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Foundations of Physics

The Equiareal Archimedean Synchronization Method of the Quantum Symplectic Phase Space: I. Spinorial Amplitudes, Transition Probability, and Areal Measure of Time

Elias Zafiris

Published Apr 1, 2022• 1 citation
Publisher
DOI
10.1007/s10701-022-00560-2

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In the search for an explanation of the thermodynamic arrow of time in our universe, cosmologists have proposed central-time models. These recent, speculative models run into a probabilistic paradox that involves a failure of conglomerability. We analyze the non-conglomerability of these models with the help of standard probability theory, as well as three alternative formalisms: finitely additive probability, non-Archimedean probability (NAP), and non-normalized quasi-probability. We find that all formalisms exhibit failures of conglomerability on infinite partitions when applied to cosmological models that invoke non-normalizable measures, such as uniform measures on infinite phase spaces. NAP theory, however, does assure conglomerability on partitions of arbitrary cardinality that meet an additional condition. In any case, failures of conglomerability need not be intrinsically problematic for the development of central-time models.
Plausibility measures are structures for reasoning in the face of uncertainty that generalize probabilities, unifying them with weaker structures like possibility measures and comparative probability relations. So far, the theory of plausibility measures has only been developed for classical sample spaces. In this paper, we generalize the theory to test spaces, so that they can be applied to general operational theories, and to quantum theory in particular. Our main results are two theorems on when a plausibility measure agrees with a probability measure, i.e. when its comparative relations coincide with those of a probability measure. For strictly finite test spaces we obtain a precise analogue of the classical result that the Archimedean condition is necessary and sufficient for agreement between a plausibility and a probability measure. In the locally finite case, we prove a slightly weaker result that the Archimedean condition implies almost agreement.