Canonical pairs in finite-dimensional Hilbert space

• URL: http://arxiv.org/abs/2508.19783v1
• Date: Wed, 27 Aug 2025 11:04:35 GMT
• Title: Canonical pairs in finite-dimensional Hilbert space
• Authors: Ralph Adrian E. Farrales, Eric A. Galapon,
• Abstract summary: A pair of Hermitian operators is canonical if they satisfy the canonical commutation relation.<n>It has been believed that no such canonical pair exists in finite-dimensional Hilbert space.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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