A Vourdas 2007 J. Phys. A: Math. Theor. 40 R285 doi:10.1088/1751-8113/40/33/R01
Quantum systems with finite Hilbert space: Galois fields in quantum mechanics
REVIEW ARTICLEA Vourdas
Show affiliationsTOPICAL REVIEW
A 'Galois quantum system' in which the position and momentum take values in the Galois field GF(pℓ) 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(pℓ) × GF(pℓ) phase space and the corresponding Heisenberg–Weyl group are studied. Symplectic transformations are shown to form the Sp(2, GF(pℓ)) 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 ![[{\bb Z}_p]^\ell](http://ej.iop.org/images/1751-8121/40/33/R01/jpa250345ieqn1.gif)
is discussed.
- PACS
- MSC
-
44A12 Radon transform (See also 92C55)
81Rxx Groups and algebras in quantum theory
81S30 Phase space methods including Wigner distributions, etc.
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
- Subjects
- Dates
-
Issue 33 (17 August 2007)
Received 11 May 2007
Published 1 August 2007
-
Quantum systems with finite Hilbert space: Galois fields in quantum mechanics
A Vourdas 2007 J. Phys. A: Math. Theor. 40 R285
-
Non-equilibrium statistical mechanics of liquid crystals: relaxation, viscosity and elasticity
C J Chan and E M Terentjev 2007 J. Phys. A: Math. Theor. 40 R103
-
The application of self-validation to wireless sensor networks
Michael A Collett et al 2008 Meas. Sci. Technol. 19 125201
-
Dephasing by electron–electron interactions in a ballistic Mach–Zehnder interferometer
Clemens Neuenhahn and Florian Marquardt 2008 New J. Phys. 10 115018
-
Mesoscopic conductance fluctuations in InAs nanowire-based SNS junctions
T S Jespersen et al 2009 New J. Phys. 11 113025
-
Angular diffraction
B Jack et al 2008 New J. Phys. 10 103013
-
From a quantum to a classical description of intense laser–atom physics with Bohmian trajectories
X Y Lai et al 2009 New J. Phys. 11 113035
-
Lattice vibrations in CaV2O5 and their manifestations: a theoretical study based on density functional theory
J Spitaler et al 2009 New J. Phys. 11 113009
-
Tracking nanoparticles in an optical microscope using caustics
Eann A Patterson and Maurice P Whelan 2008 Nanotechnology 19 105502
-
Operator fidelity approach to the quantum phase transition of the spin-1/2 XX chain with three-spin interaction and the (1/2,1) XXZ mixed-spin chain
Zhe Sun et al 2009 New J. Phys. 11 113005
Users also read
What's this?- Quantum systems with finite Hilbert space
- Symplectic transformations and quantum tomography in finite quantum systems
- Galois quantum systems
Related review articles
What's this?- Integrable structure of box–ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
- Lectures on localization and matrix models in supersymmetric Chern–Simons-matter theories
- Exact superconformal and Yangian symmetry of scattering amplitudes