On the Quantization of a Unitary Field Theory

Abstract

In a unitary field theory particles appear not as singularities but as small volumes in which energy and charge of the field are concentrated. In a theory of this nature, which is necessarily non-linear, all properties of the particles, such as their equations of motion, follow from the field equations. It is pointed out in this paper that the non-linearity present in the usual classical field theories is sufficient to permit the existence of particle-like solutions under certain conditions and that it is therefore possible to derive a class of classical unitary theories by postulating that solutions of the usual classical theories are physically admissible only if they are free from singularities. We have studied the quantization of such a theory without specializing the Lagrangian. Momentum (${G}_{\ensuremath{\alpha}}$) and angular momentum (${M}_{\ensuremath{\alpha}\ensuremath{\beta}}$) appear naturally and a relativistic definition of the position (${X}_{\ensuremath{\alpha}}$) of a particle may be given in terms of ${M}_{\ensuremath{\alpha}\ensuremath{\beta}}$ and ${G}_{\ensuremath{\alpha}}$. The commutators of these particle observables (${X}_{\ensuremath{\alpha}},{G}_{\ensuremath{\alpha}},{M}_{\ensuremath{\alpha}\ensuremath{\beta}}$) with each other are calculated and found similar to those postulated in the Snyder formalism which quantizes space-time; these commutators reduce to the usual ones for non-relativistic velocities. (The quantized Born-Infeld theory did not agree with the usual quantum theory of particles even in the non-relativistic limit, because it was not strictly unitary.) The connection between velocity and momentum, and the equation of motion of particles which follow from the field equations are the usual ones. Operators for mass and charge may be defined; these mutually commute and permit the usual classification of the elementary particles according to mass and charge. Magnetic moment and charge commute with the total electromagnetic field. Particle observables and observables describing the external field may not commute in general.