Canonical pairs in finite-dimensional Hilbert space

2025/08/27 by Farrales, Ralph Adrian E., Galapon, Eric A. · 1 citation
#FOS: Physical sciences #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2508.19783

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.

Citations

• Instantaneous tunneling time within the theory of time-of-arrival operators
• Moyal deformation of the classical arrival time
• Partial and full tunneling processes across potential barriers
• Quantum corrections to the Weyl quantization of the classical time of arrival
• Optical Atomic Clocks
• Pauli's theorem and quantum canonical pairs: the consistency of a bounded, self–adjoint time operator canonically conjugate to a Hamiltonian with non–empty point spectrum

Cited by

• On a discrete version of the position-momentum commutation relation

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