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  • Open Access

All Hilbert spaces are the same: Consequences for generalized coordinates and momenta

S. J. van Enk* and Daniel A. Steck†

  • Department of Physics and Oregon Center for Optical, Molecular & Quantum Sciences and University of Oregon, Eugene, Oregon 97403, USA

  • *Contact author: svanenk@uoregon.edu
  • †Contact author: dsteck@uoregon.edu
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Phys. Rev. Research 8, 033117 – Published 28 July, 2026

DOI: https://doi.org/10.1103/fzdt-p75h

Abstract

Making use of the simple fact that all separable complex Hilbert spaces of given dimension are isomorphic, we show that there are just six basic ways to define generalized coordinate operators in quantum mechanics. In each case, a canonically conjugate generalized momentum operator can be defined, but it may not be self-adjoint. Even in those cases, we show there is always either a self-adjoint extension of the operator or a Neumark extension of the Hilbert space that produces a self-adjoint momentum operator. In one of the six cases, both extensions work, thus leading to seven basic pairs of coordinate and momentum operators. We also show why there are more ways of defining basic coordinate and momentum measurements. A special role is reserved for measurements that simultaneously but imperfectly measure both.

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References (45)

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Phys. Rev. Research 8, 033117– Published 28 July, 2026
  • Received 11 February 2025
  • Accepted 10 July 2026
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