Canonical pairs in finite-dimensional Hilbert space

TL;DR

This paper demonstrates the existence of canonical pairs of Hermitian operators in finite-dimensional Hilbert spaces by focusing on subspaces where the canonical commutation relation holds, enabling new quantum mechanical constructions.

Contribution

It introduces a method to construct canonical pairs in finite dimensions by identifying subspaces where the canonical commutation relation is valid, challenging previous beliefs.

Findings

Canonical pairs exist in finite-dimensional Hilbert spaces within specific subspaces.

The study explores the uncertainty relations satisfied by these pairs.

Application to constructing time operators in finite-dimensional quantum systems.

Abstract

A pair of Hermitian operators is canonical if they satisfy the canonical commutation relation. It has been believed that no such canonical pair exists in finite-dimensional Hilbert space. Here, we obtain canonical pairs by noting that the canonical commutation relation holds in a proper subspace of the Hilbert space. For a given Hilbert space, we study the many possible canonical pairs and look into the uncertainty relation they satisfy. We apply our results by constructing time operators in finite-dimensional quantum mechanics.