Some General Properties of Para-Fermi Field Theory

Abstract

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 ($p+1\mathrm{}$) 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 $p=2\mathrm{}$ no fermion bound states are possible. In this sense it may be said that para-Fermi fields of $p=1\mathrm{}$ 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.