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arXiv:1206.0064 · Journal of Physics A Mathematical and Theoretical

Spin and rotations in Galois field quantum mechanics

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

Published Jan 29, 2013• 10 citations
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Abstract

We discuss the properties of Galois field quantum mechanics constructed on a vector space over the finite Galois field GF ( q ). In particular, we look at two-level systems analogous to spin, and discuss how SO (3) rotations could be embodied in such a system. We also consider two-particle 'spin' correlations and show that the Clauser–Horne-Shimony–Holt inequality is nonetheless not violated in this model.

DOI
10.1088/1751-8113/46/6/065304

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We discuss the properties of Galois field quantum mechanics constructed on a vector space over the finite Galois field GF ( q ). In particular, we look at two-level systems analogous to spin, and discuss how SO (3) rotations could be embodied in such a system. We also consider two-particle 'spin' correlations and show that the Clauser–Horne-Shimony–Holt inequality is nonetheless not violated in this model.
A 'Galois quantum system' in which the position and momentum take values in the Galois field GF( p ℓ ) is considered. It is comprised of ℓ-component systems which are coupled in a particular way and is described by a certain class of Hamiltonians. Displacements in the GF( p ℓ ) × GF( p ℓ ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF( p ℓ )) group. Wigner and Weyl functions are defined and their properties are studied. Frobenius symmetries, which are based on Frobenius automorphisms in the theory of Galois fields, are a unique feature of these systems (for ℓ ⩾ 2). If they commute with the Hamiltonian, there are constants of motion which are discussed. An analytic representation in the ℓ-sheeted complex plane provides an elegant formalism that embodies the properties of Frobenius transformations. The difference between a Galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed.
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.
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.