A Generalization of Quantum Statistics

We propose a new fractional statistics for arbitrary dimensions, based on an extension of Pauli’s exclusion principle, to allow for finite multi-occupancies of a single quantum state. By explicitly constructing the many-body Hilbert space, we obtain a new algebra of operators and a new thermodynamics. The new statistics is different from fractional exclusion statistics; and in a certain limit, it reduces to the case of parafermi statistics. 0∗ [email protected] 0† [email protected] 1 The thermodynamics of a macroscopic system is determined microscopically by the statistics of its constituent particles and elementary excitations. Herein lies a fundamental significance of statistics. Ever since Heisenberg’s second paper on matrix mechanics, it has been known that a many-body wavefunction is symmetric under permutations of identical bosons, but it is antisymmetric for identical fermions. The corresponding commutation and anticommutation relations bilinear in field operators result in bose and fermi statistics respectively. Particles are accordingly classified into bosons and fermions. The overriding difference between the two groups is that bosons condense while fermions exclude. But it is natural to inquire whether there are any meaningful generalizations of statistics intermediate between these two. Attempts to generalize statistics dates back at least to Green’s work in 1953 [1][2]. Green found that the principles of quantum mechanics also allow two kinds of statistics called parabose statistics and parafermi statistics of positive integral order M (the M=1 cases reduce to the familiar Bose-Einstein statistics and Fermi-Dirac statistics respectively). They are described by trilinear commutation relations among the creation and annihilation operators. Subsequently, the case of non-integral M was investigated for possible deviations from Bose and Fermi statistics, and in particular, for possible violations of Pauli’s exclusion principle [3][4]. This saga culminated with a recent study of infinite statistics [5] in which all representations of the symmetric group can occur; this case is realized by the q-mutator algebras. Another type of interpolating statistics, spearheaded by Wilczek, is provided by the concept of anyons [6]. Anyons are particles whose wavefunctions acquire an arbitrary phase when two of them are braided; they obey fractional statistics. More recently, Haldane introduced another definition of statistics based on a generalization of the Pauli principle [7][8]. Unlike the anyon fractional exchange statistics which is meaningful only in two spatial dimensions, Haldane’s fractional exclusion statistics is formulated in arbitrary dimensions. The thermodynamics based on exclusion statistics is studied in Ref.[8]. The issue whether anyons obey fractional exclusion statistics