Reciprocity Theory and the Geometrization of Stochastic Quantum Mechanics

Abstract

In Part I of this monograph we have been striving to solve the localizability problem in relativistic quantum mechanics by viewing a quantum particle as a kind of probability ‘cloud’ against a classical spacetime background. Our main aim was to arrive at a mathematical framework capable of a consistent physical interpretation due to the presence of bona fide probability densities and currents that transform in a covariant manner under the change of inertial frames. The epistemologically deeper question as to the existence of such frames, and, furthermore, as to the ultimate validity at the nuclear or even the subnuclear level of the pure classical description of spacetime in terms of a continuum — such as some differentiable manifold with a metric structure — had not been considered, or even posed.

“The question whether the structure of this [spacetime] continuum is Euclidean, or in accordance with Riemann’s general scheme, or otherwise, is…properly speaking a physical question which must be answered by experience, and not a question of a mere convention to be selected on practical grounds.” Einstein (1923; 1953, p. 193)

“Ordinary relativity is based on the invariance of the 4-dimensional distance, or its square R = x _k x^k. Can only really define distance of two particles in sub-atomic dimensions independently of their velocity? This seems to me not evident at all.” Born (1949a, p. 208)