Eigenvalues of the volume operator in loop quantum gravity

Meissner, Krzysztof A.

doi:10.1088/0264-9381/23/3/005

Cited by 18 publications

(32 citation statements)

References 12 publications

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“…[18,19]), but cannot be compared easily with our results due to the lack of a direct correspondence between LQG and LQC models.…”

Section: Discussionmentioning

confidence: 86%

Turning big bang into big bounce: II. Quantum dynamics

Małkiewicz¹,

Piechocki²

2010

Class. Quantum Grav.

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We analyze the big bounce transition of the quantum FRW model in the setting of the nonstandard loop quantum cosmology (LQC). Elementary observables are used to quantize composite observables. The spectrum of the energy density operator is bounded and continuous. The spectrum of the volume operator is bounded from below and discrete. It has equally distant levels defining a quantum of the volume. The discreteness may imply a foamy structure of spacetime at semiclassical level which may be detected in astro-cosmo observations. The nonstandard LQC method has a free parameter that should be fixed in some way to specify the big bounce transition.

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“…[18,19]), but cannot be compared easily with our results due to the lack of a direct correspondence between LQG and LQC models.…”

Section: Discussionmentioning

confidence: 86%

Turning big bang into big bounce: II. Quantum dynamics

Małkiewicz¹,

Piechocki²

2010

Class. Quantum Grav.

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“…In [48] a formula for the spectral density of the volume operator at the gauge invariant 4-valent vertex was derived in the limit of large spins, treating j 1 , . .…”

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confidence: 99%

Properties of the volume operator in loop quantum gravity: I. Results

Brunnemann¹,

Rideout²

2008

Class. Quantum Grav.

108

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We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. Our analysis considers for the first time generic graph vertices of valence greater than four. Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a 'volume gap' (a smallest non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding. We compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5-7, and argue that these sets can be used to label spatial diffeomorphism invariant states. We observe how gauge invariance connects vertex geometry and representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum of 4-valent vertices are included, for which the presence of a volume gap is shown. This paper presents our main results; details are provided by a companion paper [52]. IntroductionLoop Quantum Gravity (LQG) [1,2,3] has become a promising candidate for a quantum theory of gravity over the last 15 years. It is an attempt to canonically quantize General Relativity (GR) while preserving its key principle: background independence 1 . The resulting quantum theory is formulated as an SU(2) gauge theory. * brunnemann@math.uni-hamburg.de † drideout@perimeterinstitute.ca 1 With the term 'background independence' we imply independence on a choice of fixed background geometry. In order to allow for the possibility of topology change in quantum gravity, it would be desirable to have the theory in its final formulation be independent of the topology of the underlying manifold, however to our knowledge this has not been achieved so far within LQG. 1Upon casting GR into a Hamiltonian formulation by introducing a foliation of four dimensional spacetime (M, g µν ) 2 into spatial three dimensional hypersurfaces Σ, with the orthogonal timelike direction parametrized by t ∈ Ê, one obtains first class constraints which have to be imposed on the reformulated theory such that it obeys the dynamics of Einstein's equations and is independent of the particular choice of foliation. These constraints are the three spatial diffeomorphism or vector constraints which generate diffeomorphisms inside Σ, and the so called Hamilton constraint which generates deformations of the hypersurfaces Σ in the t-(foliation) direction. In addition one obtains three Gauss constraints due to the introduction of additional SU(2) gauge degrees of freedom.(There is also a more recent reformulation of the constraints, as proposed in the master constraint programme. See [9] and references therein.) The resulting theory is then quantized on the kinematical level in terms of holonomies h and electric fluxes E. Kinematical states are defined over collection of edges of embedded graphs. The physical states have to be constructe...

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“…The area operator and the volume operator in the context of loop quantum gravity have been considered in [44] for the first time. Further studies concerning the area operator can be found in [45], [46], [47] and concerning the volume operator in [48], [49], [50], [51], [52]. In this paper the considerations will remain restricted to the area operator.…”

Section: Holonomy Representation and Area Operatormentioning

confidence: 99%

Canonical quantum gravity on noncommutative space–time

Kober¹

2015

Int. J. Mod. Phys. A

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In this paper canonical quantum gravity on noncommutative space-time is considered. The corresponding generalized classical theory is formulated by using the moyal star product, which enables the representation of the field quantities depending on noncommuting coordinates by generalized quantities depending on usual coordinates. But not only the classical theory has to be generalized in analogy to other field theories. Besides, the necessity arises to replace the commutator between the gravitational field operator and its canonical conjugated quantity by a corresponding generalized expression on noncommutative space-time. Accordingly the transition to the quantum theory has also to be performed in a generalized way and leads to extended representations of the quantum theoretical operators. If the generalized representations of the operators are inserted to the generalized constraints, one obtains the corresponding generalized quantum constraints including the Hamiltonian constraint as dynamical constraint. After considering quantum geometrodynamics under incorporation of a coupling to matter fields, the theory is transferred to the Ashtekar formalism. The holonomy representation of the gravitational field as it is used in loop quantum gravity opens the possibility to calculate the corresponding generalized area operator.

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Eigenvalues of the volume operator in loop quantum gravity

Cited by 18 publications

References 12 publications

Turning big bang into big bounce: II. Quantum dynamics

Turning big bang into big bounce: II. Quantum dynamics

Properties of the volume operator in loop quantum gravity: I. Results

Canonical quantum gravity on noncommutative space–time

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