The Stochastic Representation of Hamiltonian Dynamics and The Quantization of Time

Matthew F. Brown

Published 2011-11-30, updated 2012-06-16Version 2

Here it is shown that the unitary dynamics of a quantum object may be obtained as the conditional expectation of a counting process of object-clock interactions. Such a stochastic process arises from the quantization of the clock, and this is derived naturally from the matrix-algebra representation of the nilpotent Newton-Leibniz time differential [Belavkin]. It is observed that this condition expectation is a rigorous formulation of the Feynman Path Integral.