On Quantization of Field Theories in Polymomentum Variables

TL;DR

This paper develops a covariant, multi-parameter quantum framework for field theories using polymomentum variables and Clifford algebra, connecting it to classical and quantum field theory principles.

Contribution

It introduces a hypercomplex covariant quantization scheme for field theories based on polymomentum variables and Clifford algebra, extending quantum mechanics.

Findings

Consistent covariant Schrödinger equation with classical limit

Relation to De Donder-Weyl Hamilton-Jacobi theory

Connection between wave function and quantum field theory functional

Abstract

Polymomentum canonical theories, which are manifestly covariant multi-parameter generalizations of the Hamiltonian formalism to field theory, are considered as a possible basis of quantization. We arrive at a multi-parameter hypercomplex generalization of quantum mechanics to field theory in which the algebra of complex numbers and a time parameter are replaced by the space-time Clifford algebra and space-time variables treated in a manifestly covariant fashion. The corresponding covariant generalization of the Schroedinger equation is shown to be consistent with several aspects of the correspondence principle such as a relation to the De Donder-Weyl Hamilton-Jacobi theory in the classical limit and the Ehrenfest theorem. A relation of the corresponding wave function (over a finite dimensional configuration space of field and space-time variables) to the Schroedinger wave functional in quantum field theory is examined in the ultra-local approximation.