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Modern Physics Letters A

GENERALIZED QUANTUM STATISTICS

Chihong Chou

Published Sep 21, 1992• 9 citations
Publisher

Abstract

In this letter, a non-anyonic generalization of quantum statistics is presented, in which Fermi-Dirac statistics (FDS) and Bose-Einstein statistics (BES) appear as two special cases. This new quantum statistics, which is characterized by the dimension of its single particle Fock space, contains three consistent parts, namely the generalized bilinear quantization, the generalized quantum mechanical description and the corresponding statistical mechanics.

DOI
10.1142/s0217732392002147

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In this letter, a non-anyonic generalization of quantum statistics is presented, in which Fermi-Dirac statistics (FDS) and Bose-Einstein statistics (BES) appear as two special cases. This new quantum statistics, which is characterized by the dimension of its single particle Fock space, contains three consistent parts, namely the generalized bilinear quantization, the generalized quantum mechanical description and the corresponding statistical mechanics.
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. Key Words : Fock space -- multioscillator algebras -- q-deformation -- quantum groups -- general...
The statistics of q-oscillators, quons, and, to some extent, of anyons are studied and the basic differences among these objects are pointed out In particular, the statistical distributions for different bosonic and fermionic q-oscillators are found for their corresponding Fock space representations in the case when the Hamiltonian is identified with the number operator. In this case and for non-relativistic particles, the single-particle temperature Green function is defined with q-deformed periodicity conditions. The equations of state for nonrelativistic and ultrarelativistic bosonic q-gases in an arbitrary space dimension are found near Bose statistics, as well as that for an anyonic gas near Bose and Fermi statistics. The first corrections to the second virial coefficients are also evaluated.
Beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles, a unified quantum statistics, which may interpolate smoothly between Bose-Einstein and Fermi-Dirac statistics, is proposed. The commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. Besides the Bose-Einstein condensation, another kind of particle condensation may appear.
Abstract It is well known that a new form of statistical mechanics has been recently developed by Einstein for an ideal gas of structureless mass-points. This starts from a discussion by Bose of the laws of temperature radiation based on the light quantum hypothesis, and has been further analysed by Schrödinger. Yet another new form has been proposed independently by Fermi and Dirac. The latter based his theory on a discussion of lightly coupled systems with the help of Schrödinger’s equation. Combined with Heisenberg’s work on the many-body problem, Dirac’s work forces us to conclude at least that the classical form of statistical mechanics must be changed. It indicates that the true form, which satisfies the laws of the new mechanics, is almost certainly that of Fermi and Dirac, which is the natural generalization of Pauli’s principle of exclusion for electronic orbits in an atom. The work of Heisenberg and Dirac already quoted has shown that Pauli’s principle and its extension are satisfied in the new mechanics by a complete self-consistent solution of the equations of motion. So far as I am aware, the discussions of the new forms have as yet dealt only with the statistics of a gas of structureless mass-points (and, of course, temperature radiation). There has, moreover, been as yet no attempt to define the entropy (and the absolute temperature) in strict analogy with rational thermodynamics by means of the equation dQ = TdS. If another definition is preferred, then this equation must be deduced from it. It has, therefore, seemed worth while to reopen the discussion by examining ab initio a quite general form of statistical mechanics of which the classical form and instein’s and Fermi-Dirac’s are special cases. This is rendered possible by using the powerful method of complex integration already applied to the classical form. The sequence of the argument is then to take the general form, which covers a very large range of ways of assigning possibilities and counting complexions, and construct on that basis exact integral expressions for the number of complexions possible to the assembly and for the average number of systems of the assembly in their various quantum states. We then derive from these the average energies and external reactions and so the form of dQ, deduce from dQ the existence of S and so define S and T. This can be done in the general form for assemblies of ideal systems just as general as can be handled in the classical way—ideal gases of molecules of any structure and crystals and radiation. Such assemblies are in all cases thermodynamical systems.
We investigate a generalization of Ar statistics discussed recently in the literature. The explicit complete set of state vectors for the Ar statistics system is given. We consider a Bargmann or an analytic function description of the Fock space corresponding to Ar statistics of bosonic kind. This brings, in a natural way, the so-called Gazeau–Klauder coherent states defined as eigenstates of the Jacobson annihilation operators. The minimization of Robertson uncertainty relation is also considered.