Integral quantization based on the Heisenberg-Weyl group

TL;DR

This paper introduces a relativistic integral quantization framework for spinless particles in Minkowski spacetime using Heisenberg-Weyl group coherent states, with potential extensions to curved spacetimes and applications to quantum harmonic oscillators.

Contribution

It develops a new relativistic integral quantization scheme based on coherent states and positive operator-valued measures, extending nonrelativistic quantization methods to relativistic contexts.

Findings

Successfully quantized the nonrelativistic harmonic oscillator

Recovered standard Hamiltonian via the proposed quantization

Established a foundation for future transition amplitude calculations

Abstract

We develop a relativistic framework of integral quantization applied to the motion of spinless particles in the four-dimensional Minkowski spacetime. The proposed scheme is based on coherent states generated by the action of the Heisenberg-Weyl group and has been motivated by the Hamiltonian description of the geodesic motion in General Relativity. We believe that this formulation should also allow for a generalization to the motion of test particles in curved spacetimes. A key element in our construction is the use of suitably defined positive operator-valued measures. We show that this approach can be used to quantize the one-dimensional nonrelativistic harmonic oscillator, recovering the standard Hamiltonian as obtained by the canonical quantization. A direct application of our model, including a computation of transition amplitudes between states characterized by fixed positions and momenta, is postponed to a forthcoming article.