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

TL;DR

The work shows that all separable Hilbert spaces are isomorphic, reducing the science of generalized coordinates to operator structure. It classifies six coordinate definitions (continuous/discrete and domain topology) and analyzes their canonically conjugate momentum operators, identifying when self-adjoint extensions or Neumark extensions are needed to obtain well-defined momentum operators, yielding seven basic coordinate–momentum pairs. It then connects these operator-theoretic results to measurement theory, showing that POVMs and Kraus operators describe realistic measurements and that coherent-state measurements provide a natural, pure joint measurement framework for position and momentum. The findings clarify how Hilbert-space structure constrains possible observables and measurements across continuous, semi-infinite, finite-interval, and finite-dimensional settings, with implications for quantum foundations and experimental design.

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 measure both.