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.

https://iopscience.iop.org/article/10.1088/0034-4885/67/3/R03/pdf

Quantum systems with finite Hilbert space

New Exclusion Limits on Dark Gauge Forces from Proton Bremsstrahlung in Beam-Dump Data

Theorem. If F is a df on [0,1], that is strictly increasing, and such that F(x) > x if O < x < ?, and F(x) < x if < x < 1, then F is not n-divisible for any n. There are similar, less simple results for df's with unbounded support. As a special case it follows that the uniform distribution is not n-divisible for any n though it is decomposable in many ways (see e.g., [2]). Kudina [1] uses a similar technique to prove the indecomposability of the arcsine df. It is well known that id df's with bounded support have zero variance, i.e., are degenerate. Similarly a df on [0,a] with variance exceeding a/(4n) is not ndivisible. This argument is extended to prove the following theorem (cf. [4], also for references).

Stochastic models for single neuron firing trains: a survey.

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 is discussed.

http://iopscience.iop.org/article/10.1088/1751-8113/40/33/R01/pdf

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

QCD jets as Markov branching processes

The Dirac Hamiltonian as a member of a hierarchy of matrices

On the scattering of pions by deuterons

Summary The probability distribution function π(n, t) of the number of registered events n in a time interval t for a counter in the case when the registered and unregistered events are followed by different dead times, is derived by using the method of product densities formulated recently by one of us (R.).

CXVII. A stochastic problem relating to counters

A class of stochastic processes associated with points randomly distributed in a line of finite extension L, is considered. A general integral equation for the function representing the probability distribution of the stochastic variable under consideration is derived and solved by using the Laplace transform technique. Examples of the above class of processes are cited. In particular, the problem of the fluctuations in brightness of the Milky Way is discussed in detail. The results of Chandrasekhar and Munch in regard to this astrophysical problem are derived in a simple and direct manner.

Alladi Ramakrishnan

Papers

The Dirac Hamiltonian as a member of a hierarchy of matrices

On the scattering of pions by deuterons

CXVII. A stochastic problem relating to counters

Stochastic processes associated with random divisions of a line

Einstein: a natural completion of Newton