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

TL;DR

This paper demonstrates that all separable complex Hilbert spaces are isomorphic, leading to six fundamental ways to define generalized coordinate operators in Quantum Mechanics, each with implications for their conjugate momenta.

Contribution

It classifies the six basic methods to define generalized coordinates and momenta in Quantum Mechanics, clarifying their extensions and measurement possibilities.

Findings

Six fundamental coordinate operator definitions identified

Conjugate momentum operators can be extended to be self-adjoint

Seven pairs of coordinate and momentum operators emerge from extensions

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.