Integral Transforms in Relativistic Quantum Constraint Mechanics

TL;DR

This paper develops Lorentz-invariant integral transforms in relativistic quantum constraint mechanics, enabling the representation of quantum states in a 3D subspace of Minkowski space and generalizing the Segal-Bargmann transform.

Contribution

It introduces Lorentz-invariant integral transforms for relativistic quantum systems, allowing 4-space representations to be constructed from 3D constraint space coordinates.

Findings

Transform methods preserve Lorentz invariance.

Complete equivalence classes of 4-space representations are constructed.

A relativistic generalization of the Segal-Bargmann transform is developed.

Abstract

In relativistic quantum constraint mechanics the state of a physical system is constrained to a 3-dimensional subspace of Minkowski 4-space. Fourier transformation can be used to relate this state between constraint spaces in 4-position and 4-momentum space. It is shown that integral transforms of this nature can be carried out using Lorentz-invariant 3-dimensional constraint space coordinates such that a complete equivalence class of 4-space representations can be constructed from the transform. This method is further applied to develop a relativistic generalization of the Segal-Bargmann transformation that leads to the representation of quantum systems in a three-dimensional subspace of Bargmann 4-space.