ec 1 99 6 Properties of Particles Obeying Ambiguous Statistics

A new class of identical particles which may exhibit both Bose and Fermi statistics with respective probabilities pb and pf is introduced. Such an uncertainity may be either an intrinsic property of a particle or can be viewed as an “experimental uncertainity”. Statistical equivalence of such particles and particles obeying parastatistics of infinite order is shown. Generalized statistical distributions are derived and statistical and thermodynamical properties of an ideal gas of the particles are investigated. The physical nature of such particles and the implications of this investigation for the statistics of extremal black holes are discussed. 05.30.-d, 04.70.Dy, 14.80.Pb, 05.70.Ce Typeset using REVTEX 1 Historically, the first attempt to generalize quantum Bose and Fermi statistics was made by Gentile [1], who proposed statistics in which up to k particles are allowed to occupy a single quantum state (instead of one for Fermi and infinite for Bose cases). Next was the work by Green [2], which is based on deformations of trilinear commutation relations for particle creation and annihilation operators. Further extension [3,4] of this concept was dubbed “parastatistics” of finite order p (or “para-Bose” and “para-Fermi” statistics), and particles obeying it were called “parons”. The order p reflects the number of particles in a symmetric or antisymmetric state, in terms of a wave-function. These theories do not satisfy the positivity norm condition, however. As a logical consequence, the parastatistics of infinite order or the so called “infinite” statistics, is equivalent to deformation of bilinear commutation relations (q-commutator) [5,6] aia † j − qa†jai = δij, (1) where q is either a real or complex parameter, a†j and aj are the particle creation and annihilation operators, and δij is the Kronecker’s delta. The particles which obey this statistics, called “quons”, have nonnegative square norms [7] for |q|2 ≤ 1, while the observables have nonlocal properties. Recent interest in nontraditional versions of particle statistics arose due to experimental observation of the fractional quantum Hall effect [8] in a two-dimensional electron gas. Quasiparticles which play a role of charge carriers were called “anyons”, because their many-body wave-function picks up any phase e under the interchange of any two particles [9]. The statistics of such particles was referred to as “fractional” statistics [10]. One-dimensional systems of particles can also exhibit similar statistical properties [11,12]. Although the concept of an anyon is essentially two-dimensional, it was elaborated by Haldane as a generalization of the Pauli exclusion principle [13], and does not need to refer to spatial dimensions. He proposed that the change of the number of available single-particle states ∆d is related linearly to the change of the number of particles ∆N as ∆d = −α∆N , where α is a “measure” of Pauli blocking, and defines the statistical properties of a system. The particular values 2 α = 1, 0 correspond to fermions and bosons, respectively. As shown recently [14], anyon fields satisfy the q-commutation relation Eq. (1), i.e. the q-commutator is the fundamental bracket of an anyon field theory. This result is a strict consequence of the N -anyon diffeomorphysm group representations and some general intertwining properties of the field. The deformation parameter q is related to the wave-function phase-shift as q = e. Statistical and thermodynamical properties of systems of particles obeying fractional statistics (i.e. complex q such as |q|2 = 1) were extensively studied in the last several years [12,15], whereas these properties of particles with real q, −1 ≤ q ≤ 1, (infinite statistics) were not investigated (with some special exceptions [16]). There is an interesting case of the infinite statistics [5], q = 0, which may be considered as a quantum analog of Boltzmann statistics, because the many-body wave-function does not exhibit any a priori symmetry. It was suggested [4] that quantum Boltzmann statistics corresponds to the statistics of identical particles with an infinite number of internal degrees of freedom, which is equivalent to the statistics of nonidentical particles, since they can be distinguished by their internal states. One should note here that it was also recently suggested [17] that extremal black holes should obey the infinite statistics with q = 0. This issue will be discussed below. In this work we investigate a system of particles which are equivalent to those obeying the q-deformed statistics of real q. Unlike the theories mentioned above, we admit only “primary” Bose-Einstein and FermiDirac statistics as effectively existing. Assume now that a particle can be identified by any other particle (e.g. an observer) as a boson with some probability pb and as a fermion with a probability pf . In other words, we introduce a classical mechanics notion of “experimental uncertainity” into a quantum system. Any particle which interacts with another cannot be “absolutely sure” of what class that particle is, boson or fermion. It “decides” whether the other is a boson or a fermion with some uncertainity, and the probability to arrive at the particular decision are pb and pf , respectively. Note, that pb + pf is not necessarily equal to one, and if not, it means that a particle does not “see” the other at all. The probability of this is 1− pb − pf . This “imperfect measurement” model may be relevant to avoiding acausality 3 paradox in superluminal particle (tachyon) systems [18]. This model can be interpreted in another manner. Assume a particle can oscillate between two types of statistics, then the model we propose represents a system of such particles averaged over time-scales much larger than the oscillation period. The probabilities pb and pf , thus, are those portions of time during which a particle resides in a Fermior Bose-type state. The model of this kind is relevant to systems of elementary particles which have mutual supersymmetrical partners (e.g. a photon and photino, etc.) when transitions between these supersymmetrical states can occur [19]. With some modifications, this model can also be applied to systems of particles with changing flavor, e.g. quarks, gluons, neutrinos, etc. There is a conjectured relation between the particles of the ambiguous statistical type and the q-deformed infinite statistics. Let bj and b † j are the annihilation and creation operators for such a particle. The particle exhibit bosonic properties with the probability pb and fernionic ones with pf . Thus, a bilinear commutation relation for bj and b † j is of boson type with the probability pb , of fermion type with the probability p 2 f , and the two particles are nonidentical with the probability 2pbpf , that is interchange of their positions result in another wave-function, i.e. bib † j = δij . We write (pb + p 2 f + 2pbpf )bib † j − (pb − pf )b†jbi = (pb + p 2 f + 2pbpf )δij. (2) This equation coincides with Eq. (1), and the deformation parameter is q = (pb−pf )/(pb+pf). In the rest of this letter we derive statistical distributions for identical particles which have ambiguous quantum exclusion properties and investigate some thermodynamical properties of an ideal gas of such particles. A grand partition function for a system of particles with properties defined by a stochastic label with a known probability distribution is a sum over all possible realizations (with a weight factor) of the partition functions corresponding to each realization. In each realization, the system effectively consists of k bosons and N − k fermions (N is the total number of particles), while the probability of this realization is pbp N−k f . The total number of states 4