Generalized Fock Spaces, New Forms of Quantum Statistics and their Algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ``infinite'', Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ``doubly-infinite'' statistics, many representations of orthostatistics, Hubbard statistics and its variations.Comment: This is a revised version of the earlier preprint: mp_arc 94-43. Published versio

Ergodic Classical-Quantum Channels: Structure and Coding Theorems

We consider ergodic causal classical-quantum channels (cq-channels) which additionally have a decaying input memory. In the first part we develop some structural properties of ergodic cq-channels and provide equivalent conditions for ergodicity. In the second part we prove the coding theorem with weak converse for causal ergodic cq-channels with decaying input memory. Our proof is based on the possibility to introduce joint input-output state for the cq-channels and an application of the Shannon-McMillan theorem for ergodic quantum states. In the last part of the paper it is shown how this result implies coding theorem for the classical capacity of a class of causal ergodic quantum channels.Comment: 19 pages, no figures. Final versio

CλC_{\lambda}-extended oscillator algebras and some of their deformations and applications to quantum mechanics

$C_{\lambda}$-extended oscillator algebras generalizing the Calogero-Vasiliev algebra, where $C_{\lambda}$ is the cyclic group of order $\lambda$, are studied both from mathematical and applied viewpoints. Casimir operators of the algebras are obtained, and used to provide a complete classification of their unitary irreducible representations under the assumption that the number operator spectrum is nondegenerate. Deformed algebras admitting Casimir operators analogous to those of their undeformed counterparts are looked for, yielding three new algebraic structures. One of them includes the Brzezi\'nski {\em et al.} deformation of the Calogero-Vasiliev algebra as a special case. In its bosonic Fock-space representation, the realization of $C_{\lambda}$-extended oscillator algebras as generalized deformed oscillator ones is shown to provide a bosonization of several variants of supersymmetric quantum mechanics: parasupersymmetric quantum mechanics of order $p = \lambda-1$ for any $\lambda$, as well as pseudosupersymmetric and orthosupersymmetric quantum mechanics of order two for $\lambda=3$.Comment: 48 pages, LaTeX with amssym, no figures, to be published in Int. J. Theor. Phy

Unified view of multimode algebras with Fock-like representations

A unified view of general multimode oscillator algebras with Fock-like representations is presented.It extends a previous analysis of the single-mode oscillator algebras.The expansion of the $a_ia_j^{\dagger}$ operators is extended to include all normally ordered terms in creation and annihilation operators and we analyze their action on Fock-like states.We restrict ourselves to the algebras compatible with number operators. The connection between these algebras and generalized statistics is analyzed.We demonstrate our approach by considering the algebras obtainable from the generalized Jordan-Wigner transformation, the para-Bose and para-Fermi algebras, the Govorkov "paraquantization" algebra and generalized quon algebra.Comment: Latex, 34 pages, no figures ( accepted in Int.J.Theor.Phys.A

New perturbation theory of low-dimensional quantum liquids II: operator description of Virasoro algebras in integrable systems

We show that the recently developed {\it pseudoparticle operator algebra} which generates the low-energy Hamiltonian eigenstates of multicomponent integrable systems also provides a natural operator representation for the the Virasoro algebras associated with the conformal-invariant character of the low-energy spectrum of the these models. Studying explicitly the Hubbard chain in a non-zero chemical potential and external magnetic field, we establish that the pseudoparticle perturbation theory provides a correct starting point for the construction of a suitable critical-point Hamiltonian. We derive explicit expressions in terms of pseudoparticle operators for the generators of the Virasoro algebras and the energy-momentum tensor, describe the conformal-invariant character of the critical point from the point of view of the response to curvature of the two-dimensional space-time, and discuss the relation to Kac-Moody algebras and dynamical separation.Comment: 35 pages, RevteX, preprint UA

Stochastic Models on a Ring and Quadratic Algebras. The Three Species Diffusion Problem

The stationary state of a stochastic process on a ring can be expressed using traces of monomials of an associative algebra defined by quadratic relations. If one considers only exclusion processes one can restrict the type of algebras and obtain recurrence relations for the traces. This is possible only if the rates satisfy certain compatibility conditions. These conditions are derived and the recurrence relations solved giving representations of the algebras.Comment: 12 pages, LaTeX, Sec. 3 extended, submitted to J.Phys.

Aspects of coherent states of nonlinear algebras

Various aspects of coherent states of nonlinear $su(2)$ and $su(1,1)$ algebras are studied. It is shown that the nonlinear $su(1,1)$ Barut-Girardello and Perelomov coherent states are related by a Laplace transform. We then concentrate on the derivation and analysis of the statistical and geometrical properties of these states. The Berry's phase for the nonlinear coherent states is also derived.Comment: 22 Pages, 30 Figure