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arXiv:quant-ph/0701225 · Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

Probability, arrow of time and decoherence

Guido Bacciagaluppi

Published Apr 21, 2007• 39 citations
arXivHTMLPDF
DOI
10.1016/j.shpsb.2006.04.007

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A model quantum cosmology is used to illustrate how arrows of time emerge in a universe governed by a time-neutral dynamical theory constrained by time asymmetric initial and final boundary conditions represented by initial and final density matrices. In a quantum universe universe arrows of time are described by the probabilities of appropriately coarse grained sets of histories of quantities like entropy that grow or decay. We show that the requirement of that these sets of histories decohere implies two things: (1) A time asymmetry between initial and final conditions that is a basis for arrows ot time. (2) How a final state of indifference that is represented by a final density matrix proportional to the unit density matrix is consistent with causality, and allows a finer-grained description of the model universe in terms of decoherent histories than any other final state.
We investigate the origin of the arrow of time in quantum mechanics in the context of quantum cosmology. The ``Copenhagen'' quantum mechanics of measured subsystems incorporates a fundamental arrow of time. Extending discussions of Aharonov, Bergmann and Lebovitz, Griffiths, and others we investigate a generalized quantum mechanics for cosmology that utilizes both an initial and a final density matrix to give a time-neutral formulation without a fundamental arrow of time. Time asymmetries can arise for particular universes from differences between their initial and final conditions. Theories for both would be a goal of quantum cosmology. A special initial condition and a final condition of indifference would be sufficient to explain the observed time asymmetries of the universe. In this essay we ask under what circumstances a completely time symmetric universe, with T-symmetric initial and final condition, could be consistent with the time asymmetries of the limited domain of our experience. We discuss the approach to equilibrium, the electromagnetic system, and T-violation in the weak interactions in such universes. The decoherence of alternatives that is necessary for the prediction of probabilities in the quantum mechanics of closed systems is the origin of a fundamental limitation on quantum cosmologies with initial and final conditions. The initial and final density matrices cannot both be pure; if they are there is no decoherence and no prediction.