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Journal of Physics Conference Series

Quantum theory with arrow of time: symmetry breaking and non-local spinor realization with non-commuting operators of energy and decay

Vadim V. Asadov, Oleg V. Kechkin

Published Aug 24, 2012• 1 citation
Publisher

Abstract

We present a modified quantum dynamics with a non-Hermitian Hamiltonian ≠ + , and a complex parameter of evolution. The theory is developed with [ + , ] = κ 2 , where κ is the new fundamental constant. It is shown that this theory loses its symmetry in the case of κ ≠ 0. We demonstrate that this theory allows a thermodynamical interpretation and possesses an 'arrow of time' for isothermal and adiabatic regimes of evolution. A realization of this theory is a modified four-dimensional Dirac spinor system, which appears non-local in the physical three-dimensional space.

DOI
10.1088/1742-6596/380/1/012011

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We present a modified quantum dynamics with a non-Hermitian Hamiltonian ≠ + , and a complex parameter of evolution. The theory is developed with [ + , ] = κ 2 , where κ is the new fundamental constant. It is shown that this theory loses its symmetry in the case of κ ≠ 0. We demonstrate that this theory allows a thermodynamical interpretation and possesses an 'arrow of time' for isothermal and adiabatic regimes of evolution. A realization of this theory is a modified four-dimensional Dirac spinor system, which appears non-local in the physical three-dimensional space.
It is shown, that quantum theory with complex evolutionary time parameter and non-Hermitian Hamiltonian structure can be used for natural unification of quantum and thermodynamic principles. The theory is postulated as analytical in respect to the parameter of evolution, which real part is identified with the `usual' physical time, whereas the imaginary one is understood as proportional to the inverse absolute temperature. Also, the Hermitian part of the Hamiltonian is put equal to conventional operator of energy. It is shown, that the anti-Hermitian Hamiltonian part, which is taken as commuting with the energy operator, is constructed from parameters of decay of the system. It is established, that quantum dynamics, predicted by this theory, is integrable in the same sense as the corresponding non-modified one, and that it possesses a well defined arrow of time in isothermal and adiabatic regimes of the evolution. It is proved, that average value of the decay operator decreases monotonously (as the function of the physical time) in these important thermodynamical regimes for the arbitrary initial data taken. We discuss possible application of the general formalism developed to construction of time-irreversible modification of a string theory.
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.
It is shown, that parity violation in quantum systems can be a natural result of their dynamical evolution. The corresponding (completely integrable) formalism is based on the use of quantum theory with complex time and non-Hermitian Hamiltonian. It is demonstrated, that starting with total symmetry between left and right states at the initial time, one obtains strictly polarized system at the time infinity. The increasing left-right asymmetry detects a presence of well-defined arrow of time in evolution of the system. We discuss possible application of the general formalism developed to construction of modified irreversible dynamics of massless Dirac fields (in framework of superstring theory, for example).
The quantum mechanics with a stationary non-Hermitian Hamiltonian and a complex evolution parameter, as well as its classical limit with nontrivial correlations have been studied. The corresponding dynamics is shown to be irreversible for the isothermal and adiabatic regimes of quantum and classical evolution. The possibility of a universal relationship between irreversibility and dynamical parity violation in the system has been established. The mechanism of gravity generation by the distribution of correlations in a free theory is demonstrated.
It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical", they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 .