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arXiv:gr-qc/0701035 · Classical and Quantum Gravity

Canonical wave packets in quantum cosmology

S S Gousheh, H R Sepangi, P Pedram, M Mirzaei

Published Aug 16, 2007• 12 citations
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Abstract

We discuss the construction of wave packets resulting from solutions of a class of Wheeler–DeWitt equations in Robertson–Walker-type cosmologies, for arbitrary curvature. We show that there always exists a 'canonical initial slope' for a given initial wavefunction, which optimizes some desirable properties of the resulting wave packet, most importantly a good classical-quantum correspondence. This can be properly denoted as a canonical wave packet. We introduce a general method for finding these canonical initial slopes which is a generalization of our earlier work.

DOI
10.1088/0264-9381/24/17/007

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We discuss the construction of wave packets resulting from solutions of a class of Wheeler–DeWitt equations in Robertson–Walker-type cosmologies, for arbitrary curvature. We show that there always exists a 'canonical initial slope' for a given initial wavefunction, which optimizes some desirable properties of the resulting wave packet, most importantly a good classical-quantum correspondence. This can be properly denoted as a canonical wave packet. We introduce a general method for finding these canonical initial slopes which is a generalization of our earlier work.
We discuss the construction of wave packets resulting from the solutions of a class of Wheeler-DeWitt equations in Robertson-Walker type cosmologies, for arbitrary curvature. We show that there always exists a “canonical initial slope ” for a given initial wave function, which optimizes some desirable properties of the resulting wave packet, most importantly good classical-quantum correspondence. This can be properly denoted as a canonical wave packet. We introduce a general method for finding these canonical initial slopes which is generalization of our earlier work. PACS numbers: 04.20.Ex, 04.60.Kz, 04.25.-g 1
We study a class of spherically symmetric Stephani cosmological models in the presence of a self-interacting scalar field in both classical and quantum domains. We discuss the construction of ‘canonical ’ wave packets resulting from the solutions of a class of Wheeler-DeWitt equations in the Stephani Universe. We suggest appropriate initial conditions which result in wave packets containing some desirable properties, most importantly good classical and quantum correspondence. We also study the situation from de-Broglie Bohm interpretation of quantum mechanics to recover the notion of time and compare the classical and Bohmian results. We exhibit that the usage of the canonical prescription and appropriate choices of expansion coefficients result in the suppression of the quantum potential and coincidence between classical and Bohmian results. We show that, in some cases, contrary to Friedmann-Robertson-Walker case, the bound state solutions also exist for all positive values of the cosmological constant. Pacs: 98.80.Qc, 04.40.Nr, 04.60.Ds; 1
We study a class of spherically symmetric Stephani cosmological models in the presence of a self-interacting scalar field in both classical and quantum domains. We discuss the construction of 'canonical' wavepackets resulting from the solutions of a class of Wheeler–DeWitt equations in the Stephani Universe. We suggest appropriate initial conditions which result in wavepackets containing some desirable properties, most importantly good classical and quantum correspondence. We also study the situation from the de Broglie–Bohm interpretation of quantum mechanics viewpoint to recover the notion of time and compare the classical and Bohmian results. We show the usage of the canonical prescription and appropriate choices of expansion coefficients to result in the suppression of the quantum potential and coincidence of classical and Bohmian results. We show that, in some cases, contrary to the Friedmann–Robertson–Walker case, the bound state solutions also exist for all positive values of the cosmological constant.
We construct a Bohmian quantum cosmological model for a spatially flat Friedmann-Robertson-Walker universe filled with a single scalar field whose potential provides a unified description of cold dark matter and dark energy at the background level. Starting from the Einstein-Hilbert action supplemented by a scalar field, we derive the minisuperspace Lagrangian and the associated canonical Hamiltonian formulation. By means of a nontrivial canonical transformation, the minisuperspace dynamics is mapped into that of a two-dimensional hyperbolic oscillator with a fixed frequency ratio, rendering the Wheeler-DeWitt equation exactly solvable by separation of variables. The resulting Wheeler-DeWitt solutions are expressed in terms of parabolic cylinder functions and are parametrised by a continuous separation constant, reflecting the constrained nature of the theory and the absence of a standard Schrödinger time parameter. Adopting the de Broglie-Bohm formulation, we derive deterministic guidance equations in minisuperspace and construct a well-defined Bohmian Hubble parameter directly in terms of the pilot-wave phase. Finally, we present a Wheeler-DeWitt-derived toy wave function for which the Bohmian trajectories and the associated cosmological expansion history can be obtained analytically, reproducing the late-time ΛCDM behaviour while exhibiting quantum modifications at earlier epochs.
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.