Path Integrals on Euclidean Space Forms

TL;DR

This paper introduces a path integral quantization approach for flat compact manifolds using holomorphic functions in complexified spaces, aligning with known results in Euclidean space.

Contribution

It develops a novel quantization method for flat compact manifolds utilizing reproducing kernel Hilbert spaces of holomorphic functions.

Findings

Defines a Feynman propagator based on reproducing properties.

Results agree with known expressions in Euclidean space.

Provides a new framework for quantization on Euclidean space forms.

Abstract

In this paper we develop a quantization method for flat compact manifolds based on path integrals. In this method the Hilbert space of holomorphic functions in the complexification of the manifold is used. This space is a reproducing kernel Hilbert space. A definition of the Feynman propagator, based on the reproducing property of this space, is proposed. In the $\mathbb{R}^n$ case the obtained results coincide with the known expressions.