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Journal of Russian Laser Research

Boundary Conditions for the Dynamical Equations of Quantum Mechanics

Arno Böhm, Hai Viet Bui, Piotr Kielanowski

Published Jan 1, 2014• 0 citations
Publisher
DOI
10.1007/s10946-014-9396-5

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Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time‐asymmetric boundary conditions of time‐symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann‐Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time‐asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.
A recently proposed stochastic hidden variable model for quantum mechanics has been claimed to involve "retrocausality" due to the appearance of equations of motion with future-time boundary conditions. We formulate an equivalent system of forward-time equations of motion that gives rise to the same trajectories as solutions, but involves only initial-time boundary conditions. The forward-time dynamics involves a guidance term for the dynamical variables, determined by the phase-space distribution corresponding to a quantum wavefunction. We show, however, that this particular guidance term can be recovered as the mean-field limit of averaged pairwise interactions among an ensemble of finitely many particles.