Bounded perturbations of the Heisenberg commutation relation via dilation theory

TL;DR

This paper extends dilation theory to analyze bounded perturbations of the Heisenberg commutation relation, providing new insights into the structure of unitary groups and their generators in quantum mechanics.

Contribution

It introduces a dilation distance framework for unitary groups and applies it to the Heisenberg relation, extending previous results to higher dimensions and improving bounds.

Findings

Finite dilation distance implies bounded perturbation of generators.

Reproduces and generalizes Haagerup-Rordam's result on canonical operators.

Provides improved bounds in higher-dimensional cases.

Abstract

We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d-tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao's higher-dimensional generalization of Haagerup and Rordam's result, and in typical cases we significantly improve control of the bound when the dimension grows.