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arXiv:2211.00501 · Entropy

Algebraical Entropy and Arrow of Time

Merab Gogberashvili

Published Oct 25, 2022• 3 citations
arXivHTMLPDF

Abstract

Usually, it is supposed that irreversibility of time appears only in macrophysics. Here, we attempt to introduce the microphysical arrow of time assuming that at a fundamental level nature could be non-associative. Obtaining numerical results of a measurement, which requires at least three ingredients: object, device and observer, in the non-associative case depends on ordering of operations and is ambiguous. We show that use of octonions as a fundamental algebra, in any measurement, leads to generation of unavoidable 18.6 bit relative entropy of the probability density functions of the active and passive transformations, which correspond to the groups G2 and SO(7), respectively. This algebraical entropy can be used to determine the arrow of time, analogically as thermodynamic entropy does.

DOI
10.3390/e24111522

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Usually, it is supposed that irreversibility of time appears only in macrophysics. Here, we attempt to introduce the microphysical arrow of time assuming that at a fundamental level nature could be non-associative. Obtaining numerical results of a measurement, which requires at least three ingredients: object, device and observer, in the non-associative case depends on ordering of operations and is ambiguous. We show that use of octonions as a fundamental algebra, in any measurement, leads to generation of unavoidable 18.6 bit relative entropy of the probability density functions of the active and passive transformations, which correspond to the groups G2 and SO(7), respectively. This algebraical entropy can be used to determine the arrow of time, analogically as thermodynamic entropy does.
What is the physical origin of the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of disorder, or entropy, as it appears in the statistical interpretation of the second law of thermodynamics. In this article, it is argued that the philosophical point of view that the arrow of time is a fundamental property of Nature is incompatible with the subjective information-theoretical interpretation of probability and entropy in statistical mechanics, where they arise due to the ignorance of the observer. This conclusion, however, relies on the premise that time's arrow has a statistical mechanical origin. If this assumption is abandoned, then there is no conflict between the belief in the fundamental character of time's arrow and the belief in the subjective information-theoretical interpretation.
What is the physical origin for the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of disorder, or entropy, as it appears in the statistical interpretation of the second law of thermodynamics. In this article, it is argued that the philosophical point of view that the arrow of time is a fundamental property of Nature is incompatible with the subjective information-theoretical interpretation of probability and entropy in statistical mechanics, where they arise due to the ignorance of the observer. This conclusion, however, relies on the premise that time's arrow has a statistical mechanical origin. If this assumption is abandoned, then there is no conflict between the belief in the fundamental character of time's arrow and the belief in the subjective information-theoretical interpretation.
This work investigates the fundamental connection between quantum mechanics and thermodynamics through the concept of the arrow of time. While the second law of thermodynamics identifies the arrow of time with the monotonic increase of entropy in isolated systems, closed quantum systems evolve unitarily and conserve von Neumann entropy, making the origin of irreversibility less apparent. Here we establish a no-go theorem proving that correlations in unknown states of closed quantum systems cannot be universally removed by any physical operation. This result demonstrates that correlation generation is inherently irreversible, providing a microscopic and universal basis for the arrow of time. In thermodynamic settings, this irreversibility ensures that heat flows spontaneously from hotter to colder subsystems, reproducing the Clausius formulation of the second law. More generally, the irreversible growth of correlations explains entropy increase, the approach to equilibrium, and decoherence as observable manifestations of the same principle. Our findings ground a quantum-mechanical foundation for the second law of thermodynamics and demonstrate the universality of the arrow of time.
Abstract Based on the hypothesis that the (non-reversible) arrow of time is intrinsic in any system, no matter how small, the consequences are discussed. Within the framework of local quantum physics it is shown how such a semi-group action of time can consistently be extended to that of the group of spacetime translations in Minkowski space. In presence of massless excitations, however, there arise ambiguities in the theoretical extensions of the time translations to the past. The corresponding loss of quantum information on states upon time is determined. Finally, it is explained how the description of operations in classical terms combined with constraints imposed by the arrow of time leads to a quantum theoretical framework. These results suggest that the arrow of time is fundamental in nature and not merely a consequence of statistical effects on which the Second Law is based.
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What is the physical origin of the arrow of time? It is a commonly held belief in the physics community that it relates to the increase of entropy as it appears in the statistical interpretation of the second law of thermodynamics. At the same time, the subjective information-theoretical interpretation of probability, and hence entropy, is a standard viewpoint in the foundations of statistical mechanics. In this article, it is argued that the subjective interpretation is incompatible with the philosophical point of view that the arrow of time is a fundamental property of Nature. The subjectivist can only uphold this philosophy if the role played by the second law of thermodynamics in defining time's arrow is abandoned.