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

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

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

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

``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

Quantum geometry with intrinsic local causality

The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact two-dimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group G_q over all compact (finite genus) oriented 2-surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism.Comment: Latex 33 pages, 7 Figure, epsfi

A candidate for a background independent formulation of M theory

A class of background independent membrane field theories are studied, and several properties are discovered which suggest that they may play a role in a background independent form of M theory. The bulk kinematics of these theories are described in terms of the conformal blocks of an algebra G on all oriented, finite genus, two-surfaces. The bulk dynamics is described in terms of causal histories in which time evolution is specified by giving amplitudes to certain local changes of the states. Holographic observables are defined which live in finite dimensional states spaces associated with boundaries in spacetime. We show here that the natural observables in these boundary state spaces are, when G is chosen to be Spin(D) or a supersymmetric extension of it, generalizations of matrix model coordinates in D dimensions. In certain cases the bulk dynamics can be chosen so the matrix model dynamics is recoverd for the boundary observables. The bosonic and supersymmetric cases in D=3 and D=9 are studied, and it is shown that the latter is, in a certain limit, related to the matrix model formulation of M theory. This correspondence gives rise to a conjecture concerning a background independent form of M theory in terms of which excitations of the background independent membrane field theory that correspond to strings and D0 branes are identified.Comment: Latex 46 pages, 21 figures, new results included which lead to a modification of the statement of the basic conjecture. Presentation improve

Loop Quantum Gravity

The problem of finding the quantum theory of the gravitational field, and thus understanding what is quantum spacetime, is still open. One of the most active of the current approaches is loop quantum gravity. Loop quantum gravity is a mathematically well-defined, non-perturbative and background independent quantization of general relativity, with its conventional matter couplings. The research in loop quantum gravity forms today a vast area, ranging from mathematical foundations to physical applications. Among the most significative results obtained are: (i) The computation of the physical spectra of geometrical quantities such as area and volume; which yields quantitative predictions on Planck-scale physics. (ii) A derivation of the Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale discreteness. This discreteness emerges naturally from the quantum theory and provides a mathematically well-defined realization of Wheeler's intuition of a spacetime ``foam''. Long standing open problems within the approach (lack of a scalar product, overcompleteness of the loop basis, implementation of reality conditions) have been fully solved. The weak part of the approach is the treatment of the dynamics: at present there exist several proposals, which are intensely debated. Here, I provide a general overview of ideas, techniques, results and open problems of this candidate theory of quantum gravity, and a guide to the relevant literature.Comment: Review paper written for the electronic journal `Living Reviews'. 34 page