Decoherent Histories Quantum Mechanics
Generalizations for Quantum Spacetime
Models, Examples, Implications
Generalized Quantum Theory of Recollapsing Homogeneous Cosmologies [127]
(with David Craig) This paper provides a physical example of the generalization of decoherent histories quantum theory described above. A sum-over-histories generalized quantum theory is developed for homogeneous minisuperspace type A Bianchi cosmological models, focussing on the particular example of the classically recollapsing Bianchi IX universe. The decoherence functional for such universes is exhibited. We show how the probabilities of decoherent sets of alternative, coarse-grained histories of these model universes can be calculated. We consider in particular the probabilities for classical evolution defined by a suitable coarse-graining. For a restricted class of initial conditions and coarse grainings we exhibit the approximate decoherence of alternative histories in which the universe behaves classically and those in which it does not. For these situations we show that the probability is near unity for the universe to recontract classically if it expands classically. We also determine the relative probabilities of quasi-classical trajectories for initial states of WKB form, recovering for such states a precise form of the familiar heuristic “J · dΣ” rule of quantum cosmology, as well as a generalization of this rule to generic initial states.
Generalized Quantum Theory in Evaporating Black Hole Spacetimes [115]
Generalized Quantum Theory and Black Hole Evaporation [119]
These two papers are essentially the same. Usually quantum theory is formulated in terms of the evolution of states through spacelike surfaces. However, a generalization of this formulation is needed for field theory in spacetimes that are not foliable by spacelike surfaces, or in quantum gravity where geometry is not definite but a quantum variable. In particular, a generalization of usual quantum theory is needed for field theory in the spacetimes that model the process of black hole evaporation. This paper discusses a spacetime generalization of usual quantum theory that is applicable to evaporating black hole spacetimes. In this generalization, information is not lost in the process of evaporation. Rather, complete information is distributed about four-dimensional spacetime. Black hole evaporation is thus not in conflict with the principles of quantum theory when suitably generally stated.
Time and Time Functions in Parametrized Non-Relativistic Quantum Mechanics [106]
By expanding its configuration space non-relativistic quantum mechanics can be tranformed into a theory invariant under reparametizatoins of the time in the same way that general relativity is. Its predictions are not changed but hte expanded formulation provides a model framework for illustrating the ideas on this set of pages.This paper investigates ``evolving constants'' method of defining the quantum dynamics parametrized non-relativistic quantum mechanics (PNRQM). The wide range of time functions that are available to define evolving constants raises issues of interpretation, consistency, and the degree to which the resulting quantum theory coincides with, or generalizes, the usual non-relativistic theory. The allowed time functions must be restricted for the predictions of PNRQM to coincide with those of usual quantum theory. They must be restricted to have a notion of quantum evolution in a time-parameter connected to spacetime geometry. They must be restricted to prevent the theory from making inconsistent predictions for the probabilities of histories. Suitable restrictions can be introduced in PNRQM but these seem unlikely to apply to a reparametrization invariant theory like general relativity.
For more investigations of PNRQM see Chapter VII.3 of the Les Houches lectures [107].
Comparing Formulations of Generalized Quantum Mechanics for Reparametrization Systems [114]
(with Donald Marolf) A class of decoherence schemes is described for implementing the principles of generalized quantum theory in reparametrization-invariant ‘hyperbolic’ models such as minisuperspace quantum cosmology. The connection with sum-over-histories constructions is exhibited and the physical equivalence or in- equivalence of different such schemes is analyzed. The discussion focuses on comparing constructions based on the Klein-Gordon product with those based on the induced (a.k.a. Rieffel, Refined Algebraic, Group Averaging, or Spectral Analysis) inner product. It is shown that the Klein-Gordon and induced products can be simply related for the models of interest. This fact is then used to establish isomorphisms between certain decoherence schemes based on these products.
Observables in Effective Gravity [134]
(with Steve Giddings and Don Marolf) We address the construction and interpretation of diffeomorphism-invariant observables in a low-energy effective theory of quantum gravity. The observables we consider are constructed as integrals over the space of coordinates, in analogy to the construction of gauge-invariant observables in Yang-Mills theory via traces. As such, they are explicitly non-local. Nevertheless we describe how, in suitable quantum states and in a suitable limit, the familiar physics of local quantum field theory can be recovered from appropriate such observables, which we term ‘pseudo-local.’ We consider measurement of pseudo-local observables, and describe how such measurements are limited by both quantum effects and gravitational interactions. These limitations support suggestions that theories of quantum gravity associated with finite regions of spacetime contain far fewer degrees of freedom than do local field theories.
Classical Physics and Hamiltonian Quantum Mechanics as Relics of the Big Bang [92]
An early paper arguing that the Hamiltonian quantum mechanics was no the general form of quantum theory but rather an approximate feature of the universe emergent after the Planck time becuse of the character of the specific intial condition and dynamics of our universe.
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