Towards the Born-Weyl Quantization of Fields

TL;DR

This paper develops a covariant quantization framework for fields based on the De Donder-Weyl formalism, introducing a graded Poisson bracket and a covariant Schr"odinger equation, connecting to classical and quantum field theories.

Contribution

It introduces a covariant Born-Weyl quantization approach using graded Poisson brackets and a hypercomplex Schr"odinger equation in field theory.

Findings

Derivation of a covariant Schr"odinger equation for fields.

Connection between the covariant formalism and Hamilton-Jacobi equations.

Outline of relation to the functional Schr"odinger picture.

Abstract

Elements of the quantization in field theory based on the covariant polymomentum Hamiltonian formalism (the De Donder-Weyl theory), a possibility of which was originally discussed in 1934 by Born and Weyl, are developed. The approach is based on a recently proposed graded Poisson bracket on differential forms in field theory (see e.g. hep-th/9709229). A covariant analogue of the Schr\"odinger equation for a hypercomplex wave function on the space of field and space-time variables is put forward. It is shown to lead to the De Donder-Weyl Hamilton-Jacobi equations in quasiclassical limit. A possible relation to the functional Schr\"odinger picture in quantum field theory is outlined.