Statistical Mechanics of Fields and the 'Apeiron'

Abstract

IN view of the difficulties encountered in the quantum theory of fields, we have developed a new approach to this subject which bears a much closer resemblance to ordinary quantum mechanics of particles than the existing theories (Heisenberg and Pauli), We consider the field in a finite volume Ω as a mechanical system described by its total energy and momentum. The latter are obtained from the classical expressions for the energy-momentum densities simply by multiplication with Ω. Each field component is considered as an operator (for the whole volume Ω) and is not regarded as a function of the co-ordinates at all. Between the field components we have established very simple commutation laws, analogous to the ordinary law pq—qp = ℏ/i which depend on the transformation character of the field considered. For example, for the scalar meson field (potential v, its gradient f, its time derivative g; * means Hermitean adjoint; [a,b] means ab – ba):