Path Integral in Modular Space

TL;DR

This paper develops a novel Feynman path integral formulation for the quantum harmonic oscillator in modular space, revealing new features like winding modes, Aharonov-Bohm phases, and non-classical locality, with potential applications to various physical systems.

Contribution

It introduces a modular path integral framework with a new action, symmetries, and a general prescription for constructing modular actions from Hamiltonians.

Findings

Path integral incorporates winding modes and Aharonov-Bohm phases.

Saddle points are sequences of superposition states.

New symmetries and a Legendre transform-like prescription are identified.

Abstract

The modular spaces are a family of polarizations of the Hilbert space that are based on Aharonov's modular variables and carry a rich geometric structure. We construct here, step by step, a Feynman path integral for the quantum harmonic oscillator in a modular polarization. This modular path integral is endowed with novel features such as a new action, winding modes, and an Aharonov-Bohm phase. Its saddle points are sequences of superposition states and they carry a non-classical concept of locality in alignment with the understanding of quantum reference frames. The action found in the modular path integral can be understood as living on a compact phase space and it possesses a new set of symmetries. Finally, we propose a prescription analogous to the Legendre transform, which can be applied generally to the Hamiltonian of a variety of physical systems to produce similar modular actions.