HYBRID QUANTUM COSMOLOGY: COMBINING LOOP AND FOCK QUANTIZATIONS

As a necessary step towards the extraction of realistic results from Loop Quantum Cosmology, we analyze the physical consequences of including inhomogeneities. We consider in detail the quantization of a gravitational model in vacuo which possesses local degrees of freedom, namely, the linearly polarized Gowdy cosmologies with the spatial topology of a three-torus. We carry out a hybrid quantization which combines loop and Fock techniques. We discuss the main aspects and results of this hybrid quantization, which include the resolution of the cosmological singularity, the polymeric quantization of the internal time, a rigorous definition of the quantum constraints and the construction of their solutions, the Hilbert structure of the physical states, and the recovery of a conventional Fock quantization for the inhomogeneities.

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Hybrid Loop Quantum Cosmology: An Overview

Frontiers in Astronomy and Space Sciences ◽

10.3389/fspas.2021.624824 ◽

2021 ◽

Vol 8 ◽

Author(s):

Beatriz Elizaga Navascués ◽

Guillermo A. Mena Marugán

Keyword(s):

Quantum Cosmology ◽

Degrees Of Freedom ◽

Hybrid Approach ◽

Loop Quantum Cosmology ◽

Main Challenge ◽

Cosmological Observations ◽

Linearly Polarized ◽

Homogeneous Background ◽

Gowdy Spacetimes ◽

Physical Consequences

Loop Quantum Gravity is a nonperturbative and background independent program for the quantization of General Relativity. Its underlying formalism has been applied successfully to the study of cosmological spacetimes, both to test the principles and techniques of the theory and to discuss its physical consequences. These applications have opened a new area of research known as Loop Quantum Cosmology. The hybrid approach addresses the quantization of cosmological systems that include fields. This proposal combines the description of a finite number of degrees of freedom using Loop Quantum Cosmology, typically corresponding to a homogeneous background, and a Fock quantization of the field content of the model. In this review we first present a summary of the foundations of homogeneous Loop Quantum Cosmology and we then revisit the hybrid quantization approach, applying it to the study of Gowdy spacetimes with linearly polarized gravitational waves on toroidal spatial sections, and to the analysis of cosmological perturbations in preinflationary and inflationary stages of the Universe. The main challenge is to extract predictions about quantum geometry effects that eventually might be confronted with cosmological observations. This is the first extensive review of the hybrid approach in the literature on Loop Quantum Cosmology.

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COSMIC TIME IN QUANTUM COSMOLOGY: INFLATION-COMPACTIFICATION AS A QUANTUM EFFECT

International Journal of Modern Physics D ◽

10.1142/s0218271893000180 ◽

1993 ◽

Vol 02 (02) ◽

pp. 221-247 ◽

Cited By ~ 15

Author(s):

E.I. GUENDELMAN ◽

A.B. KAGANOVICH

Keyword(s):

Cosmological Constant ◽

Quantum Cosmology ◽

Quantum Effect ◽

Cosmic Time ◽

Expectation Values ◽

Cosmological Singularity ◽

Inflationary Phase ◽

Definition Of ◽

The Universe ◽

Kaluza Klein

We consider 1+D-dimensional, toroidally compact Kaluza-Klein theories. In the context of the minisuperspace approach of quantum cosmology, we solve the Wheeler-DeWitt equation in the presence of a negative cosmological constant and dust. Then, it is found that the quantum effects stabilize the volume of the Universe, so that there can be an avoidance of the cosmological singularity. Although cosmic time does not appear explicitly in the Wheeler-DeWitt equation, we find that a cosmic time dependence appears for the expectation values of certain variables. This result is obtained when proper care of some subtle points concerning the definition of averages in this model is taken. The stabilization of the volume, when there is anisotropy in the evolution of the Universe (which turns out to be quantized), is consistent with another effect we find: the existence of a “quantum inflationary phase” for some dimensions and simultaneously the existence of a “quantum deflationary contraction” for the rest.

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Non-Oscillatory Power Spectrum From States of Low Energy in Kinetically Dominated Early Universes

Frontiers in Astronomy and Space Sciences ◽

10.3389/fspas.2021.702543 ◽

2021 ◽

Vol 8 ◽

Author(s):

Mercedes Martín-Benito ◽

Rita B. Neves ◽

Javier Olmedo

Keyword(s):

Power Spectrum ◽

Quantum Cosmology ◽

Initial Conditions ◽

Power Spectra ◽

Low Energy ◽

Loop Quantum Cosmology ◽

Test Function ◽

Vacuum States ◽

Oscillatory Power ◽

Definition Of

Recently, States of Low Energy (SLEs) have been proposed as viable vacuum states of primordial perturbations within Loop Quantum Cosmology (LQC). In this work we investigate the effect of the high curvature region of LQC on the definition of SLEs. Shifting the support of the test function that defines them away from this regime results in primordial power spectra of perturbations closer to those of the so-called Non-oscillatory (NO) vacuum, which is another viable choice of initial conditions previously introduced in the LQC context. Furthermore, through a comparison with the Hadamard-like SLEs, we prove that the NO vacuum is of Hadamard type as well.

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Using Trajectories in Quantum Cosmology

Universe ◽

10.3390/universe4080089 ◽

2018 ◽

Vol 4 (8) ◽

pp. 89 ◽

Cited By ~ 1

Author(s):

Patrick Peter

Keyword(s):

Wave Function ◽

Quantum Cosmology ◽

Degrees Of Freedom ◽

Effective Means ◽

Quantum Mechanical ◽

Quantum Mechanical System ◽

Single Scale ◽

Trajectory Approach ◽

Full Knowledge ◽

Physical Consequences

Quantum cosmology based on the Wheeler De Witt equation represents a simple way to implement plausible quantum effects in a gravitational setup. In its minisuperspace version wherein one restricts attention to FLRW metrics with a single scale factor and only a few degrees of freedom describing matter, one can obtain exact solutions and thus acquire full knowledge of the wave function. Although this is the usual way to treat a quantum mechanical system, it turns out however to be essentially meaningless in a cosmological framework. Turning to a trajectory approach then provides an effective means of deriving physical consequences.

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ON THE SEMICLASSICAL LIMIT OF LOOP QUANTUM COSMOLOGY

International Journal of Modern Physics D ◽

10.1142/s0218271812500769 ◽

2012 ◽

Vol 21 (09) ◽

pp. 1250076 ◽

Cited By ~ 20

Author(s):

ALEJANDRO CORICHI ◽

EDISON MONTOYA

Keyword(s):

Coherent State ◽

Quantum Cosmology ◽

Coherent States ◽

Semiclassical Limit ◽

Loop Quantum Cosmology ◽

Squeezed States ◽

Gaussian States ◽

Exactly Solvable ◽

Spatial Topology ◽

Frw Model

We consider a k = 0 Friedman–Robertson–Walker (FRW) model within loop quantum cosmology (LQC) and explore the issue of its semiclassical limit. The model is exactly solvable and allows us to construct analytical (Gaussian) coherent-state solutions for each point on the space of classical states. We propose physical criteria that select from these coherent states, those that display semiclassical behavior, and study their properties in the deep Planck regime. Furthermore, we consider generalized squeezed states and compare them to the Gaussian states. The issue of semiclassicality preservation across the bounce is studied and shown to be generic for all the states considered. Finally, we comment on some implications these results have, depending on the topology of the spatial slice. In particular, we consider the issue of the recovery, within our class of states, of a scaling symmetry present in the classical description of the system when the spatial topology is noncompact.

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Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology

Zeitschrift für Naturforschung A ◽

10.1515/zna-2019-0223 ◽

2019 ◽

Vol 74 (12) ◽

pp. 1069-1098 ◽

Cited By ~ 2

Author(s):

Leonardo Chataignier

Keyword(s):

Quantum Cosmology ◽

Degrees Of Freedom ◽

Constraint Equation ◽

Positive Definite ◽

Time Variable ◽

Inner Product ◽

Time Coordinate ◽

Gauge Fixing ◽

Klein Gordon ◽

Definition Of

AbstractWe make a critical review of the semiclassical interpretation of quantum cosmology and emphasise that it is not necessary to consider that a concept of time emerges only when the gravitational field is (semi)classical. We show that the usual results of the semiclassical interpretation and its generalisation known as the Born–Oppenheimer approach to quantum cosmology can be obtained by gauge fixing, both at the classical and quantum levels. By “gauge fixing,” we mean a particular choice of the time coordinate, which determines the arbitrary Lagrange multiplier that appears in Hamilton’s equations. In the quantum theory, we adopt a tentative definition of the (Klein–Gordon) inner product, which is positive definite for solutions of the quantum constraint equation found via an iterative procedure that corresponds to a weak coupling expansion in powers of the inverse Planck mass. We conclude that the wave function should be interpreted as a state vector for both gravitational and matter degrees of freedom, the dynamics of which is unitary with respect to the chosen inner product and time variable.

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Hybrid models in loop quantum cosmology

International Journal of Modern Physics D ◽

10.1142/s0218271816420074 ◽

2016 ◽

Vol 25 (08) ◽

pp. 1642007 ◽

Cited By ~ 21

Author(s):

Beatriz Elizaga Navascués ◽

Mercedes Martín-Benito ◽

Guillermo A. Mena Marugán

Keyword(s):

Quantum Cosmology ◽

Degrees Of Freedom ◽

Hybrid Approach ◽

Loop Quantum Cosmology ◽

Hamiltonian Constraint ◽

Hybrid Strategy ◽

Fock Representation ◽

Zero Modes ◽

Inhomogeneous Models ◽

Global Zero

In the framework of Loop Quantum Cosmology (LQC), inhomogeneous models are usually quantized by means of a hybrid approach that combines loop quantization techniques with standard quantum field theory methods. This approach is based on a splitting of the phase space in a homogeneous sector, formed by global, zero-modes and an inhomogeneous sector, formed by the remaining, infinite number of modes, that describe the local degrees of freedom. Then, the hybrid quantization is attained by adopting a loop representation for the homogeneous gravitational sector, while a Fock representation is used for the inhomogeneities. The zero-mode of the Hamiltonian constraint operator couples the homogeneous and inhomogeneous sectors. The hybrid approach, therefore, is expected to provide a suitable quantum theory in regimes where the main quantum effects of the geometry are those affecting the zero-modes, while the inhomogeneities, still being quantum, can be treated in a more conventional way. This hybrid strategy was first proposed for the simplest cosmological midisuperspaces: the Gowdy models, and it has been later applied to the case of cosmological perturbations. This paper reviews the construction and main applications of hybrid LQC.

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Shape Dynamics and the Linking Theory

10.1093/oso/9780198789475.003.0012 ◽

2018 ◽

Author(s):

Flavio Mercati

Keyword(s):

Phase Space ◽

Degrees Of Freedom ◽

Line Element ◽

Hamiltonian Formulation ◽

Operational Definition ◽

Extended Phase Space ◽

Extended Phase ◽

Problem Of Time ◽

Definition Of ◽

Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

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