Statistics for Particles Having Internal Quantum State

A new kind of quantum statistics which interpolates between Bose and Fermi statistics is proposed beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles. The quantum 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. The application of this quantum statistics for the ensemble of extremal black holes are discussed. PACS numbers: 05.30.-d, 03.65.Ca Typeset using REVTEX ∗E-mail: [email protected] 1 Although there were some attempts to propose a generalized quantum statistics (GQS) [1], yet Bose and Fermi statistics were believed to be unique for quite a long time. However, since the model by Wilczek [2] which is system of particles with an AharonovBohm type of interaction in two dimensions, particles called anyons have been a subject of intense study and a number of different physical applications have been investigated, such as fractional quantum Hall effect (FQHE) [3] and high-tempreture superconductivity [4]. The concept of anyons, which is based on the wave function arises a factor e as it exchanges two particles (exchange statistics), is essentially two-dimensional. Another way to define GQS has been formulated by Haldane [5], which is based on the rate at the number of the available states in a system of fixed size decrease as more and more particles add to it (exclusion statistics). This statistics, formulated without any reference to spatial dimensions, captures the essential features of the anyon statistics peculiar to two-dimensional systems. Recently, a variant notion of GQS bases on deformations of the bilinear Bose and Fermi commutation relations. Particles obeying a simple case of this type statistics (the so-called ”infinite” statistics) are called quons [6], which obey the minimally deformed commutator [ai, a † j]q = δij (0.1) where [A,B]q ≡ AB−qBA, and q is a c-number, |q| ≤ 1. The equivalence of anyon statistics and quon statistics, Eq.(0.1) with q = e, was proved [7] via the properties of the N -anyon permutation group. More recently, a new model of GQS, in which identical particles exhibit both Bose and Fermi statistics with respective probabilities pb and pf , is introduced by Medvedev [8]. In this letter we investigate a new exchange statistics beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles. Three decades ago, it was shown by Finkelstein and Rubenstein [9] that, in nonlinear field theories, soliton statistics can be determined from the fact that the quantum state is a functional on the space of field configurations. If the soliton has no internal states, the eigenspaces of exchange operator are superselection sector: Bosons are forever bosons 2 and fermions are forever fermions; if the soliton does have internal state then the exchange operator may or may not change the field configuration, depending on whether or not the solitons are the same state [10]. These ideas were firstly applied to quantum gravity in a series of beautiful papers by Friedman and Sorkin [11]. Recently, Strominger applied these ideas to the problem of charged extremal black hole statistics and he argued that the charged extremal black holes will obey the infinite statistics with q = 0 on the condition that none of them are in the same internal state. Basing on the developed ideas above, we assume at first that the wave function of such particles is more composition, a functional, according to its internal degrees of freedom, and suppose then that a phase factor the wave functional of the many-particle system arises as single exchange of a pair of particles is an operator [13], which is dependent on the intrinsic property of the particles, rather than an usual c-number. Furthermore, we consider that all the particles of the system are in the same state, and then the exchange operator will be independent of the pairs of particles and commute with any operator in the system simply because it dose neither change the physical configuration nor mix up internal states again. This operator is marked by q̂ in following. Consider a system of N noninteracting such identical particles, represented by wave functional Ψ(x1, · · · , xj , · · · , xi, · · · , xN ). Single exchange any two particles we get Ψ(x1, · · · , xj , · · · , xi, · · · , xN) = q̂Ψ(x1, · · · , xi, · · · , xj, · · · , xN) (0.2) It is easy to show that the operator q̂ is both Hermitian and unitary [14], i.e., it satisfies q̂ = q̂† and q̂† q̂ = q̂ q̂† = 1 (0.3) The eigenvalues of q̂ take on +1 or −1 only, and the eigenequation of q̂ is q̂ | ± 1, j > = ±1 | ± 1, j > (0.4) where | ± 1, j > compose a complete orthonormal set, with j denoting the degeneration degrees of freedom; and then the internal quantum state can be written 3