendix, we briefly introduce jet bundles and fix the relevant notation. For simplicity, we restrict ourselves to fields taking values in vector bundles. However, the discussion could be straightforwardly generalized to general smooth bundles. More details, as well as a coordinate independent definition, can be found in the standard literature. [83][84][85] Fix a vector bundle F → M , with dim M = n, with fibers modeled on a vector space U , and consider an adapted coordinate patch R n × U , with coordinates (x i , u a ). Extend this patch to a k-jet patch R n × U × U n k by adding extra copies of U , with new coordinates (x i , u a , u a i , u a ij , . . . , u a i1···i k ), which formally denote the derivatives of ∂ i1i2··· φ a (x) of a section φ at x. To keep track of all the derivatives, we introduce multi-index notation. A multi-index I = i 1 i 2 · · · i k replaces the corresponding set of symmetric covariant coordinate indices (the multi-index does not change when the defining i's are permuted). The order of this multi-index is given by |I| = k, with |∅| = 0. To augment a multi-index by adding another index, we use the notation Ij = jI = i 1 · · · i k j. Thus we can write higher order derivatives as ∂ i1···i k φ(x) = ∂ I φ(x), the higher order jet coordinates as u a i1···i k = u a I and the total set of coordinates on a k-jet patch as (x i , u a I ), |I| ≤ k. In particular the empty multi-index I = ∅ corresponds to u a ∅ = u a . Since the higher derivatives are symmetric in all indices, the number of extra coordinates is given by n k = k l=1 dim S k R n , with S k denoting the symmetric tensor product. Given two different coordinate patches on F , we define the transition maps between the corresponding k-jet patches according to the usual calculus chain rule applied to higher order derivatives. These k-jet patches can be glued together into the total space of the k-jet bundle J k F → M , which includes J 0 F ∼ = F .
, but not naturally. Jet bundles come with natural projections J k F → J k−1 F , which simply discard all derivatives of order k. This projection gives J k F the structure of an affine bundle over the base J k−1 F , with fibers modeled on the vector bundle (F ⊗ M S k T * M ) k−1 → J k−1 F (see Def. Appendix A.1 next). The bundle J k F → J k−1 F is affine because, in general, bundle morphisms of J k F → J k F induced by vector bundle automorphisms of F are not linear but affine.
Given a vector bundle E → M it can be pulled back to the k-jet bundle along the projection J k F → M . We introduce a convenient notation for this pullback.
Definition Appendix A.1. We denote by (E) k → J k F the pullback of E → M to J k F , which then fits into the pullback commutative square Any smooth section φ : M → F automatically gives rise to its k-jet prolongation or k-prolongation j k φ : M → J k F . Namely j k φ is a section of the bundle J k F → M that is defined in a local adapted coordinate patch as One can think of the k-prolongation symbol as a differential operator of order k. In fact, any (not necessarily linear) differential operator of order k, can be written as a composition of j k with an order 0 (not necessarily linear) . Note that we are slightly abusing notation by denoting both the differential operator and the bundle morphism by the same symbol f . Further, we can define an l-prolongation of a differential operator f of order k, which is then a differential operator of order k + l, by composing with j l : . Prolongation is discussed briefly using coordinate-wise operations in Sec. Appendix B.1. The k-jet prolongation j k φ can now be thought of as a special case of bundle morphisms, that is, j k φ = p k φ, where on the right hand side we interpret φ as the base fixing bundle morphism to F → M from the trivial We can also define two special kinds of vector fields. A vector fieldξ is horizontal if its action in local coordinates iŝ (A.14) for some ξ i = ξ i (x, u a I ). In particular, the vector field∂ j , with ξ i = δ i j , is horizontal. Note that [∂ i ,∂ j ] = 0. A vector fieldψ is evolutionary if its action in local coordinates isψ ) for multi-index I = i 1 i 2 · · · i k (the order of application of these vector fields does not matter since they commute). Note that the ψ a can be seen as the fiber coordinate components of a section of the bundle (F ) ∞ → J ∞ F . These definitions can be checked to be coordinate independent.
One can show that for a horizontal vector fieldξ on J ∞ F there exists a vector field ξ φ on M such that their actions on scalar functions are intertwined by the pullback along the jet prolongation j ∞ φ of a section φ : M → F , 16) for any scalar function f on J ∞ F . Namely, in local coordinates, On the other hand, evolutionary vector fieldsψ satisfy the identities for any form α ∈ Ω * (J ∞ F ) and section ψ : M → F . Actually, ψ could be a section of (F ) k → J k F , that is, it could depend on φ a (x) and its derivatives and not only on x ∈ M . The only corresponding change in the above formula would be to replace εψ by ε(j k φ) * ψ. Ostensibly, L ψ should stand for the Lie derivative on the infinite dimensional manifold of sections of F → M , where the section ψ is identified with the vector field whose action on local coordinates is L ψ φ a (x) = ψ a (x). However, since we do not delve into the differential geometry of infinite dimensional manifolds here, we keep the symbol L ψ (j ∞ φ) * primitive and defined as above.
Integrations or differentiations by parts are carried out using the following basic identity The proof follows from basic differential topology. The obstruction is of a global topological nature [66, §7] and is related to the fact that not every embedded submanifold can be represented as the zero-set of a section of a vector bundle. Clearly, the equation form is not unique. For instance, applying any invertible transformation to the equations f = 0 gives another equation form f ′ = 0, which describes exactly the same PDE system. We refer to E → M as the equation bundle and to f or the pair (f, E) as the equation form of the PDE system E. A section φ : M → F , also referred to as a field configuration, is said to satisfy the PDE system E if the k-jet prolongation of φ is contained in E, j k φ(x) ∈ E x ⊂ J k x (F, M ). Then, equivalently, j k φ is a section of E → M . We denote the space of all solution sections by S(F ) ⊂ Γ(F ) or S E (F ) when the PDE system needs to be mentioned explicitly. Using the above proposition, we can equivalently say that φ is a solution of the PDE system E if Expressing the k-jet in local coordinates, j k φ(x) = (x, φ a (x), ∂ i φ a (x), . . .), it is clear that f (x, φ a (x), ∂ i φ a (x), . . .) = 0 is a system of partial differential equations in the usual sense of the term. Starting with a PDE system in the usual sense, its geometric form as a sub-bundle of the jet bundle can be obtained by a converse of the above lemma. At this point, the regularity assumptions on both E and f become important. Namely, the transversality properties of f ensure that the zero set of f = 0 is a submanifold of J k F and vice versa. The linear and affine structures on J k F give us the possibility of defining the notion of linear and quasilinear PDE systems.
Definition Appendix B.2. A PDE system E ⊂ J k F is called linear if E → M is a vector sub-bundle of the vector bundle J k F → M . The PDE system is called quasilinear if E → J k−1 F is an affine sub-bundle of the affine bundle J k F → J k−1 F . The connection to the usual meanings of these terms can be seen through adapted equation forms.
Lemma Appendix B.1. The PDE system E ⊂ J k F is linear iff it has an equation form (f, E), where f : J k F → E is a morphism of vector bundles over M .
The PDE system E ⊂ J k F is quasilinear iff it has an equation form (f, E), where f : J k F → E is a morphism of affine bundles, which fits into the commutative diagram where the vertical maps define the affine bundles, with the vector bundle E → M naturally considered an affine one.
Definition Appendix B.3. Consider two field bundles F i → M , i = 1, 2, and two PDE systems E i ⊆ J ki F i . Denote the corresponding spaces of smooth solution sections by S i (F i ). The PDE systems E 1 and E 2 are said to be equivalent if there exist bundle morphisms e ij : J li F i → F j , i = j, such that φ i ∈ S i (F i ) and φ j = e ij • j li φ i implies φ j ∈ S j (F j ), (B.7) as well as that e 12 • j l1 and e 21 • j l2 are mutual inverses when restricted to the solution spaces S 1 (F 1 ) and S 2 (F 2 ).
We can easily extend the notion of equivalence to equation forms of PDE systems. In that case two different equations forms that define the same PDE system manifold are trivially equivalent. Note that neither the field bundles nor the orders of the PDE systems need to be same for equivalence to hold. Let us restrict to the case that is of importance elsewhere in this review, namely of F 1 = F 2 = F and e 12 and e 21 respectively equal to the canonical projections J l1 F → F and J l2 F → F , which are in a sense trivial. In this case, it is certainly sufficient that E 1 = E 2 for equivalence to hold, but it is not necessary. In fact E 1 and E 2 could be of different orders. To obtain necessary conditions for equivalence, we need to consider prolongation of PDE systems and the possible resulting integrability conditions.
A discussion of these notions in the setting of the jet bundle description of PDE systems can be rather technical. On the other hand, the theory of equivalence of PDE systems formulated in these terms has become quite mature and has yielded some important results. The technical details of this theory can be found for example in Refs. 86, 87. Below we give a brief non-technical introduction to this theory and state some simplified results relevant for hyperbolic systems.
The step by step derivation and inclusion of integrability conditions into a PDE is called prolongation. It is easiest to define prolongation in equation form and in local coordinates. Consider an equation form (f, E) of a PDE system E ⊆ J k F , as well as local coordinates (x i , u a ) on F and (x i , v A ) on E. If the section φ : M → F satisfies the PDE system, we have the following system of equations holding in local coordinates In the literature on relativity, causal structure is most often studied as a subset of Lorentzian geometry. 92 The term Lorentzian geometry refers to the study of structures induced on spacetime manifolds by the presence of a Lorentzian metric. One of these structures consists of the cones of null vectors. In particular, it is these cones that determine causal relationships between points in Lorentzian spacetimes. While causal relationships themselves can be defined solely in terms of the null cones, the reason they deserve the name causal is that they also describe the maximal speed at which disturbances can travel in solutions of hyperbolic PDEs with wave-like 58, 63 principal symbols. However, a similar property holds for more general classes of hyperbolic equations, even those that have no relation to a Lorentzian metric. It stands to reason then that causal relationships should be definable in terms of the intrinsic geometry of such PDEs. Indeed, if we consider cones of so-called characteristic covectors 93 and define causal relationships in terms of them, surprisingly few changes are necessary, with Lorentzian null cones appearing as special cases for the class of wave-like PDEs mentioned above. It stands to reason to give the study of such cones the name characteristic geometry. On the other hand, we find it convenient to generalize even further and consider simply a priori given cones (in the tangent and cotangent space), thus abstracting and clarifying the geometric notions that go into the definitions and basic properties of causal relations. Thus, we shall actually be studying conal geometry or what have sometimes been called conal manifolds. 94,95 This abstraction highlights the fact that a basic tool in the study of causal structures should be differential topology, rather than pseudo-Riemannian geometry. Characteristic geometry and its abstraction to conal geometry are discussed in a fuller and more integrated way in Ref. 60. Some cues have been taken from previous attempts to abstract the notion of causal structure or causal order in Lorentzian geometry. 92,[96][97][98] The generalization from Lorentzian cones to more general ones, for the purposes of describing causality in quantum field theory has been considered before, 99-101 but not in a concrete way.
Each point of a conal manifold, referred to here as a cone bundle, is smoothly assigned an open cone (a set invariant under multiplication by positive scalars) of tangent or cotangent vectors.
Definition Appendix C.1. A smooth bundle C → M of finite dimensional manifolds is termed a cone bundle if there exists an enveloping vector bundle E → M and an inclusion bundle morphism ι : C ⊂ E, such that each fiber C x , x ∈ M , is an open convex cone in the corresponding fiber E x (where we have implicitly identified C with its image ι(C) ⊂ E). A bundle map χ : C → C ′ is a cone bundle morphism if, given corresponding enveloping vector bundles E → M and E ′ → M ′ , there exists a vector bundle morphism ψ : E → E ′ such that χ = ψ| C . Namely, the following diagrams exist and commute: (C.1) Before proceeding, we need some terminology concerning cones and operations on them. These notions are often used in convex geometry. 102 Definition Appendix C.2. Given a finite dimensional vector space V and a convex cone C ⊂ V , denote its closure byC and its open interior byC. We define the convex dual (a.k.a. polar dual ) C * ⊂ V * as the set We define the strict convex dual C ⊛ ⊂ V * as the set The attribute strict may be dropped from the description of C ⊛ when it is clear from context. It is easy to check the following Proposition Appendix C.1. Consider a convex cone C.
(i) The convex dual C * is always closed and also convex. In addition, C * * =C. The strict convex dual C ⊛ is always open and convex. (ii) C * \ {0} is non-empty iff C is contained in a closed half space. C ⊛ is non-empty iff C contains no affine line (it is salient). (iii) If C is open and salient, then C ⊛⊛ = C. (iv) The inclusion of cones C 1 ⊆ C 2 implies the reverse inclusion of their duals, C * 1 ⊇ C * 2 and C ⊛ 1 ⊇ C ⊛ 2 . (v) The convex dual of the intersection of closures of cones C 1 and C 2 is the convex union (convex hull of the union) of their duals, (C 1 ∩C 2 ) * = C * 1 + C * 2 , where the right hand side is written as a Minkowski sum, which for cones coincides with the convex hull of the union. The converse identity holds as well, (C 1 +C 2 ) * = C * 1 ∩ C * 2 . Similarly, if C 1 and C 2 are open and salient, then (C 1 ∩ C 2 ) ⊛ = C ⊛ 1 + C ⊛ 2 and (C 1 + C 2 ) ⊛ = C ⊛ 1 ∩ C ⊛ 2 .
We extend these operations to cone bundles by acting fiberwise. So, if C → M is a cone sub-bundle of a vector bundle E → M , then the convex dual cone bundle C * → M is the cone sub-bundle of E * → M such that each fiber C * x is the convex dual of the corresponding fiber C x , for x ∈ M . Similarly, we can define the strict convex dual cone bundle C ⊛ → M . The operations of intersection (∩) and convex union (+) are extended to cone bundles in the same way. Clearly if C ⊛ → M is a smooth bundle, it is also a cone bundle. On the other hand, C * → M is usually not (since it is usually not open), though for brevity we shall sometimes refer to it as a cone bundle anyway.
In the next definition, we make a slight break with the usual terminology concerning covectors in Lorentzian geometry. A covector naturally defines a codimension-1 subspace, its annihilator, of the tangent space. In the presence of a metric, a covector can be canonically identified with a vector. If this vector is timelike, the corresponding codimension-1 subspace is called spacelike. However, a direct means of identifying covectors with vectors is missing in general. On the other hand, the link between covectors and codimension-1 tangent subspaces is metric independent and since the term spacelike still makes sense for tangent subspaces, we transfer it to corresponding covectors naming them spacelike as well. At this point, one may recall some standard notions of Lorentzian geometry, as long as they are defined only in terms of timelike cones, and apply them to the geometry of cone bundles. Many of the standard theorems translate as well, some directly and others with some extra effort. We restrict ourselves to those that are relevant to the issues at hand. Fix a chronal cone bundle C → M on a spacetime manifold M . Note that any chronal cone bundle is time oriented, since by definition the fibers consist of single cones rather than double cones like in the Lorentzian case. By assumptions the cones are directed into the future. When (x, y) ∈ I + , we say that x chronologically precedes y and also write x ≪ y.
If we replace C byC in the above definitions, we obtain causal curves and the causal precedence relation J + ⊆ M × M , denoted x < y. The inverse relations are written I − and J − .