Obstacles to the quantization of general relativity using symplectic structures

TL;DR

This paper explores the challenges of applying symplectic and covariant Hamiltonian methods to quantize general relativity, highlighting obstacles and limitations of current approaches.

Contribution

It provides an overview of polysymplectic approaches and demonstrates why existing geometric quantization methods face difficulties with general relativity.

Findings

No difficulties in applying toy models to GR

Standard covariant Hamiltonian tools are inadequate for GR

Future work prospects are discussed

Abstract

In this paper I give overviews of the polysymplectic approach to covariant Hamiltonian field theory and the simplest geometric quantization of classical particle theories. I then give a synopsis of a recently proposed toy model for applying this geometric quantization map to polysymplectic field theory. I show that no special difficulties arise when this toy model is applied to general relativity. I then sketch the reasons why the standard tools of covariant Hamiltonian field theory are not up to the challenge of GR, so that the resulting quantum theory cannot be taken seriously. A few remarks are given about prospects for future work.