Galois quantum systems, irreducible polynomials and Riemann surfaces Available to Purchase

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.