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Journal of Mathematical Physics

Statistics with number operator C C*

Steven Robbins

Published Jan 1, 1982• 1 citation
Publisher

Abstract

The number operator a*a, where a is an annihilation operator, plays a fundamental role in the statistics of bosons and fermions. However, it is possible for other statistics to have the same number operator. We have previously shown that for one degree of freedom there is a type of statistics having this number operator corresponding to each p which is a positive integer or ∞. Fermions are obtained when p = 1 and bosons are obtained when p = ∞. No state can have more than p particles. In order to treat many degrees of freedom it is necessary to first consider the case of one degree of freedom differently. We show here that for more degrees of freedom a similar situation occurs but for each case other than bosons and fermions there is a positive integer q, such that no state can have more than q particles, even when the number of degrees of freedom is infinite. Thus these statistics are probably not physically realizable except in an approximate way.

DOI
10.1063/1.525206

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The number operator a*a, where a is an annihilation operator, plays a fundamental role in the statistics of bosons and fermions. However, it is possible for other statistics to have the same number operator. We have previously shown that for one degree of freedom there is a type of statistics having this number operator corresponding to each p which is a positive integer or ∞. Fermions are obtained when p = 1 and bosons are obtained when p = ∞. No state can have more than p particles. In order to treat many degrees of freedom it is necessary to first consider the case of one degree of freedom differently. We show here that for more degrees of freedom a similar situation occurs but for each case other than bosons and fermions there is a positive integer q, such that no state can have more than q particles, even when the number of degrees of freedom is infinite. Thus these statistics are probably not physically realizable except in an approximate way.
A generalized framework for particle statistics is developed within which it becomes clear how bosons can combine to form fermions without violating the normal connection between spin and statistics. In the situation where internal and external particle degrees of freedom are separable, the possible statistical types are found and their combining rules derived. Here the statistical "interaction" between two particles is characterized either by the familiar sign or by an integer ("twist number"). However, further possibilities arise if the separability of external from internal degrees of freedom fails, as it can in gauge theories. In spatial dimensions other than three the "fermion from boson" phenomenon does not occur.
Abstract We propose a new generalization of the standard (anti-)commutation relations for creation and annihilation operators of bosons and fermions. These relations preserve the usual symmetry properties of bosons and fermions. Only the standard (anti-)commutator relation involving one creation and one annihilation operator is deformed by introducing fractional coefficients, containing a positive integer parameterp. The Fock space is determined by the classical definition. The new relations are chosen in such a way that the total occupation number in the system has the maximum valuep. From the actions of creation and annihilation operators in the Fock space, a group theoretical framework is determined, and from here the correspondence with known particle statistics is established.
Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.
As with the boson field, the creation and annihilation operators for the paraboson field are unbounded, and the usual paraboson relations therefore, cannot be regiorously satisfied. There are no Weyl relations for parabosons, and so the paraboson relations must be treated in unbounded form. A new approach for treating such unbounded relations is developed, and it is found that the paraboson field is determined by the requirement that the appropriate operator is the one-dimensional number operator. A similar result is shown th hold for the boson field.
The degrees of freedom studied so far have been either real or complex variables. These variables commute under multiplication, in the sense that two numbers a, b satisfy ab = ba ; commuting variables are generically called bosonic variables, or bosonic degrees of freedom. Typical of the bosonic case are the degrees of freedom for a collection of quantum mechanical particles. Interactions of fundamental particles are generally mediated by bosonic fields such as the Maxwell electromagnetic field, whereas mass is usually carried by particles that are fermions, the most familiar being the electron. Two key features distinguish fermions from bosons: Fermions obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. This is the reason the concept of intensity does not apply to a fermion. A high intensity electric field is a reflection of the presence of a large number of photons, which are bosons, in the same quantum state; for photons, any number of photons can be in the same quantum state. In contrast, an electron is either in a quantum state or it is not; in particular, ignoring spin, an electron exists at a point or there is no electron there. The state function of a multi-bosonic system is totally symmetric in that the exchange of any two bosonic degrees of freedom yields the same state function. In contrast, a multi-fermion system is totally anti-symmetric: the exchange of any two fermion degrees of freedom gives the same state – but with a negative sign .
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.