Probabilities in quantum theory and probability theory
A viewpoint is presented which, taken in probability theory and/or in quantum theory, will help place certain facts in a unified perspective and clarify their apparently enigmatic status. These facts are: (1) while the probabilities in probability theory are squared moduli of complex numbers; (2) the measure-theoretic apparatus of probability theory cannot accommodate the quantization rules for physical measurables and, above all; (3) the notion of limits on the accuracy attainable in simultaneous measurements on canonically conjugate variables does not appear in the theorems and algorithms of probability theory. The viewpoint rests on the intuitive but fundamental idea that measurable functions have non-unique inverses, and that an inner product representation of measures can be obtained whenever the function is differentiable. This idea is generalized to functions measurable on real rectangles and on complex plane. The formulas derived for measures are formally the same as those in quantum mechanics. The remainder of the development is based on the notion of a measurable-preserving point-to-point mapping from the range of these special measurable functions to an appropriate subset of a Hilbert space. Physical interpretations are attempted for each of the measure-theoretic constructs.
- Research Organization:
- Univ. of Waterloo, Ontario
- OSTI ID:
- 6789394
- Journal Information:
- Hadronic J.; (United States), Vol. 2:4
- Country of Publication:
- United States
- Language:
- English
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