Quantum mechanics with real numbers: entanglement, superselection rules and gauges

TL;DR

This paper demonstrates how imaginary numbers in quantum mechanics can be removed by enlarging the Hilbert Space and applying superselection rules, simplifying the mathematical framework while maintaining physical consistency.

Contribution

It introduces a method to eliminate imaginary numbers in quantum physics through Hilbert Space enlargement and superselection rules, with practical examples like a qubit and interferometer.

Findings

Imaginary numbers can be replaced with real numbers in quantum models.

The procedure preserves quantum phenomena such as entanglement.

The approach parallels constrained quantization in electromagnetic theory.

Abstract

We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert Space followed by an imposition of - what effectively amounts to - a superselection rule. We illustrate this procedure with a qubit and apply it to the Mach-Zehnder interferometer. The procedure is somewhat reminiscent of the constrained quantization of the electromagnetic field, where, in order to manifestly comply with relativity, one enlargers the Hilbert Space by quantizing the longitudinal and scalar modes, only to subsequently introduce a constraint to make sure that they are actually not directly observable.