Canonical pairs in finite-dimensional Hilbert space
Quantum Physics
2026-02-25 v1
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.
Keywords
Cite
@article{arxiv.2508.19783,
title = {Canonical pairs in finite-dimensional Hilbert space},
author = {Ralph Adrian E. Farrales and Eric A. Galapon},
journal= {arXiv preprint arXiv:2508.19783},
year = {2026}
}