Multi-Path and Multi-Particle Tests of Complex vs. Hyper-Complex Quantum Theory

TL;DR

This paper develops a mathematical framework for multi-path and multi-particle interference experiments to test whether quantum mechanics is based on complex numbers or more general hyper-complex systems, extending previous single-particle tests.

Contribution

It introduces a formal matrix approach for multi-path and multi-particle tests, enabling direct probing of the underlying number system in quantum mechanics beyond complex numbers.

Findings

Proposes a new mathematical formalism for interference tests.

Extends tests to multi-particle and multi-path scenarios.

Provides a method to distinguish complex from hyper-complex quantum theories.

Abstract

The axioms of quantum mechanics provide limited information regarding the structure of the Hilbert space, such as the underlying number system. The latter is generally regarded as complex, but generalizations of complex numbers, so-called hyper-complex numbers, cannot be ruled out in theory. Therefore, specialized experiments to test for hyper-complex quantum mechanics are needed. To date, experimental tests are limited to single-particle interference exploiting a closed phase relation in a three-path interferometer called the Peres test. The latter distinguishes complex quantum mechanics from quaternionic quantum mechanics. Here, we propose a general matrix formalism putting the Peres test on a solid mathematical ground. On this basis, we introduce multi-path and multi-particle interference tests, which provide a direct probe for any dimension of the number system of quantum mechanics.