Quantum Mechanical Time

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Abstract
Some suggestions on the probabilistic structure of quantization are first obtained by studying local properties of random variables such as continuity and differentiability with respect to a topology in the sample space. This analysis is extended to analytic functions on the complex plane and conditions are formulated under which probability densities are expressed as sums of absolute squares of complex numbers. Physical restrictions are introduced through the Schrödinger equation for a single bound‐particle system. As a result, observations on a physical system become identified with the random selection of points in a topological measure space and the physical observables such as energy, position, and momentum, as well as time, are identified with measurable functions on appropriate spaces. Time, considered as a function, appears to be multiple‐valued with spacings between multiple values and a detailed functional structure that characterizes and is characterized by the physical system under observation and the physical observables being measured. Its detailed functional structure is related to the physically measurable probabilities of quantum theory and it is seen to serve in the capacity of a conditioning random variable in the computation of quantum mechanical expectations.

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