The decoherence functional in quantum mechanics and quantum cosmology

This dissertation contains a sequence of three investigations into the relationship between quantum mechanics and boundary conditions in cosmology. The lens through which each of these investigations is focussed is the decoherence functional, the generalized notion of (algebraic) quantum state of a closed quantum system. It not only measures the quantum interference between the alternative histories of a closed system, thereby determining when probabilities may consistently be assigned in an exhaustive set of alternative histories, but also supplies the probabilities of histories in such consistent sets. The decoherence functional is the central structural element of the generalized quantum mechanics required for a "quantum theory of the universe." Chapter two formulates generalized quantum theory in terms of the "geometric" language of linear operators in Hilbert space. Using this language I derive general results concerning the relationship between the decoherence functional of a closed quantum system, and the sets of quantum histories about which it can make predictions, that apply to any physical theory that may be formulated as a generalized quantum theory. The third chapter employs generalized quantum theory in the construction of a model minisuperspace quantum cosmology of closed homogeneous universes. The central prediction of the resulting sum-over-histories theory is that, in an appropriate semiclassical limit and with an appropriate choice of boundary conditions, universes which recollapse classically also do so quantum mechanically. (Not all quantizations of minisuperspace models can make this apparently natural prediction.) Chapter four examines another aspect of the relationship between the boundary conditions on a closed universe and its semiclassical behaviour: the arrow of time. I study the implications of a particular choice of boundary condition on a closed quantum universe, the symmetric condition that matter in the universe be smoothly distributed and in local thermodynamic equilibrium whenever a closed universe is small. This time symmetric boundary condition has the observational implication that the bath of optical radiation from extragalactic sources must be at least twice that due to the galaxies to our past alone. This prediction is barely consistent with present observations, and may soon be ruled out entirely.