A finite-dimensional canonical formalism in the classical field theory

Abstract

A canonical formalism based on the geometrical approach to the calculus of variations is given. The notion of multi-phase space is introduced which enables to define whole the canonical structure (physical quantities, Poisson bracket, canonical fields) without use of functional derivatives. All definitions are of pure geometrical (finite dimensional) character.

The observable algebra$\mathcal{O}$ (physical quantities algebra) obtained here is much smaller then the algebra of all (sufficiently smooth) functionals on the space of states, derived from the standard infinite-dimensional formulation. As it is known, the latter is much too large for purposes of quantization. As the examples prove, our algebra$\mathcal{O}$ could be an adequate start-point for quantization.

For simplifying the language the notion of observable-valued distribution is introduced. Many concrete physical examples are given. E.g. it is shown that some problems connected with gauge in electrodynamics are automatically solved in this approach. The introduced language allows to obtain the Noether theorem in a most natural way.