Geometric Quantization of free fields in space of motions

TL;DR

This paper presents a geometric quantization method for free fields using Kähler polarization, resulting in Lorentz-invariant canonical commutators consistent with canonical quantization.

Contribution

It introduces a covariant symplectic current approach to geometric quantization of free fields, ensuring invariance under Lorentz and discrete symmetries.

Findings

Canonical commutators match those from canonical quantization

Quantization method is Lorentz and parity invariant

Provides a geometric framework for free field quantization

Abstract

Via K$\ddot{a}$hker polarization we geometrically quantize free fields in the spaces of motions, namely the space of solutions of equations of motion. We obtain the correct results just as that given by the canonical quantization. Since we follow the method of covariant symplectic current proposed by Crnkovic, Witten and Zuckerman et al, the canonical commutator we obtained are naturally invariant under proper Lorentz transformation and the discrete parity and time transverse transformations, as well as the equations of motion.