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arXiv:hep-th/0608148

Arrow of time in generalized quantum theory and its classical limit dynamics

Vadim V. Asadov, Oleg V. Kechkin

Published Aug 21, 2006• 0 citations
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Abstract

In this paper we have studied a generalized quantum theory and its consistent classical limit, which possess a well-defined arrow of time in their dynamics. The original quantum theory is defined as analytically dependent on complex time and specified by non-Hermitian Hamiltonian structure.
DOI
10.48550/arxiv.hep-th/0608148

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We study a quantum theory with complex time parameter and non-Hermitian Hamiltonian structure. In this theory, the real part of the complex time is equal to `usual' physical time, whereas the imaginary one is proportional to inverse absolute temperature of the system. Then, the Hermitian part of the Hamiltonian coincides with conventional operator of energy; the anti-Hermitian part, which is taken as a symmetry operator, defines decay parameters of the theory. We integrate the equations of motion in a Hamiltonian proper basis, and detect a classical dynamics of the corresponding quantum numbers, using their continuous approximation and the zero limit of the Plank's constant. It is proved, that this dynamics possesses a well defined arrow of time in the isothermal and adiabatic regimes of the thermodynamical evolution of the system.
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It is brought forward that viable theories of the physical world that have no variable at all that can play the role of time, do not exist; some notion of time is one of the very first ingredients a candidate theory should possess. Almost by definition, time has an arrow. In contrast, time reversibility, or even the possibility to run the equations of motion backwards in time, is not at all a primary requirement. This means that the direction of the arrow of time may well be uniquely defined in the theory, even locally. It is explained that a rigorous definition of time, as well as a formulation of the causality and locality concepts, can only be given when one has a model for the physical phenomena described. The only viable causality condition is one that is symmetric under time reversal. We explain these statements in terms of the author’s favoured deterministic cellular automaton interpretation of quantum mechanics, also to be referred to as 'vector space analysis', and expand on these ideas. It is also summarised how our more rigorous causality condition affects Bell’s theorems. What distinguishes quantum systems from classical ones is our fundamental inability to control the microscopic details of the initial state when phenomena are studied in the light of some theoretical model.