A natural symplectic form for every field theory

In this article, a natural weakly symplectic form on a natural configuration space associated to the fiber bundle of a given field theory is constructed. Although the form is in general not strongly symplectic, in the case of a vector bundle it gives rise to a Poisson bracket on a dense subspace of the differentiable L-functions on this space. If one has a local field theory whose Lagrangian’s kinetic term is a nondegenerate bilinear form on the total space, then the Poisson bracket produces the usual commutation relations. Classical field theories are described by sections of fiber bundles as states of a physical system and functions 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]) restricts 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]). Here we want to construct a weakly symplectic structure on the infinite-dimensional manifold Γ(π) of smooth sections with compact support of a fiber bundle with bundle projection π : E → Σ over a spacelike submanifold of spacetime (resp. of the world sheet in the case of string theory). Compact support means here that we fix a section of π playing the role of the zero section and consider only sections differing from it in a compact subset of Σ. The definition of a symplectic form for sections of a trivial bundle with a symplectic form on its fiber that is used in this construction 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. For fruitful discussions and advices, I want to thank Jürgen Jost, Peter Albers, Stefan Fredenhagen, Michael Holicki, Frank Klinker, Marc Nardmann, Tobias Preusser, Markus Roth, and Arnold Wassmer. Special thanks I would like to give to Hông Vân Lê for her great and friendly help and the correction of some bad mistakes.