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The Ninth Marcel Grossmann Meeting

Affine coherent states in quantum cosmology

Przemysław Małkiewicz

Published Nov 16, 2017• 1 citation
Publisher

Abstract

A brief summary of the application of coherent states in the examination of quantum dynamics of cosmological models is given. We discuss quantization maps, phase space probability distributions and semiclassical phase spaces. The implementation of coherent states based on the affine group resolves the hardest singularities, renders self-adjoint Hamiltonians without boundary conditions and provides a completely consistent semiclassical description of the involved quantum dynamics. We consider three examples: the closed Friedmann model, the anisotropic Bianchi Type I model and the deep quantum domain of the Bianchi Type IX model.

DOI
10.1142/9789813226609_0340

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A brief summary of the application of coherent states in the examination of quantum dynamics of cosmological models is given. We discuss quantization maps, phase space probability distributions and semiclassical phase spaces. The implementation of coherent states based on the affine group resolves the hardest singularities, renders self-adjoint Hamiltonians without boundary conditions and provides a completely consistent semiclassical description of the involved quantum dynamics. We consider three examples: the closed Friedmann model, the anisotropic Bianchi Type I model and the deep quantum domain of the Bianchi Type IX model.
We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on group representation and probabilistic aspects of these constructions. Simple phase space examples illustrate the procedure: plane (Weyl-Heisenberg symmetry), half-plane (affine symmetry). Interesting applications to quantum cosmology (“smooth bouncing”) for Friedmann-Robertson-Walker metric are presented and those for Bianchi I and IX models are mentioned.
We have discovered a class of dynamically stable coherent states for motion on the half-line. The regularization of the half-line boundary and the consequent quantum motion are expounded within the framework of covariant affine quantization, although alternative approaches are also feasible. The former approach is rooted in affine coherent states and offers a consistent semiclassical representation of quantum motion. However, this method has been known to possess two shortcomings: (a) the dependence of affine coherent states on the choice of a vector, denoted as 'fiducial vector' (which remains unspecified), introduces significant arbitrariness in boundary regularization, and (b) regardless of the choice of 'fiducial vector,' affine coherent states fail to evolve parametrically under the Schrödinger equation, thus limiting the accuracy of the semiclassical description. This limitation, in particular, hampers their suitability for approximating the evolution of compound observables. We demonstrate that a distinct and more refined definition of affine coherent states can simultaneously address both of these issues. In other words, these new affine coherent states exhibit parametric evolution only when the 'fiducial vector,' denoted as ∣ψ0>|ψ_0>∣ψ0​>, possesses a highly specific character, such as being an eigenstate of a well-defined Hamiltonian. Our discovery holds significant relevance in the field of quantum cosmology, particularly in scenarios where the positive variable is the scale factor of the universe, and its regularized motion plays a crucial role in avoiding the big-bang singularity.
We have discovered a class of dynamically stable coherent states for motion on the half line. The regularization of the half line boundary and the consequent quantum motion are expounded within the framework of covariant affine quantization, although alternative approaches are also feasible. The former approach is rooted in affine coherent states and offers a consistent semiclassical representation of quantum motion. However, this method has been known to possess two shortcomings: (a) the dependence of affine coherent states on the choice of a vector, denoted as the ``fiducial vector'' (which remains unspecified), introduces significant arbitrariness in boundary regularization, and (b) regardless of the choice of fiducial vector, affine coherent states fail to evolve parametrically under the Schr\"odinger equation, thus limiting the accuracy of the semiclassical description. This limitation, in particular, hampers their suitability for approximating the evolution of compound observables. We demonstrate that a distinct and more refined definition of affine coherent states can simultaneously address both of these issues. In other words, these new affine coherent states exhibit parametric evolution only when the fiducial vector, denoted as $|{\ensuremath{\psi}}_{0}⟩$, possesses a highly specific character, such as being an eigenstate of a well-defined Hamiltonian. Our discovery holds significant relevance in the field of quantum cosmology, particularly in scenarios where the positive variable is the scale factor of the universe, and its regularized motion plays a crucial role in avoiding the big-bang singularity.
In this paper we study quantum dynamics of the bouncing cosmological model. We focus on the model of the flat Friedmann-Robertson-Walker universe with a free scalar field. The bouncing behavior, which replaces the classical singularity, appears due to the modification of general relativity using the methods of loop quantum cosmology. We show that there exists a unitary transformation that enables one to describe the system as a free particle with a Hamiltonian equal to the canonical momentum. We examine properties of the various quantum states of the universe: the boxcar state, the standard coherent state, and the soliton-like state, as well as Schr\"odinger's cat states constructed from these states. Characteristics of the states---such as quantum moments and Wigner functions---are investigated. We show that each of these states have, for some range of parameters, a proper semiclassical limit fulfilling the correspondence principle. The decoherence of the superposition of two universes is described and possible interpretations in terms of triad orientation and the Belinsky-Khalatnikov-Lifshitz conjecture are given. We also examine negative-part sectors of the Wigner functions for the considered states. Their respective areas and localizations in the phase space feature an adequate estimate of nonclassicality. In particular, we show that regions where the Wigner function for the Glauber coherent cat state assumes its negative values have area equal to one fourth of the Planck constant. Based on the examined examples, we also conjecture that regions with negative values for the Wigner function are never smaller than one fourth of the Planck constant for states satisfying the uncertainty principle.
We explain the powerful role that operator-valued measures can play in quantizing any set equipped with a measure, for instance a group (resp. group coset) with its invariant (resp. quasi-invariant) measure. Coherent state quantization is a particular case. Such integral quantizations are illustrated with two examples based on the Weyl-Heisenberg group and on the affine group respectively. An interesting application of the affine quantization in quantum cosmology is mentioned, and we sketch a construction of new coherent states for the hydrogen atom.