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TURKISH JOURNAL OF PHYSICS

Deformed Quantum Statistics

Akira Inomata

Published Jan 1, 1997• 0 citations
Publisher

Abstract

To understand possible physical consequences of quantum deformation, we investigate statistical behaviors of a quon gas. The quon is an object which obeys the minimally deformed commutator (or q-mutator): a\, a^{\dagger} - q a^{\dagger}a=1 with -1\leq q\leq 1. Although q=1 and q=-1 appear to correspond respectively to boson and fermion statistics, it is not easy to create a gas which unifies the boson gas and the fermion gas. We present a model which is able to interpolates between the two limits. The quon gas shows the Bose-Einstein condensation near the Boson limit in two dimensions.

DOI
10.55730/1300-0101.2374

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To understand possible physical consequences of quantum deformation, we investigate statistical behaviors of a quon gas. The quon is an object which obeys the minimally deformed commutator (or q-mutator): a, a^{\dagger} - q a^{\dagger}a=1 with -1\leq q\leq 1. Although q=1 and q=-1 appear to correspond respectively to boson and fermion statistics, it is not easy to create a gas which unifies the boson gas and the fermion gas. We present a model which is able to interpolates between the two limits. The quon gas shows the Bose-Einstein condensation near the Boson limit in two dimensions.
Thermodynamical properties of a model quon gas are reported. The quons considered are the objects obeying the deformed commutator aa†−qa†a=1 with −1⩽q⩽1. Although q=1 and q=−1 correspond respectively to boson and fermion statistics, it is not trivial to generate a quon gas which interpolates between the boson gas and the fermion gas without Pauli’s exclusion principle. The proposed model, interpolating between the two limits, can realize the Maxwell-Boltzmann limit at q=0 and simulate a free anyon gas. Under the restriction −1⩽q⩽1, the gas is deformed but bosonic in nature. It shows the Bose-Einstein condensation even in two dimensions.
I discuss the statistics of "quons" (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the qqq-deformed commutation relation (the quon algebra or qqq-mutator) which interpolates between fermions and bosons. Topics discussed include representations of the quon algebra, proof of the TCP\mathrm{TCP}TCP theorem and clustering, violation of the usual locality properties, and experimental constraints on violations of the Pauli exclusion principle (i.e., Fermi statistics) and of Bose statistics.
After a brief mention of Bose and Fermi oscillators and of particles which obey other types of statistics, including intermediate statistics, parastatistics, paronic statistics, anyon statistics and infinite statistics, I discuss the statistics of ``quons'' (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the qqq-deformed commutation relation (the quon algebra or q-mutator) which interpolates between fermions and bosons. I emphasize that the operator for interaction with an external source must be an effective Bose operator in all cases. To accomplish this for parabose, parafermi and quon operators, I introduce parabose, parafermi and quon Grassmann numbers, respectively. I also discuss interactions of non-relativistic quons, quantization of quon fields with antiparticles, calculation of vacuum matrix elements of relativistic quon fields, demonstration of the TCP theorem, cluster decomposition, and Wick's theorem for relativistic quon fields, and the failure of local commutativity of observables for relativistic quon fields. I conclude with the bound on the parameter qqq for electrons due to the Ramberg-Snow experiment.
An outstanding idea originally introduced by Greenberg is to investigate whether there is equivalence between intermediate statistics, which may be different from anyonic statistics, and q-deformed particle algebra. Also, a model to be studied for addressing such an idea could possibly provide us some new consequences about the interactions of particles as well as their internal structures. Motivated mainly by this idea, in this work, we consider a q-deformed Fermi gas model whose statistical properties enable us to effectively study interpolating statistics. Starting with a generalized Fermi-Dirac distribution function, we derive several thermostatistical functions of a gas of these deformed fermions in the thermodynamical limit. We study the high-temperature behavior of the system by analyzing the effects of q deformation on the most important thermostatistical characteristics of the system such as the entropy, specific heat, and equation of state. It is shown that such a deformed fermion model in two and three spatial dimensions exhibits the interpolating statistics in a specific interval of the model deformation parameter 0 < q < 1. In particular, for two and three spatial dimensions, it is found from the behavior of the third virial coefficient of the model that the deformation parameter q interpolates completely between attractive and repulsive systems, including the free boson and fermion cases. From the results obtained in this work, we conclude that such a model could provide much physical insight into some interacting theories of fermions, and could be useful to further study the particle systems with intermediate statistics.
A q-deformed field theory at finite temperature is presented and its related thermo field dynamics is constructed. This enables us to introduce the quon field, whose Lagrangian leads to a q-deformed partition function. Accordingly, the general thermostatistical properties of a gas model of quon fields, such as the q-deformed statistical distribution function describing an intermediate-statistics behavior and the equation of state in two and three dimensions, are investigated. For low temperatures, the conditions under which Bose–Einstein-like condensation would occur in the quon gas model are discussed. It is shown that the critical temperature of such a gas is higher than that of the usual Bose gas for values of the model parameter q in the range [Formula: see text]. For high temperatures, possible anyonic behavior of the present quon gas model in two spatial dimensions is studied through an analysis of the effect of deformation on the second and third virial coefficients in the equation of state. The results obtained in this work reveal the ability of the model for analyzing a parastatistical behavior of systems with quasiparticles and put forward it as a possible candidate for effectively modeling the properties of some exotic quantum states, such as in the case of dark matter constituents.
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.