A self-contained theory of Dirac fermions interacting with (pseudo-) scalar bosons at infinite momentum has been constructed. The fields are coupled with a $\overline{\psi }\Gamma \psi \phi$ interaction where $\Gamma =1\mathrm{}$ or ${\gamma }^5$, depending on the parity of the boson. The theory is designed to give the correct relativistic dynamics when the space of intermediate states is restricted to those states which can be interpreted as the infinite-momentum limit of sets of particles with finite momentum, that is, those states which correspond to the particle configurations observed in nature. The intermediate states which were eliminated are those containing particles moving away from the infinite-momentum observer. The dynamical effect of these states is incorporated into a counterterm in the Hamiltonian. The effect of this counterterm is to produce a set of four-point vertices in the old-fashioned Hamiltonian perturbation expansion. The perturbation-theory rules for this model were described in an earlier paper. A general formalism for handling interactions at infinite momentum is presented here. It is used to derive the counterterm for our model theory. A complete set of generators for the Poincaré group in a basis appropriate to the infinite-momentum limit is demonstrated. These generators satisfy the commutation algebra of the Poincaré group, proving the relativistic invariance of the theory and its internal consistency.

• Received 22 January 1970

DOI:https://doi.org/10.1103/PhysRevD.1.2795