In previous papers a Bose representation of fermions and parafermions was constructed in such a way that (1) the only fundamental quantum fields were Bose fields and (2) no property of the Fermi or para-Fermi systems (as e.g., the Pauli principle for fermions) be lost. The use of that construction and also the proofs of the theorems were made difficult by the complexity of many of the mathematical expressions we used previously. In the present paper we simplify much the above-mentioned construction. Moreover no results (concerning the Bose representation) proved in the previous papers are lost in the present reformulation, but all results are more general: In the previous papers we made the construction of the Bose representation of fermions for the case of pure Fock representations of Quantum Field Theory. The present work is consistent with Fock as well as with non-Fock representations of the Fermi anticommutation relations.
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.