Schrödinger Second Quantization. I. Fundamentals

Abstract

A Hamilton-Jacobi theory, based on a Lorentz scalar Hamiltonian function, is developed for first-quantized relativistic fields. Following Schr\"odinger's original method for the analogous classical-mechanics formulation, the new Hamilton-Jacobi equation is regarded as the short-wavelength limit of a new wave equation. In the new wave equation, the fields themselves appear as independent coordinates; e.g., in the treatment of three scalar fields, the current density goes over into a current operator which is the "$z$ component of angular momentum" operator in the three-dimensional space of these fields. The scalar Hamiltonian function becomes an operator whose eigenvalues are tentatively identified with particle rest masses. A treatment of interacting scalar Maxwell fields indicates that the new wave equation may allow simple calculations of mass shifts due to interactions, as well as transition quantum numbers, selection rules, and lifetimes. It is shown that the commutators of this theory are essentially the same as "integrated" quantum-field-theory commutators. The interpretation in terms of mass spectra is regarded as inconclusive, primarily because a treatment of the Dirac spinor field has yet to be made.