On Forces and Interactions between Fields

Abstract

After a discussion of certain properties of multivalued functions is given, these functions are used to give a scalar representation of electromagnetic theory. In this representation, the four-vector potential A appears as the four-gradient of a multivalued function φ. The problem of solving the time-dependent Schrödinger equation for a particle of charge e in an electromagnetic field is equivalent to the problem of solving the corresponding field-free equation in a space of multivalued functions ψ̄, which is connected to the space of single-valued functions ψ by the relation ψ̄=Uψ where U = exp (—ieφ). Further, if L0(ψ̄, A) is the Lagrangian for a noninteracting particle field ψ̄ and photon field A, then L0(Uψ, A), regarded as a functional of ψ rather than ψ̄, is the Lagrangian for interacting fields ψ and A. This suggests that other interactions (e.g., strong and weak interactions) should be introduced by using a transformation T which is more general than U. More fundamental reasons for believing that this is the case arise when one considers multivalued coordinate transformations. It is shown that any curved Riemannian space may be connected with a flat space by a multivalued coordinate transformation. This is possible because the metric transforms like a tensor under these transformations, but the Riemann curvature symbol does not. If one writes the equations of motion of a free, classical, relativistic particle in flat-space coordinates and assumes that the equations transform covariantly under multivalued coordinate transformations (this is a natural generalization of the equivalence principle), one obtains equations of motion in the curved-space coordinates which describe a particle in a gravitational and electromagnetic field. Unfortunately, however, the electromagnetic field at any point of space-time depends on the four-velocity of the test particle at that point. This defect, together with the known existence of essentially quantum-mechanical forces and the fact that the uncertainty principle prevents one from measuring velocity when position is measured exactly, suggests that one should write the Klein-Gordon (or Dirac) equation for a free particle in flat-space coordinates, and then make a change of independent variables into curved-space coordinates. When this is done, it is seen that a corresponding change of dependent variable ψ̄=Tψ is required in order that the curved-space wave function shall be a single-valued function of the curved-space coordinates. The fully transformed equation describes a particle in a gravitational and electromagnetic field as well as certain other fields whose properties are not fully understood.