Bianchi cosmologies: the role of spatial topology

Ashtekar, Abhay; Samuel, Joseph

doi:10.1088/0264-9381/8/12/005

Cited by 104 publications

(185 citation statements)

References 18 publications

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“…As in isotropic models, one absorbs powers of the coordinate volume to obtain variables φ i I and p I i . The kinematics is the same for all class A models, except possibly for slight differences in the diffeomorphism constraint [150,151]. Connection components define a distinguished triple of su(2) elementsφ i I τ i , one for each independent direction of space.…”

Section: Homogeneitymentioning

confidence: 99%

Homogeneous loop quantum cosmology

Bojowald

2003

Class. Quantum Grav.

167

364

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Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e. the fact that the backward evolution of a classical space-time inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding space-time is then modified. One particular realization is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. Main effects are introduced into effective classical equations which allow to avoid interpretational problems of quantum theory. They give rise to new kinds of early universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function which allows to extend space-time beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of space-time arising in loop quantum gravity and its application to cosmology sheds new light on more general issues such as time.

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Section: Homogeneitymentioning

confidence: 99%

Homogeneous loop quantum cosmology

Bojowald

2003

Class. Quantum Grav.

167

364

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“…For simplicity, we specialize to the vacuum models and will focus upon the Bianchi I and Bianchi IX cases. Many of the results here, including the algebra of the momentum constraints, have been obtained by Ashtekar and Samuel [40].…”

Section: Appendix A: An Alternative Quantization Of Class a Bianchi Mmentioning

confidence: 72%

Applications of a new proposal for solving the ‘‘problem of time’’ to some simple quantum cosmological models

Higuchi

Wald

1995

Phys. Rev. D

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We apply a recent proposal for defining states and observables in quantum gravity to some simple models. First, we consider the toy model of a Klein-Gordon particle in an external potential in Minkowski spacetime and compare our proposal to the theory obtained by deparametrizing with respect to a choice of time slicing prior to quantization. We show explicitly that the dynamics defined by the deparametrization approach depends upon the choice of time slicing. On the other hand, our proposal automatically yields a well defined dynamics -manifestly independent of the choice of time slicing at intermediate times -but there is a "memory effect": After the potential is turned off, the dynamics no longer returns to the standard, free particle dynamics. Next, we apply our proposal to the closed Robertson-Walker quantum cosmology with a homogeneous massless scalar field. We choose our time variable to be the size of the universe, so the only dynamical variable is the scalar field. It is shown that the resulting theory has the expected semi-classical behavior up to the classical turning point from expansion to contraction, i.e., given a classical solution which expands for much longer than the Planck time, there is a quantum state whose dynamical evolution closely approximates this classical solution during the expanding phase. However, 1 when the "time" takes a value larger than the classical maximum, the scalar field becomes "frozen" at the value it had when it entered the classically forbidden region. The Taub model, with and without a homogeneous scalar field, also is studied, and similar behavior is found. In an Appendix, we derive the form of the Wheeler-DeWitt equation on minisuperspace for the Bianchi models by performing a proper quantum reduction of the momentum constraints; this equation differs from the usual form of the Wheeler-DeWitt equation, which is obtained by solving the momentum constraints classically, prior to quantization.

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“…The case of Bianchi Type I geometries, has been repeatedly treated in the literatureboth at the classical level [8] and the quantum level [9]. In all these works, the scale factor matrix is taken to be diagonal.…”

Section: Introductionmentioning

confidence: 99%

“…In the Schrödinger representation: 9) with the relevant operators satisfying the basic Canonical Commutation Relations (CCR) -corresponding to the classical ones:…”

Section: Introductionmentioning

confidence: 99%

Conditional symmetries and the quantization of Bianchi Type I vacuum cosmologies with and without cosmological constant

Christodoulakis

Gakis

Papadopoulos

2002

Class. Quantum Grav.

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In this work, the quantization of the most general Bianchi Type I geometry, with and without a cosmological constant, is considered. In the spirit of identifying and subsequently removing as many gauge degrees of freedom as possible, a reduction of the initial 6-dimensional configuration space is presented. This reduction is achieved by imposing as additional conditions on the wave function, the quantum version of the -linear in momenta-classical integrals of motion (conditional symmetries). The vector fields inferred from these integrals induce, through their integral curves, motions in the configuration space which can be identified to the action of the automorphism group of Type I, i.e. GL(3, ℜ). Thus, a wave function depending on one degree of freedom, namely the determinant of the scale factor matrix, is found. A measure for constructing the Hilbert space is proposed. This measure respects the above mentioned symmetries, and is also invariant under the classical property of covariance under arbitrary scalings of the Hamiltonian

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Bianchi cosmologies: the role of spatial topology

Cited by 104 publications

References 18 publications

Homogeneous loop quantum cosmology

Homogeneous loop quantum cosmology

Applications of a new proposal for solving the ‘‘problem of time’’ to some simple quantum cosmological models

Conditional symmetries and the quantization of Bianchi Type I vacuum cosmologies with and without cosmological constant

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