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Journal of Physics A Mathematical and Theoretical

Quantum systems with finite Hilbert space: Galois fields in quantum mechanics

Apostol Vourdas

Published Aug 1, 2007• 83 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/40/33/r01

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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.
A finite quantum system in which the position and momentum take values in the Galois field GF( p ℓ ) is constructed from a smaller quantum system in which the position and momentum take values in , using field extension. The Galois trace is used in the definition of the Fourier transform. The Heisenberg–Weyl group of displacements and the Sp(2, GF( p ℓ )) group of symplectic transformations are studied. A class of transformations inspired by the Frobenius maps in Galois fields is introduced. The relationship of this 'Galois quantum system' with its subsystems in which the position and momentum take values in subfields of GF( p ℓ ) is 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.
Quantum systems in which the position and momentum take values in the ring calZd{\\cal Z}_dcalZd​ and which are described with ddd-dimensional Hilbert space, are considered. When ddd is the power of a prime, the position and momentum take values in the Galois field $GF(p^ \\ell)$, the position-momentum phase space is a finite geometry and the corresponding `Galois quantum systems' have stronger properties. The study of these systems uses ideas from the subject of field extension in the context of quantum mechanics. The Frobenius automorphism in Galois fields leads to Frobenius subspaces and Frobenius transformations in Galois quantum systems. Links between the Frobenius formalism and Riemann surfaces, are discussed.
Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU (2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.
A quantum system with positions and momenta in GF(pℓ) is considered. Such a system can be constructed from ℓ smaller systems, in which the positions and momenta take values in Zp, if the Hamiltonian of this ℓ-partite system is compatible with GF(pℓ). The concept of compatibility of a Hamiltonian with GF(pℓ) allows the quantum formalism in the ℓ-partite system to be expressed in terms of Galois arithmetic. Transformations of the basis in GF(pℓ) produce unitary transformations of the quantum states, which form a representation of GL(ℓ,Zp). They are used to define which subset of the general set of Hamiltonians in the ℓ-partite system is compatible with GF(pℓ).