Quantum field theory of particles of indefinite mass. I

Abstract

A quantum field theory of particles of indefinite mass is derived using rigorous correspondence arguments starting with a classical theory of particles of indefinite mass. The classical theory can be recovered in a suitable limiting case as h/→0 by means of Ehrenfest’s theorem. Our deviation leads to a quantum mechanical wave equation which turns out to be basically the same equation investigated earlier by Fock, Nambu, and others, but differs from this earlier equation in our use of a new evolution parameter—herein called ’’evolution-time’’—defined as proper time divided by the classical mass. Owing to our use of evolution time as the development parameter of the system our Fock equation is without any reference to a mass parameter, in contrast to the older Fock equation. The indefiniteness of the particle mass frees the ordinary time, t—herein called ’’observer’s time’’—of any fixed relationship to the evolution time, and the two times becomes quite independent parameters. The Hamiltonian of our system turns out to be (minus half) the total mass squared of the system and is a constant of the motion. In order to guarantee negative definiteness of the Hamiltonian of the second quantized system, it is necessary to quantize our Fock equation using Fermi–Dirac statistics. A real scalar field is described by an equation which is second-order in the evolution time, obtained by iterating our first-order Fock equation. The second-order Fock equation must be second quantized using Bose–Einstein statistics, in order to preserve the interpretation of the Fourier amplitudes as creation and annihilation operators. The propagator for the second-order Fock equation has a Pauli–Villars type sum of terms, in which one term describe the propagation of timelike states, the other term describes space-like states.