Consistent quantum prediction and decoherence in quantum cosmology

David Craig, Parampreet Singh

We describe a complete, coherent framework within which probabilities may be consistently extracted from amplitudes for quantum histories in loop quantum cosmology. The decoherence functional is constructed for a loop quantization of a flat Friedmann-Lemaǐtre-Robertson-Walker cosmological model, thereby permitting consistent quantum predictions to be made in a mathematically precise model of quantum cosmology. Consistent families of quantum histories are exhibited. Singularity avoidance by generic quantum states in loop-quantized models is contrasted with the failure of parallel Wheeler-DeWitt-quantized universes to avoid the big-bang singularity. It is shown that all loop quantum states approach arbitrarily large volume Wheeler-DeWitt states in both the asymptotic "past" and "future" of emergent time. The critical role played by decoherence of histories in these predictions is illustrated.

Masa‐aki Sakagami, Takashi Okamura, Hiroto Kubotani

Claus Kiefer

David Craig

We review the application of the consistent (or decoherent) histories formulation of quantum theory to canonical loop quantum cosmology. Conventional quantum theory relies crucially on “measurements” to convert unrealized quantum potentialities into physical outcomes that can be assigned probabilities. In the early universe and other physical contexts in which there are no observers or measuring apparatus (or indeed, in any closed quantum system), what criteria determine which alternative outcomes may be realized and what their probabilities are? In the consistent histories formulation it is the vanishing of interference between the branch wave functions describing alternative histories — as determined by the system’s decoherence functional — that determines which alternatives may be assigned probabilities. We describe the consistent histories formulation and how it may be applied to canonical loop quantum cosmology, describing in detail the application to homogeneous and isotropic cosmological models with scalar matter. We show how the theory may be used to make definite physical predictions in the absence of “observers”. As an application, we demonstrate how the theory predicts that loop quantum models “bounce” from large volume to large volume, while conventional “Wheeler–DeWitt”-quantized universes are invariably singular. We also briefly indicate the relation to other work.

David Craig, Parampreet Singh

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 model's 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.

David Craig, Parampreet Singh