Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory

In the two previous papers of this series we defined a new combinatorical approach to quantum gravity, Algebraic Quantum Gravity (AQG). We showed that AQG reproduces the correct infinitesimal dynamics in the semiclassical limit, provided one incorrectly substitutes the non -- Abelean group SU(2) by the Abelean group $U(1)^3$ in the calculations. The mere reason why that substitution was performed at all is that in the non -- Abelean case the volume operator, pivotal for the definition of the dynamics, is not diagonisable by analytical methods. This, in contrast to the Abelean case, so far prohibited semiclassical computations. In this paper we show why this unjustified substitution nevertheless reproduces the correct physical result: Namely, we introduce for the first time semiclassical perturbation theory within AQG (and LQG) which allows to compute expectation values of interesting operators such as the master constraint as a power series in $\hbar$ with error control. That is, in particular matrix elements of fractional powers of the volume operator can be computed with extremely high precision for sufficiently large power of $\hbar$ in the $\hbar$ expansion. With this new tool, the non -- Abelean calculation, although technically more involved, is then exactly analogous to the Abelean calculation, thus justifying the Abelean analysis in retrospect. The results of this paper turn AQG into a calculational discipline

Eigenvalues of the volume operator in loop quantum gravity

We present a simple method to calculate certain sums of the eigenvalues of the volume operator in loop quantum gravity. We derive the asymptotic distribution of the eigenvalues in the classical limit of very large spins which turns out to be of a very simple form. The results can be useful for example in the statistical approach to quantum gravity.Comment: 12 pages, version accepted in Class. Quantum Gra

Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation

The properties of the Volume operator in Loop Quantum Gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. The present analysis benefits from the general simplified formula for matrix elements of the Volume operator derived in gr-qc/0405060, making it feasible to implement it on a computer as a matrix which is then diagonalized numerically. The resulting eigenvalues serve as a database to investigate the spectral properties of the volume operator. Analytical results on the spectrum at 4-valent vertices are included. This is a companion paper to arXiv:0706.0469, providing details of the analysis presented there.Comment: Companion to arXiv:0706.0469. Version as published in CQG in 2008. More compact presentation. Sign factor combinatorics now much better understood in context of oriented matroids, see arXiv:1003.2348, where also important remarks given regarding sigma configurations. Subsequent computations revealed some minor errors, which do not change qualitative results but modify some numbers presented her

Properties of the Volume Operator in Loop Quantum Gravity I: Results

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 on 4-valent vertices are included, for which the presence of a volume gap is proved. This paper presents our main results; details are provided by a companion paper arXiv:0706.0382v1.Comment: 36 pages, 7 figures, LaTeX. See also companion paper arXiv:0706.0382v1. Version as published in CQG in 2008. See arXiv:1003.2348 for important remarks regarding the sigma configurations. Subsequent computations have revealed some minor errors, which do not change the qualitative results but modify some of the numbers presented her

Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity

We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator in loop quantum gravity (LQG). These structures encode geometrical information of the embedding of arbitrary valence vertices of a graph in 3-dimensional Riemannian space, and can be represented by sign strings containing relative orientations of embedded edges. We demonstrate that these signature factors are a special representation of the general mathematical concept of an oriented matroid. Moreover, we show that oriented matroids can also be used to describe the topology (connectedness) of directed graphs. Hence the mathematical methods developed for oriented matroids can be applied to the difficult combinatorics of embedded graphs underlying the construction of LQG. As a first application we revisit the analysis of [4-5], and find that enumeration of all possible sign configurations used there is equivalent to enumerating all realizable oriented matroids of rank 3, and thus can be greatly simplified. We find that for 7-valent vertices having no coplanar triples of edge tangents, the smallest non-zero eigenvalue of the volume spectrum does not grow as one increases the maximum spin \jmax at the vertex, for any orientation of the edge tangents. This indicates that, in contrast to the area operator, considering large \jmax does not necessarily imply large volume eigenvalues. In addition we give an outlook to possible starting points for rewriting the combinatorics of LQG in terms of oriented matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos corrected, presentation slightly extende

New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory

We quantise the new connection formulation of D+1 dimensional General Relativity developed in our companion papers by Loop Quantum Gravity (LQG) methods. It turns out that all the tools prepared for LQG straightforwardly generalise to the new connection formulation in higher dimensions. The only new challenge is the simplicity constraint. While its "diagonal" components acting at edges of spin network functions are easily solved, its "off-diagonal" components acting at vertices are non trivial and require a more elaborate treatment.Comment: 36 pages. v2: Journal version. Discussion on simplicity constraints extended. Conclusion and outlook extended. Minor clarification

Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity

We perform a canonical, reduced phase space quantisation of General Relativity by Loop Quantum Gravity (LQG) methods. The explicit construction of the reduced phase space is made possible by the combination of 1. the Brown -- Kuchar mechanism in the presence of pressure free dust fields which allows to deparametrise the theory and 2. Rovelli's relational formalism in the extended version developed by Dittrich to construct the algebra of gauge invariant observables. Since the resulting algebra of observables is very simple, one can quantise it using the methods of LQG. Basically, the kinematical Hilbert space of non reduced LQG now becomes a physical Hilbert space and the kinematical results of LQG such as discreteness of spectra of geometrical operators now have physical meaning. The constraints have disappeared, however, the dynamics of the observables is driven by a physical Hamiltonian which is related to the Hamiltonian of the standard model (without dust) and which we quantise in this paper.Comment: 31 pages, no figure

On (Cosmological) Singularity Avoidance in Loop Quantum Gravity

Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of the phase space of classical General Relativity to spatially homogeneous situations which is then quantized by the methods of LQG. Thus, LQC is a quantum mechanical toy model (finite number of degrees of freedom) for LQG(a genuine QFT with an infinite number of degrees of freedom) which provides important consistency checks. However, it is a non trivial question whether the predictions of LQC are robust after switching on the inhomogeneous fluctuations present in full LQG. Two of the most spectacular findings of LQC are that 1. the inverse scale factor is bounded from above on zero volume eigenstates which hints at the avoidance of the local curvature singularity and 2. that the Quantum Einstein Equations are non -- singular which hints at the avoidance of the global initial singularity. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in full LQG, if curvature singularity avoidance is realized, then not in this simple way. In fact, it turns out that the boundedness of the inverse scale factor is neither necessary nor sufficient for curvature singularity avoidance and that non -- singular evolution equations are neither necessary nor sufficient for initial singularity avoidance because none of these criteria are formulated in terms of observable quantities.After outlining what would be required, we present the results of a calculation for LQG which could be a first indication that our criteria at least for curvature singularity avoidance are satisfied in LQG.Comment: 34 pages, 16 figure

Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity

Degenerate geometrical configurations in quantum gravity are important to understand if the fate of classical singularities is to be revealed. However, not all degenerate configurations arise on an equal footing, and one must take into account dynamical aspects when interpreting results: While there are many degenerate spatial metrics, not all of them are approached along the dynamical evolution of general relativity or a candidate theory for quantum gravity. For loop quantum gravity, relevant properties and steps in an analysis are summarized and evaluated critically with the currently available information, also elucidating the role of degrees of freedom captured in the sector provided by loop quantum cosmology. This allows an outlook on how singularity removal might be analyzed in a general setting and also in the full theory. The general mechanism of loop quantum cosmology will be shown to be insensitive to recently observed unbounded behavior of inverse volume in the full theory. Moreover, significant features of this unboundedness are not a consequence of inhomogeneities but of non-Abelian effects which can also be included in homogeneous models.Comment: 28 pages, 1 figure; v2: extended discussion of singularity removal and summar

The Holst Spin Foam Model via Cubulations

Spin foam models are an attempt for a covariant, or path integral formulation of canonical loop quantum gravity. The construction of such models usually rely on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional "Regge-like" constraints arising from simplicial triangulations, and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a "cubulation" of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: first, the structure of its amplitudes differs from the standard spin foam models. Second, the tetrad n-point functions admit a "Wick-like" structure. Third, the restriction to simple representations does not automatically occur -- unless one makes use of the time gauge, just as in the classical theory.Comment: 25 pages, 1 figure; v3: published version. arXiv admin note: substantial text overlap with arXiv:0911.213