The nonrelativistic theory of a single para-Fermi field of order
p is investigated. General properties of state vectors are studied in detail, and it is shown that the state-vector space can be spanned by what we shall call standard state vectors. A restriction on the form of interaction Hamiltonians is derived from the requirement that our formalism be described by local Lagrangian field theory. This restriction on interaction Hamiltonians gives rise to a conservation law of a physical quantity to be called
A, which resembles the magnitude of angular momentum with respect to its rule of addition. The conservation law of
A leads then to absolute selection rules for reactions, which are a generalization of those obtained elsewhere. The problem of bound states made up of our para-Fermi field is also studied, and all bound states are classified into (
p+1) categories according to their statistical behaviors. It is found that for
p< 3 all bound states can be described by ordinary parafield theory, whereas for
p> 4 such is no longer the case. Furthermore, it can be shown that in the theory of
p=2 no fermion bound states are possible. In this sense it may be said that para-Fermi fields of
p=1 and 3 occupy a very privileged position in para-Fermi theory in general. The main results in this paper are stated as 12 theorems. It is expected that the whole argument will be valid in a relativistic theory as well.
A new generalization of quantum statistics is described, which is different from parastatistics, though it includes Fermi statistics and parafermi statistics of order two. It can be applied to the quantization of nonlinear field theories, without violating the correspondence principle. Like parastatistics, it allows the occupation of a given dynamical state by more than one particle of half-odd-integral spin. A special feature is that quark-like particles are naturally associated in modules, which have many of the characteristics of baryons and mesons. The group theoretical properties of the new statistics, and the implied classification of states, are briefly examined.
It has long been known that, under the laws of quantum mechanics, particles in two spatial dimensions may exhibit non-abelian braid statistics. In this note we briefly discuss particular states of matter, so-called paired and clustered quantum Hall states, which offer a concrete realization of this scenario: quasi-holes over these quantum Hall states satisfy non-abelian braid statistics. The associated degeneracies have their origin in combinatorial properties of (para-)fermions in conformal field theory. We also discuss the exclusion statistics of edge excitations in these quantum Hall systems.