Natural symplectic structures for each field theory

In this article, a natural symplectic form on the space of all smooth sections with compact support of an arbitrary fiber bundle admitting a global section is constructed. If one has a local field theory whose Lagrangian’s kinetic term is a nondegenerate bilinear form on the total space, then the corresponding Poisson bracket produces the usual commutation relations for sections of low momentum. Field theories are described by sections of fiber (almost always vector) bundles as states of a physical system and functionals on these sections as physical observables. In the spirit of geometric quantization (cf. [1] for a good overview), to get commutator relations for the quantum analogues of these observables, one needs a Poisson bracket for such functionals, i.e. a symplectic structure on the space of sections. There are attempts to get such a structure for particular cases of field theories one of which can be found in [1]. Chernoff and Marsden ( [2]) use the natural symplectic form on the tangent space of a space of sections which is not enough for our purposes because we want to have a bracket for observables on the space itself. Kijowski ( [3]) shrinks the set of possible observables to Poincaré generators, field strength and its time-derivative. Most general approaches deal with functions on the total space of a related jet bundle instead of functions on the space of sections (cf. e.g. [4], [5], [6], [7]). The strategy will be to first define a form on sections of the bundle given by dπ instead of the original bundle π and then to pull it back via the jet embedding. The definition of the form, at least for trivial bundles, can already be found in [8] (p.185), but there, for the question of closedness, the reader is referred to [9] which does not contain any proof of closedness. Also the existence of a symplectic form on the total space is just assumed, so the problem of pull-back does not appear. Now we want to construct a symplectic structure on the infinite-dimensional manifold Γ(π) of smooth sections with compact support of a given vector bundle with bundle projection π : E → M ′ over spacetime (resp. over the world sheet in the case of string theory). Compact support means here that we fix a section of π (whose existence is, of course, a non-trivial condition) playing the role of the zero section and consider only sections differing from it in a compact subset of M . First, we go over to the bundle dπ = π : TE =: E → TM ′ =: M , whose total space is equipped with a symplectic structure via a nondegenerate bilinear form on