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arXiv:1208.5189 · Journal of Physics A Mathematical and Theoretical

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

Lay Nam Chang, Zachary Lewis, Djordje Minić, Tatsu Takeuchi

Published Nov 19, 2013• 11 citations
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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.

DOI
10.1088/1751-8113/46/48/485306

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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.
In this work, the consequences of several modifications to quantum mechanics are examined. These modifications, motivated by string theory, fall into two categories: ones in which the canonical commutation relations between position and momentum are deformed and ones in which the space of states used are vector spaces over Galois fields instead of complex Hilbert spaces. The particular deformation of the canonical commutation relations used leads to a minimum value of the uncertainty in position which is interpreted as a minimum length scale. Both harmonic and anharmonic oscillators are studied in this framework with normalizable, positive energy eigenstates found in both cases. The quantum uncertainty relations and classical counterparts to these states are discussed. Creating modified quantum theories by replacing the Hilbert spaces of canonical quantum mechanics with vector spaces defined over several finite, Galois fields is accomplished. Correlation functions are calculated in these theories and the maximum values are shown to not behave as would be expected by the standard, Bell-like, bounding inequality theorems. The interpretations and implications of these theories are discussed.
Finite quantum systems in which the position and momentum take values in the Galois field GF(pℓ), are studied. Ideas from the subject of field extension are transferred in the context of quantum mechanics. The Frobenius automorphisms in Galois fields lead naturally to the “Frobenius formalism” in a quantum context. The Hilbert space splits into “Frobenius subspaces” which are labeled with the irreducible polynomials associated with the ypℓ−y. The Frobenius maps transform unitarily the states of a Galois quantum system and leave fixed all states in some of its Galois subsystems (where the position and momentum take values in subfields of GF(pℓ)). An analytic representation of these systems in the ℓ-sheeted complex plane shows deeper links between Galois theory and Riemann surfaces.