Characterization of point transformations in quantum mechanics

TL;DR

This paper provides a mathematical characterization of point transformations in quantum mechanics, demonstrating that they correspond to selfadjoint operators with continuous spectra satisfying canonical commutation relations.

Contribution

It establishes that all point transformations in quantum mechanics are represented by selfadjoint operators with the real line as their spectrum, preserving canonical commutation relations.

Findings

Point transformations correspond to selfadjoint operators in $L^2(\

\mathbb{R}^n)$.

These operators have continuous spectra coinciding with $\\mathbb{R}$.

Abstract

We characterize point transformations in quantum mechanics from the mathematical viewpoint. To conclude that the canonical variables given by each point transformation in quantum mechanics correctly describe the extended point transformation, we show that they are all selfadjoint operators in $L^2(\mathbb{R}^n)$ and that the continuous spectrum of each coincides with $\mathbb{R}$. They are also shown to satisfy the canonical commutation relations.