On Generalizing Boson Field Theories

Abstract

If a boson field theory is generalized to admit in the field Lagrangian all the derivatives of the field coordinates up to the σth in a particularly symmetrical way, then we find that in the final expression for the interaction energy of fermions the usual interaction is replaced by a weighted sum of $\sigma$ such interactions, where the relative weight factor and the boson mass associated with each interaction are given uniquely by algebraic relationships involving the constants appearing in the field Lagrangian. We thus are able to formulate a simple principle for obtaining the interaction energy according to a multiple-boson theory of the form suggested by our generalizations if the interaction energy is known for the one-boson case. The principle is then applied to the non-relativistic and relativistic interaction terms for electrons according to electrodynamic (photon) field theory, and for nucleons according to the meson field theories investigated by Kemmer. We find that in almost every case the weight factors are such that not only are the inadmissible $R^{-3\mathrm{}}$ and $R^{-2\mathrm{}}$ singularities removed, but also in three- and four-boson theory, the objectionable $R^{-1\mathrm{}}$ singularities. A closely related result is the fact that generalization justifies the neglect of short wave-lengths in the evaluation of the interaction and self-energy integrals.