An approach to nonstandard quantum mechanics

TL;DR

This paper introduces a nonstandard analysis framework for quantum mechanics using hyperfinite-dimensional spaces, enabling a unified treatment of bound and continuum states, and provides rigorous formulas for scattering theory.

Contribution

It develops a novel nonstandard analysis approach to quantum mechanics, extending the standard formalism and applying it to scattering theory with explicit formulas.

Findings

Unified treatment of bound and continuum states.

Rigorous derivation of Møller wave operators and S-Matrix.

Framework extends standard quantum mechanics formalism.

Abstract

We use nonstandard analysis to formulate quantum mechanics in hyperfinite-dimensional spaces. Self-adjoint operators on hyperfinite-dimensional spaces have complete eigensets, and bound states and continuum states of a Hamiltonian can thus be treated on an equal footing. We show that the formalism extends the standard formulation of quantum mechanics. To this end we develop the Loeb-function calculus in nonstandard hulls. The idea is to perform calculations in a hyperfinite-dimensional space, but to interpret expectation values in the corresponding nonstandard hull. We further apply the framework to non-relativistic quantum scattering theory. For time-dependent scattering theory, we identify the starting time and the finishing time of a scattering experiment, and we obtain a natural separation of time scales on which the preparation process, the interaction process, and the detection process take place. For time-independent scattering theory, we derive rigorously explicit formulas for the M{\o}ller wave operators and the S-Matrix.