Graphical Evolution of Spin Network States

The evolution of spin network states in loop quantum gravity can be described by introducing a time variable, defined by the surfaces of constant value of an auxiliary scalar field. We regulate the Hamiltonian, generating such an evolution, and evaluate its action both on edges and on vertices of the spin network states. The analytical computations are carried out completely to yield a finite, diffeomorphism invariant result. We use techniques from the recoupling theory of colored graphs with trivalent vertices to evaluate the graphical part of the Hamiltonian action. We show that the action on edges is equivalent to a diffeomorphism transformation, while the action on vertices adds new edges and re-routes the loops through the vertices.Comment: 24 pages, 21 PostScript figures, uses epsfig.sty, Minor corrections in the final formula in the main body of the paper and in the formula for the Tetrahedral net in the Appendi

General Relativity in terms of Dirac Eigenvalues

The eigenvalues of the Dirac operator on a curved spacetime are diffeomorphism-invariant functions of the geometry. They form an infinite set of ``observables'' for general relativity. Recent work of Chamseddine and Connes suggests that they can be taken as variables for an invariant description of the gravitational field's dynamics. We compute the Poisson brackets of these eigenvalues and find them in terms of the energy-momentum of the eigenspinors and the propagator of the linearized Einstein equations. We show that the eigenspinors' energy-momentum is the Jacobian matrix of the change of coordinates from metric to eigenvalues. We also consider a minor modification of the spectral action, which eliminates the disturbing huge cosmological term and derive its equations of motion. These are satisfied if the energy momentum of the trans Planckian eigenspinors scale linearly with the eigenvalue; we argue that this requirement approximates the Einstein equations.Comment: 6 pages, RevTe

The projector on physical states in loop quantum gravity

We construct the operator that projects on the physical states in loop quantum gravity. To this aim, we consider a diffeomorphism invariant functional integral over scalar functions. The construction defines a covariant, Feynman-like, spacetime formalism for quantum gravity and relates this theory to the spin foam models. We also discuss how expectation values of physical quantity can be computed.Comment: 15 pages, 2 figures, substantially revised versio

Black Hole Entropy from Loop Quantum Gravity

We argue that the statistical entropy relevant for the thermal interactions of a black hole with its surroundings is (the logarithm of) the number of quantum microstates of the hole which are distinguishable from the hole's exterior, and which correspond to a given hole's macroscopic configuration. We compute this number explicitly from first principles, for a Schwarzschild black hole, using nonperturbative quantum gravity in the loop representation. We obtain a black hole entropy proportional to the area, as in the Bekenstein-Hawking formula.Comment: 5 pages, latex-revtex, no figure

Degenerate Plebanski Sector and Spin Foam Quantization

We show that the degenerate sector of Spin(4) Plebanski formulation of four-dimensional gravity is exactly solvable and describes covariantly embedded SU(2) BF theory. This fact ensures that its spin foam quantization is given by the SU(2) Crane-Yetter model and allows to test various approaches of imposing the simplicity constraints. Our analysis strongly suggests that restricting representations and intertwiners in the state sum for Spin(4) BF theory is not sufficient to get the correct vertex amplitude. Instead, for a general theory of Plebanski type, we propose a quantization procedure which is by construction equivalent to the canonical path integral quantization and, being applied to our model, reproduces the SU(2) Crane-Yetter state sum. A characteristic feature of this procedure is the use of secondary second class constraints on an equal footing with the primary simplicity constraints, which leads to a new formula for the vertex amplitude.Comment: 34 pages; changes in the abstract and introduction, a few references adde

``Sum over Surfaces'' form of Loop Quantum Gravity

We derive a spacetime formulation of quantum general relativity from (hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the 3-geometry in proper time. We show that the perturbation expansion of this operator is finite and computable order by order. By giving a graphical representation a' la Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) 2d surfaces in 4d. The contribution of one surface to the sum is given by the product of one factor per branching point of the surface. Therefore branching points play the role of elementary vertices of the theory. Their value is determined by the matrix elements of the hamiltonian constraint, which are known. The formulation we obtain can be viewed as a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Ooguri-Crane-Yetter 4d topological field theory, with a few key differences that illuminate the relation between quantum gravity and TQFT. Finally, we suggests that certain new terms should be added to the hamiltonian constraint in order to implement a ``crossing'' symmetry related to 4d diffeomorphism invariance.Comment: Seriously revised version. LaTeX, with revtex and epsfi

Quantization of static space-times

A 4-dimensional Lorentzian static space-time is equivalent to 3-dimensional Euclidean gravity coupled to a massless Klein-field. By canonically quantizing the coupling model in the framework of loop quantum gravity, we obtain a quantum theory which actually describes quantized static space-times. The Kinematical Hilbert space is the product of the Hilbert space of gravity with that of imaginary scalar fields. It turns out that the Hamiltonian constraint of the 2+1 model corresponds to a densely defined operator in the underlying Hilbert space, and hence it is finite without renormalization. As a new point of view, this quantized model might shed some light on a few physical problems concerning quantum gravity.Comment: 14 pages, made a few modifications, added Journal-re

Targeting Inflammatory Mediators in Epilepsy: A Systematic Review of Its Molecular Basis and Clinical Applications

Introduction: Recent studies prompted the identification of neuroinflammation as a potential target for the treatment of epilepsy, particularly drug-resistant epilepsy, and refractory status epilepticus. This work provides a systematic review of the clinical experience with anti-cytokine agents and agents targeting lymphocytes and aims to evaluate their efficacy and safety for the treatment of refractory epilepsy. Moreover, the review analyzes the main therapeutic perspectives in this field. Methods: A systematic review of the literature was conducted on MEDLINE database. Search terminology was constructed using the name of the specific drug (anakinra, canakinumab, tocilizumab, adalimumab, rituximab, and natalizumab) and the terms “status epilepticus,” “epilepsy,” and “seizure.” The review included clinical trials, prospective studies, case series, and reports published in English between January 2016 and August 2021. The number of patients and their age, study design, specific drugs used, dosage, route, and timing of administration, and patients outcomes were extracted. The data were synthesized through quantitative and qualitative analysis. Results: Our search identified 12 articles on anakinra and canakinumab, for a total of 37 patients with epilepsy (86% febrile infection-related epilepsy syndrome), with reduced seizure frequency or seizure arrest in more than 50% of the patients. The search identified nine articles on the use of tocilizumab (16 patients, 75% refractory status epilepticus), with a high response rate. Only one reference on the use of adalimumab in 11 patients with Rasmussen encephalitis showed complete response in 45% of the cases. Eight articles on rituximab employment sowed a reduced seizure burden in 16/26 patients. Finally, one trial concerning natalizumab evidenced a response in 10/32 participants. Conclusion: The experience with anti-cytokine agents and drugs targeting lymphocytes in epilepsy derives mostly from case reports or series. The use of anti-IL-1, anti-IL-6, and anti-CD20 agents in patients with drug-resistant epilepsy and refractory status epilepticus has shown promising results and a good safety profile. The experience with TNF inhibitors is limited to Rasmussen encephalitis. The use of anti-α4-integrin agents did not show significant effects in refractory focal seizures. Concerning research perspectives, there is increasing interest in the potential use of anti-chemokine and anti-HMGB-1 agents

The complete spectrum of the area from recoupling theory in loop quantum gravity

We compute the complete spectrum of the area operator in the loop representation of quantum gravity, using recoupling theory. This result extends previous derivations, which did not include the ``degenerate'' sector, and agrees with the recently computed spectrum of the connection-representation area operator.Comment: typos corrected in eqn.(21). Latex with IOP and epsf styles, 1 figure (eps postscript file), 12 pages. To appear in Class. Quantum Gra

Area spectrum in Lorentz covariant loop gravity

We use the manifestly Lorentz covariant canonical formalism to evaluate eigenvalues of the area operator acting on Wilson lines. To this end we modify the standard definition of the loop states to make it applicable to the present case of non-commutative connections. The area operator is diagonalized by using the usual shift ambiguity in definition of the connection. The eigenvalues are then expressed through quadratic Casimir operators. No dependence on the Immirzi parameter appears.Comment: 12 pages, RevTEX; improved layout, typos corrected, references added; changes in the discussion in sec. IIIB and