Integral quantization based on the Heisenberg-Weyl group

TL;DR

This work develops a relativistic integral quantization (IQ) framework for spinless particles in Minkowski spacetime using the four-dimensional Heisenberg-Weyl group. By constructing coherent states and employing positive operator-valued measures (POVMs), the authors map classical observables to self-adjoint quantum operators while allowing time to be treated as a quantum observable, with potential extensions to curved spacetimes. The approach reproduces canonical results for simple systems (e.g., the one-dimensional harmonic oscillator) and provides covariant quantization under Poincaré transformations, along with a formalism for relativistic Hamiltonians and transition amplitudes in configuration space; it also outlines generalizations to multi-particle systems and curved geometries, offering a pathway toward quantum gravity and quantum-geometric assessments of geodesic motion. The framework is poised to yield quantum-propagation amplitudes and mass-shell constrained dynamics in a covariant, group-theoretic setting, with possible observational implications for black-hole shadows and quantum cosmology.

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.