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Semiclassical analysis of the Loop Quantum Gravity volume operator: I. Flux Coherent States
, 2008
"... The volume operator plays a pivotal role for the quantum dynamics of Loop Quantum Gravity (LQG), both in the full theory and in truncated models adapted to cosmological situations coined Loop Quantum Cosmology (LQC). It is therefore crucial to check whether its semiclassical limit coincides with the ..."
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The volume operator plays a pivotal role for the quantum dynamics of Loop Quantum Gravity (LQG), both in the full theory and in truncated models adapted to cosmological situations coined Loop Quantum Cosmology (LQC). It is therefore crucial to check whether its semiclassical limit coincides with the classical volume operator plus quantum corrections. In the present article we investigate this question by generalizing and employing previously defined coherent states for LQG which derive from a cylindrically consistently defined complexifier operator which is the quantization of a known classical function. These coherent states are not normalizable due to the non separability of the LQG Hilbert space but they define uniquely define cut – off states depending on a finite graph. The result of our analysis is that the expectation value of the volume operator with respect to coherent states depending on a graph with only n−valent verticies reproduces its classical value at the phase space point at which the coherent state is peaked only if n = 6. In other words, the semiclassical sector of LQG defined by those states is described by graphs with cubic topology! This has some bearing on current
Spin Foams and Canonical Quantization
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2012
"... This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the threedimensional context, where the two approaches are in good agreement, we show how the canonical quantization à la Witten of Riemanni ..."
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Cited by 7 (0 self)
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This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the threedimensional context, where the two approaches are in good agreement, we show how the canonical quantization à la Witten of Riemannian gravity with a positive cosmological constant is related to the Turaev–Viro spin foam model, and how the Ponzano–Regge amplitudes are related to the physical scalar product of Riemannian loop quantum gravity without cosmological constant. In the fourdimensional case, we recall a Lorentzcovariant formulation of loop quantum gravity using projected spin networks, compare it with the new spin foam models, and identify interesting relations and their pitfalls. Finally, we discuss the properties which a spin foam model is expected to possess in order to be consistent with the canonical quantization, and suggest a new model illustrating these results.
The Holst Spin Foam Model via Cubulations
, 2008
"... Spin Foam Models (SFM) are an attempt at a covariant or path integral formulation of canonical Loop Quantum Gravity (LQG). Traditionally, SFM rely on 1. the Plebanski formulation of GR as a constrained BF Theory. 2. simplicial triangulations as a UV regulator and 3. a sum over all triangulations via ..."
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Cited by 5 (2 self)
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Spin Foam Models (SFM) are an attempt at a covariant or path integral formulation of canonical Loop Quantum Gravity (LQG). Traditionally, SFM rely on 1. the Plebanski formulation of GR as a constrained BF Theory. 2. simplicial triangulations as a UV regulator and 3. a sum over all triangulations via group field techniques (GFT) in order to get rid off triangulation dependence. Subtle tasks for current SFM are to establish 1. the correct quantum implementation of Plebanski’s constraints. 2. the existence of a semiclassical sector implementing additional Regge constraints arising from simplicial triangulations and 3. the physical inner product of LQG via GFT. We propose a new approach which deals with these issues as follows: 1. The simplicity constraints are correctly implemented by starting directly from the Holst action which is also a proper starting point for canonical LQG. 2. Cubulations are chosen rather than triangulations as a regulator. 3. We give a direct interpretation of our spin foam model as a generating functional of 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, it transpires that 1. this model’s amplitudes differ from those of current SFM, 2. tetrad npoint functions reveal
Null twisted geometries
 Phys.Rev. D89 (2014) 084070 [1311.3279
"... We define and investigate a quantization of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrization of the theory in terms of twistors, which has already proved useful in discussing the interpretation of spin networks as the quantizatio ..."
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We define and investigate a quantization of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrization of the theory in terms of twistors, which has already proved useful in discussing the interpretation of spin networks as the quantization of twisted geometries. The classical formalism can be extended in a natural way to null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra with spacelike faces, and SU(2) by the little group ISO(2). The main difference is that the simplicity constraints present in the formalism are all first class, and the symplectic reduction selects only the helicity subgroup of the little group. As a consequence, information on the shapes of the polyhedra is lost, and the result is a much simpler, Abelian geometric picture. It can be described by a Euclidean singular structure on the twodimensional spacelike surface defined by a foliation of spacetime by null hypersurfaces. This geometric structure is naturally decomposed into a conformal metric and scale factors, forming locally conjugate pairs. Proper actionangle variables on the gaugeinvariant phase space are described by the eigenvectors of the Laplacian of the dual graph. We also identify the variables of the phase space amenable to characterize the extrinsic geometry of the foliation. Finally, we quantize the phase space and its algebra using Dirac’s algorithm, obtaining a notion of spin networks for null hypersurfaces. Such spin networks are labeled by SO(2) quantum numbers, and are embedded nontrivially in the unitary, infinitedimensional irreducible representations of the Lorentz group. 1
Contents
, 2006
"... The proof planning systems available today are sequential systems. The hypothesis of this project is that the engineering of a proof planning system that is capable of dynamic, distributed, parallel proof planning will empower the paradigm of proof planning by unlocking its latent potential for thes ..."
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The proof planning systems available today are sequential systems. The hypothesis of this project is that the engineering of a proof planning system that is capable of dynamic, distributed, parallel proof planning will empower the paradigm of proof planning by unlocking its latent potential for these features. The system will be built on IsaPlanner, which is a proof planning system developed by Lucas Dixon (Dixon, 2005), based on the theorem prover, Isabelle. Alice, is an extension of Standard ML, with additional features for supporting concurrent, distributed programming. This is the choice for the implementation language for the purpose of this project. The methodology for the evaluation and the choice of test cases will also be outlined.