David Craig

Consistent Probabilities in Wheeler-DeWitt Quantum Cosmology

We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the models Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to study the singular behavior of quantum Wheeler-DeWitt universes. Within this framework, rigorous formulas are given for calculating the corresponding probabilities from the wave function when those probabilities may be consistently defined, thus replacing earlier heuristics for interpreting the wave function of the universe with explicit constructions. It is shown according to a rigorously formulated standard, and in a quantum-mechanically consistent way, that in this quantization these models are generically singular. Independent of the choice of state we show that the probability for these Wheeler-DeWitt quantum universes to ever encounter a singularity is unity. In addition, the relation between histories formulations of quantum theory and relational Dirac observables is clarified.

Dynamical eigenfunctions and critical density in loop quantum cosmology

We offer a new, physically transparent argument for the existence of the critical, universal maximum matter density in loop quantum cosmology for the case of a flat Friedmann?Lema?tre?Robertson?Walker cosmology with scalar matter. The argument is based on the existence of a sharp exponential ultraviolet cutoff in momentum space on the eigenfunctions of the quantum cosmological dynamical evolution operator (the gravitational part of the Hamiltonian constraint), attributable to the fundamental discreteness of spatial volume in loop quantum cosmology. The existence of the cutoff is proved directly from recently found exact solutions for the eigenfunctions for this model. As a consequence, the operators corresponding to the momentum of the scalar field and the spatial volume approximately commute. The ultraviolet cutoff then implies that the scalar momentum, though not a bounded operator, is in effect bounded on subspaces of constant volume, leading to the upper bound on the expectation value of the matter density. The maximum matter density is universal (i.e. independent of the quantum state) because of the?linear scaling of the cutoff with volume. These heuristic arguments are supplemented by a new proof in the volume representation of the existence of the maximum matter density. The techniques employed to demonstrate the existence of the cutoff also allow us to extract the large-volume limit of the exact eigenfunctions, confirming earlier numerical and analytical work showing that the eigenfunctions approach superpositions of the eigenfunctions of the Wheeler?DeWitt quantization of the same model. We argue that generic (not just semiclassical) quantum states approach symmetric superpositions of expanding and contracting universes.

The Vertex Expansion in the Consistent Histories Formulation of Spin Foam Loop Quantum Cosmology

Assignment of consistent quantum probabilities to events in a quantum universe is a fundamental challenge which every quantum cosmology/gravity framework must overcome. In loop quantum cosmology, this issue leads to a fundamental question: What is the probability that the universe undergoes a non-singular bounce? Using the consistent histories formulation, this question was successfully answered recently by the authors for a spatially flat FRW model in the canonical approach. In this manuscript, we obtain a covariant generalization of this result. Our analysis is based on expressing loop quantum cosmology in the spin foam paradigm and using histories defined via volume transitions to compute the amplitudes of transitions obtained using a vertex expansion. We show that the probability for bounce turns out to be unity.

A Consistent Histories Formulation of Wheeler‐DeWitt Quantum Cosmology

We summarize a fully worked‐out consistent histories formulation of a quantum cosmological model, and illustrate with an example calculation of the probability that the quantum universe becomes singular. Specifically, we take a complete Wheeler‐DeWitt canonical quantization of a flat Friedmann‐Robertson‐Walker cosmological model sourced with a free massless scalar field, and construct the decoherence functional which determines which quantum histories may be consistently assigned probabilities. As an example, we exhibit the probability that the quantum universe is singular in the sense that it assumes zero volume. The classical solutions of this model are all singular; we show that the quantum solutions are as well. We comment on the crucial role played by quantum decoherence in arriving at this conclusion.

Cosmological dynamics in spin-foam loop quantum cosmology: challenges and prospects

We explore the structure of the spin foam-like vertex expansion in loop quantum cosmology and discuss properties of the corresponding amplitudes, with the aim of elucidating some of the expansions useful properties and features. We find that the expansion is best suited for consideration of conceptual questions and for investigating short-time, highly quantum behavior. In order to study dynamics at cosmological scales, the expansion must be carried to very high order, limiting its direct utility as a calculational tool for such questions. Conversely, it is unclear that the expansion can be truncated at finite order in a controlled manner.

Noncontextuality in Quantum Measure Theory

The Clauser‐Horne‐Shimony‐Holt‐Bell inequalities are necessary conditions for a set of no‐signalling probabilities for two measurers each with two alternative experiments each with two possible outcomes to admit a joint probability distribution. An analogue of these inequalities in the context of quantum measure theory is presented. Talk given by Fay Dowker.