Biorthogonal quantum mechanics: super-quantum correlations and expectation values without definite probabilities

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.

Irreversibility of the renormalization group flow in non-unitary quantum field theory

“…In the context of non-hermitian quantum mechanics this is termed a biorthogonal basis [54,55,56]. In particular, we have 1 = n |R n L n |.…”

Quantum ${{\mathbb{F}}_{{\rm un}}}$: theq= 1 limit of Galois field quantum mechanics, projective geometry and the field with one element

“…While we have not discussed the limit q → 1 within the context of the Biorthogonal Quantum Mechanics we defined in Ref. 51, the premise is quite apparent there as well. The approach there was an alternative method of constructing a quantum-like theory on F N q .…”

Quantum Systems Based Upon Galois Fields — From Sub-Quantum to Super-Quantum Correlations

“…This was the prime motivation for our search for a simple model with super-quantum correlations in the context of discrete quantum theories over Galois fields. [2][3][4][5] Note that the CHSH inequality relies on the knowledge of expectation values and not probabilities. Though predicting probabilities and predicting expectation values may seem like the same thing, it turns out they are not necessarily when "mutations" are introduced.…”

“…In the following, we review our recent work on "mutant" quantum theories constructed on discrete and finite vector spaces over Galois fields. [2][3][4][5] (See also Refs. 6-9.)…”