The quantum Cauchy functional and space-time approach to relativistic quantum mechanics

A. A. Beilinson

Published 2015-01-15Version 1

We show that the quantum complex Cauchy functional on bump functions (see \cite{16}), yields a generalized complex Cauchy process, which is a generalized functional on the bump functionals on Borel cylindrical sets of a real Hilbert space, whose support is locally compact for the uniform convergence topology with derivatives in $L_2 (0,t)$. The retarded Green's functions of the Dirac electron and Einstein photon viewed as complex matrix-vaued functionals on bump functionals, also yield generalized complex matrix-valued processes of Cauchy-Dirac and Cauchy-Maxwell, which are generalized functional on the bump functionals on Borel cylindrical sets of a real Hilbert space, but their supports are compact for the uniform convergence topology with derivatives in $L_2 (0,t)$. We study the way the classical relativistic mechanics of particle comes from the quantum mechanics of the free Dirac particle.