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Physics Essays

Complementarity, quantization, and uncertainty: An analytical perspective of set theory and probability

Aalok Pandya

Published Jun 2, 2014• 0 citations
Publisher
DOI
10.4006/0836-1398-27.2.281

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Abstract The electromagnetic field is quantized for free space and for dispersive dielectric media. Different quantum states of the field (thermal, laser, squeezed) are discussed and their photon-statistical properties derived. The density matrix operator for the field is introduced along with a detailed discussion of the diagonal representation of the field density operator and its properties. Field commutation relations, correlation functions, and uncertainty relations are derived and discussed. Also discussed are the historically important Bohr-Rosenfeld analysis of field commutation and uncertainty relations; energy-time uncertainty relations; simultaneous (Arthurs-Kelly) measurement of non-commuting observables; and complementarity from the perspective of probability amplitudes and probabilities for distinguishable and indistinguishable processes.
In this paper we discuss the rationale why sub(super)‐additive probabilities in a psychological setting could be explained via the use of quantum probability interference. We propose to measure the complementarity of two variables: i) time of processing (by experiment participants) of (non‐moving) images and ii) the ability (by experiment participants) of recognizing deformations of (non‐moving) pictures. We argue in the paper why we can not find this complementarity using the Heisenberg Uncertainty Principle. The paper provides for the details on the experimental set up to test the complementarity.
The contextual probabilistic quantization procedure is formulated. This approach to quantization has much broader field of applications, compared with the canonical quantization. The contextual probabilistic quantization procedure is based on the notions of probability context and the principle of complementarity of probabilities. The general definition of probability context is given. The principle of complementarity of probabilities, which combines the ideas of the Bohr complementarity principle and the technique of noncommutative probability, is introduced. The principle of complementarity of probabilities is the criterion of possibility of the contextual quantization.