Mathematical structure of loop quantum cosmology

Ashtekar, Abhay; Bojowald, Martin; Lewandowski, Jerzy

doi:10.4310/atmp.2003.v7.n2.a2

Cited by 726 publications

(1,443 citation statements)

References 34 publications

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“…In isotropic models, µ 0 c has to be small compared to one for the classical constraint as a polynomial in c to be a good approximation of functions of exponentials exp(iµ 0 c/2) = exp(iµ 0 V 1/3 0c /2) used for the quantization. As noted before, however, in the presence of a cosmological constant c can become arbitrarily large even in classical regimes (while µ 0 has been argued to be of order one by comparing with the lowest area eigenvalue [26]). …”

Section: Cosmological Constantmentioning

confidence: 87%

“…Following the loop quantization, these basic objects are represented on a Hilbert space with an orthonormal basis {|µ } µ∈R of states given as functions of the connection compo-nent by c|µ = e iµc/2 [26]. Exponentials e iµ ′ c/2 , analogous to holonomies of the full theory, then act by multiplication, e iµ ′ c/2 |µ = |µ + µ ′…”

Section: Isotropic Loop Quantum Cosmologymentioning

confidence: 99%

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Loop quantum cosmology and inhomogeneities

2006

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Inhomogeneities are introduced in loop quantum cosmology using regular lattice states, with a kinematical arena similar to that in homogeneous models considered earlier. The framework is intended to encapsulate crucial features of background independent quantizations in a setting accessible to explicit calculations of perturbations on a cosmological background. It is used here only for qualitative insights but can be extended with further more detailed input. One can thus see how several parameters occuring in homogeneous models appear from an inhomogeneous point of view. Their physical roles in several cases then become much clearer, often making previously unnatural choices of values look more natural by providing alternative physical roles. This also illustrates general properties of symmetry reduction at the quantum level and the roles played by inhomogeneities. Moreover, the constructions suggest a picture for gravitons and other metric modes as collective excitations in a discrete theory, and lead to the possibility of quantum gravity corrections in large universes.

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Section: Cosmological Constantmentioning

confidence: 87%

Section: Isotropic Loop Quantum Cosmologymentioning

confidence: 99%

Loop quantum cosmology and inhomogeneities

2006

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“…Extrinsic curvature plays an important role since in a flat isotropic model it appears in holonomies on which loop quantizations are based in such a way that only e iαk with α ∈ R can be represented as operators, but not k itself [15]. Large values of k would either require one to use extremely small α in the relevant operators, or imply unexpected deviations from classical behavior.…”

Section: Introductionmentioning

confidence: 99%

“…This feature of lattice refinements was not mimicked in the first formulations of loop quantum cosmology [20,21,14,22,15] since the main focus was to understand small-volume effects such as classical singularities [23,24]. In this context, lattice refinements appear irrelevant because only a few action steps of the Hamiltonian, rather than long evolution, are sufficient to probe a singularity.…”

Section: Introductionmentioning

confidence: 99%

Lattice refining loop quantum cosmology, anisotropic models, and stability

Cartin²,

2007

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A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed.

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“…Because our FRW metric is spatially flat, we have Γ i a = 0 and hence A i a = γK i a . Via fixing the degrees of freedom of local gauge and diffeomorphism, we finally obtain the connection and densitized triad by symmetrical reduction as [40]: p) and (φ, π). The basic Poisson brackets between them can be simply read as…”

Section: A Loop Quantum Brans-dicke Cosmologymentioning

confidence: 99%

Loop quantum modified gravity and its cosmological application

Zhang

2013

Front. Phys.

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A general nonperturvative loop quantization procedure for metric modified gravity is reviewed. As an example, this procedure is applied to scalar-tensor theories of gravity. The quantum kinematical framework of these theories is rigorously constructed. Both the Hamiltonian and master constraint operators are well defined and proposed to represent quantum dynamics of scalar-tensor theories. As an application to models, we set up the basic structure of loop quantum Brans-Dicke cosmology. The effective dynamical equations of loop quantum Brans-Dicke cosmology are also obtained, which lay a foundation for the phenomenological investigation to possible quantum gravity effects in cosmology.

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Mathematical structure of loop quantum cosmology

Cited by 726 publications

References 34 publications

Loop quantum cosmology and inhomogeneities

Loop quantum cosmology and inhomogeneities

Lattice refining loop quantum cosmology, anisotropic models, and stability

Loop quantum modified gravity and its cosmological application

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