Quantum and Classical Fields in the Finite-Dimensional Formalism

TL;DR

This paper applies recent finite-dimensional quantization rules to Klein-Gordon and Dirac fields, proposes a new classical formalism and quantization rules, and discusses the resulting multi-particle interpretation with harmonic-oscillator-like degrees of freedom.

Contribution

It introduces an improved classical and quantum formalism for finite-dimensional quantum field theory, addressing previous limitations and revealing new multi-particle aspects.

Findings

New classical equations of motion and quantization rules overcome previous issues.

Quantum fields exhibit harmonic-oscillator-like degrees of freedom.

Formalism suggests a multi-particle interpretation.

Abstract

The quantization rules recently proposed by M. Navarro (and independently I.V. Kanatchikov) for a finite-dimensional formulation of quantum field theory are applied to the Klein-Gordon and the Dirac fields to obtain the quantum equations of motion of both fields. In doing so several problems arise. Solving these difficulties leads us to propose a new classical canonical formalism, which, in turn, leads us to new, improved rules of quantization. We show that the new classical equations of motion and rules of quantization overcome several known unsatisfactory features of the previous formalism. We argue that the new formalism is a general improvement with respect to the previous one. Further we show that the quantum field theory of the Dirac and Klein-Gordon field describes particles with extra, harmonic-oscillator-like degrees of freedom. We argue that these degrees of freedom should give rise to a multi-particle interpretation of the formalism.