Relativistic and Causal Theories with Four-Fermion Interaction Hamiltonians

Abstract

Possible ways of constructing field-theoretic operators satisfying the commutation relations of the inhomogeneous Lorentz group are investigated along lines laid down by Dirac. They can be satisfied in both the instant form, in which operators representing rotations and translations in space remain unchanged, and the point form, in which operators representing the homogeneous Lorentz group G41 remain unchanged, provided one can find a causal Hamiltonian density such that [H(t, x), H(t, y)] is proportional to δ(x − y) and which transforms as a scalar under G41. Less restrictive sufficient conditions in the instant form are found, similar to those of Dirac. The commutator can be proportional to derivatives of δ(x − y) if the coefficients on the derivatives satisfy a certain condition. The only way found to satisfy these conditions for an interaction Hamiltonian constructed from fields for identical spin ½ particle (in the interaction picture) is to have the commutator proportional to δ(x − y), which implies local coupling with no derivations. The possibility of having relativistic theories in the instant form without causality is also investigated for the case of a four-fermion interaction Hamiltonian constructed from creation and annihilation operators for a spin ½ particle, but no definite conclusion is arrived at.