On the Kostant-Souriau prequantization of scalar fields with polysymplectic structures

TL;DR

This paper introduces a new geometric method for quantizing scalar fields using polysymplectic structures, offering an alternative to canonical quantum field theory with some comparable fundamental results.

Contribution

It develops a purely differential geometric approach to field quantization based on polysymplectic structures, extending the Kostant-Souriau prequantization framework to scalar fields.

Findings

Operators differ from canonical quantum field theory

Reproduces some fundamental quantum results

Highlights current limitations and future directions

Abstract

In this paper, I present a novel, purely differential geometric approach to the quantization of scalar fields, with a special focus on the familiar case of Minkowski spacetimes. This approach is based on using the natural geometric structures of polysymplectic Hamiltonian field theory to produce an analog of the Kostant-Souriau prequantization map familiar from geometric quantization. I show that while the resulting operators are quite different from those of canonical quantum field theory, the approach is nonetheless able to reproduce a few of canonical quantum field theory's most fundamental results. I finish by elaborating the current limitations of this approach and briefly discussing future prospects.