Limit theorems in new probability spaces associated with quantum statistics

Norbert Wiener, Armand Siegel

A new type of representation of the wave function of quantum mechanics, and a new form of the statistical interpretation of the wave function, are presented. The variable in the new representation is a mapping of points of differential space, closely related to Hilbert space, but having an associated measure for suitably defined sets of points in it. In the new statistical interpretation, an eigenvalue of every observable defined in the system is associated with each point in differential space. This can be done in such a way that, for each observable, the measure (=probability) of the set of all differential-space points belonging to a given set of eigenvalues is equal to the quantum-mechanical probability, calculated in the usual way (Born statistical postulate), that an experiment will yield an eigenvalue in this set. Thus we obtain an interpretation of quantum mechanics in terms of probability densities or probabilities instead of probability amplitudes.

James van Aken

Paul A. Benioff

In this work, operator valued measures are used to study finite and infinite sequences of measurements. It is shown that to each such process qΔ there is uniquely associated a probability operator measure OqΔ which contains all the statistical properties of the process. In order to make this association for infinite processes, the operator valued equivalent of the Kolmogorov extension theorem is needed. This theorem is given and proved. It is then shown that for each qΔ and each set E of possible outcome sequences, there are two ways to find the probability that carrying out qΔ on a system in state ρ gives an outcome sequence φ in E. The usual method of repeating qΔ on ρ over and over again generates a sequence α of outcome sequences φ. The probability is obtained as the limit relative frequency that α(j) is in E, for j = 0, 1, …. The other, new, method is the repeated measurement of OEqΔ on ρ. The remarkable aspect of this equivalence is that the mathematical procedures of the usual method for determining if α(j) is in E or not `disappear' into the operators OEqΔ of the new method. This is discussed in some detail and examples are given.

James van Aken

Andrei Yu. Khrennikov

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A K Mishra, G. Rajasekaran