Biorthogonal Quantum Mechanics: Super-Quantum Correlations and Expectation Values without Definite Probabilities

TL;DR

This paper introduces mutant quantum mechanics over finite Galois fields that retains expectation values but has indeterminate probabilities, revealing super-quantum correlations with implications for quantum gravity.

Contribution

It presents a novel mutation of quantum mechanics over GF(3) and GF(9), showing super-quantum correlations and a different approach to expectation values and probabilities.

Findings

Super-quantum correlations with Bell bound of 4

Expectation values retained, probabilities indeterminate

Implications for quantum gravity

Abstract

We propose mutant versions of quantum mechanics constructed on vector spaces over the finite Galois fields GF(3) and GF(9). The mutation we consider here is distinct from what we proposed in previous papers on Galois field quantum mechanics. In this new mutation, the canonical expression for expectation values is retained instead of that for probabilities. In fact, probabilities are indeterminate. Furthermore, it is shown that the mutant quantum mechanics over the finite field GF(9) exhibits super-quantum correlations (i.e. the Bell-Clauser-Horne-Shimony-Holt bound is 4). We comment on the fundamental physical importance of these results in the context of quantum gravity.