The Volume Operator plays a crucial role in the definition of the quantum dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulas for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which in turn are given in terms of Racah's formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far not even numerically the spectrum could be accessed.In this article we demonstrate that by means of the Elliot -Biedenharn identity one can get rid of all the 6j symbols for any valence of the gauge invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge invariant 4 -vertex.The techniques derived in this paper could be of use also for the analysis of spin -spin interaction Hamiltonians of many -particle problems in atomic and nuclear physics. * jbrunnemann@perimeterinstitute.ca † tthiemann@perimeterinstitute.ca 1

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) the quantum Einstein equations are nonsingular which hints at the avoidance of the global initial singularity. This rests on (1) a key technique developed for LQG and (2) the fact that there are no inhomogeneous excitations. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is 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 the 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.

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) the quantum Einstein equations are nonsingular which hints at the avoidance of the global initial singularity. This rests on (1) a key technique developed for LQG and (2) the fact that there are no inhomogeneous excitations. We display the result of a calculation for LQG which proves that the (analogon of the) inverse scale factor, while densely defined, is 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 the 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.

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...

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. We find that the occurrence of a smallest non-zero eigenvalue is dependent upon the geometry of the underlying graph, and is not a property of the Volume operator itself. The present analysis benefits from the general simplified formula for matrix elements of the Volume operator derived in [24], 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 [25], providing details of the analysis presented there. e I ,e J ,e K ∈E(γ)The decoration '!' in ' ! =' simply indicates that the equality is required to hold. 5 This can always be achieved by subdividing and redirecting the edges of a graph γ, see [25].Using furthermore the antisymmetry of ǫ ijk and the fact that [J i I , J j J ] = 0 whenever I = J we can restrict the summation in (2.20) to I < J < K if we simultaneously write a factor 3! in front of the sum. The result is:We then make use of the identitywhere (J IJ ) 2 = 6 As mentioned before, one can without loss of generality always redirect edges such that there are only outgoing edges at each vertex. 7 The latter regularization [15], which we have presented here, differs by a numerical factor of 27 8 from the former [16]. The former exactly reproduces the value of Creg obtained in [22]. 10 6j-symbols are invariant under the interchange of 2 columns and the simultaneous flip of two columns, see appendix C.3. 23We can then bring (4.18) into its final formWe will subsequently discuss the properties of first the set (1), (2), (3) and secondly (4) as contained in (4.21) in two separate subsections. 0 < Y I J

The relation between standard Loop Quantum Cosmology and full Loop Quantum Gravity fails already at the first nontrivial step: The configuration space of Loop Quantum Cosmology can not be embedded into the configuration space of full Loop Quantum Gravity due to a topological obstruction. We investigate this obstruction in detail, because many topological obstructions are the source of physical effects. For this we derive the topology of a large class of subspaces of the Loop Quantum Gravity configuration space. This allows us to find the extension of the standard Loop Quantum Cosmology configuration space that admits an embedding in agreement with [1]. We then construct the embedding for flat FRW Loop Quantum Cosmology and find that it coincides asymptotically with standard LQC.

Within the DYNCAP project, the Modelica library ClaRaCCS is being developed. This library will provide a framework to model both steam power plants and carbon capture units in an integrated manner. The current status of the library is presented. The structure of the library and the general model design is outlined. Its user-friendly handling as well as its high flexibility in the modelling of individual complex scenarios are demonstrated by the concrete modelling of a furnace. The scenario of a closed steam cycle coupled to a carbon capture cycle based on an amine gas treatment is described and simulation results are briefly discussed.

We describe preliminary results of a detailed numerical analysis of the volume operator as formulated by Ashtekar and Lewandowski [2]. Due to a simplified explicit expression for its matrix elements[3], it is possible for the first time to treat generic vertices of valence greater than four. It is found that the vertex geometry characterizes the volume spectrum.