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arXiv:2502.08494 · Physical Review Research

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

S. J. van Enk, D. Steck

Published Jul 10, 2026• 0 citations
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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.

DOI
10.1103/fzdt-p75h

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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.
Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ihihih where hhh is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.
Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatization is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "transposed" of a physical transformation. What is new in the present paper with respect to quant-ph/0603011 is the operational deduction of an involution corresponding to the "complex-conjugation" for effects, whose extension to transformations allows to define the "adjoint" of a transformation when the extension is composition-preserving.
Hamiltonians in the geveralized Feshbach-Villars and Foldy-Wouthuysen representations describing an interaction of a scalar particle with electromagnetic fields in the Minkowski spacetime are self-adjoint and Hermitian (or pseudo-Hermitian) when they are presented in terms of operators of covariant derivatives. When one uses curvilinear coordinates in special relativity, the transition to the canonical momentum operator does not change these properties. When the curvilinear coordinates are applied in general relativity, the corresponding transition to the canonical momentum operator leads to the seeming non- Hermiticity of the Hamiltonians. Since the Hamiltonians remain in fact Hermitian, this seeming non-Hermiticity should not be eliminated by any nonunitary transformation.
This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.