Abstract
Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classic probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classic probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic hidden variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. Uncovering this variable confirms the possibility that it could be the stochastic archetype of quantum probability
Highlights
Using the Schö dinger picture and the causal interpretation as accounted by Holland in his quantum theory of motion, in particular [1],[2],[10],[11], and confining ourselves to a single spatial dimension to keep a focus on concepts, the probability distribution of the location of a moving particle in the spatial dimension x at time t is derived from the wavefunction ψψ(xx, tt), which is defined and continuous everywhere
Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory
There remains an unexplained coincidence in quantum mechanics, namely that mathematically the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly
Summary
The search begins with a statement of the properties that such a pair of variables must display in order to fit in with quantum mechanics. We will call the generic form of these variables the unit base variable and the unit squared amplitude variable, respectively A. The transformed unit squared amplitude variable, which we will call the squared amplitude variable, is the stochastic analogue of the deterministic squared amplitude, and its transforming scale factor must equal to the deterministic squared amplitude. The dependent unit base variable WW, which is the product of the variables A and C and is a generic stochastic analogue of the real part of a wavefunction. The generic variables WW and ZZ are present everywhere unless there is quantum activity present to trigger their transformation into the specific variables X and Y and maintain each as a stochastic process. Our central aim is to demonstrate that with these conjectures the means of Z and Y equal the variances of WWand X respectively, the property we have called the mean/variance property
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