Spin foam models and the Wheeler-DeWitt equation for the quantum 4-simplex

The asymptotics of some spin foam amplitudes for a quantum 4-simplex is known to display rapid oscillations whose frequency is the Regge action. In this note, we reformulate this result through a difference equation, asymptotically satisfied by these models, and whose semi-classical solutions are precisely the sine and the cosine of the Regge action. This equation is then interpreted as coming from the canonical quantization of a simple constraint in Regge calculus. This suggests to lift and generalize this constraint to the phase space of loop quantum gravity parametrized by twisted geometries. The result is a reformulation of the flat model for topological BF theory from the Hamiltonian perspective. The Wheeler-de-Witt equation in the spin network basis gives difference equations which are exactly recursion relations on the 15j-symbol. Moreover, the semi-classical limit is investigated using coherent states, and produces the expected results. It mimics the classical constraint with quantized areas, and for Regge geometries it reduces to the semi-classical equation which has been introduced in the beginning.Comment: 16 pages, the new title is that of the published version (initial title: A taste of Hamiltonian constraint in spin foam models

Asymptotics of 10j symbols

The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a `degenerate spin network', where the rotation group SO(4) is replaced by the Euclidean group of isometries of R^3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones.Comment: 25 pages LaTeX with 8 encapsulated Postscript figures. v2 has various clarifications and better page breaks. v3 is the final version, to appear in Classical and Quantum Gravity, and has a few minor corrections and additional reference

Gauge symmetries in spinfoam gravity: the case for "cellular quantization"

The spinfoam approach to quantum gravity rests on a "quantization" of BF theory using 2-complexes and group representations. We explain why, in dimension three and higher, this "spinfoam quantization" must be amended to be made consistent with the gauge symmetries of discrete BF theory. We discuss a suitable generalization, called "cellular quantization", which (1) is finite, (2) produces a topological invariant, (3) matches with the properties of the continuum BF theory, (4) corresponds to its loop quantization. These results significantly clarify the foundations - and limitations - of the spinfoam formalism, and open the path to understanding, in a discrete setting, the symmetry-breaking which reduces BF theory to gravity.Comment: 6 page

Higher Poincare Lemma and Integrability

We prove the non-abelian Poincare lemma in higher gauge theory in two different ways. The first method uses a result by Jacobowitz which states solvability conditions for differential equations of a certain type. The second method extends a proof by Voronov and yields the explicit gauge parameters connecting a flat local connective structure to the trivial one. Finally, we show how higher flatness appears as a necessary integrability condition of a linear system which featured in recently developed twistor descriptions of higher gauge theories.Comment: 1+21 pages, presentation streamlined, section on integrability for higher linear systems significantly improved, published versio

Direct estimation of electron density in the Orion Bar PDR from mm-wave carbon recombination lines

A significant fraction of the molecular gas in star-forming regions is irradiated by stellar UV photons. In these environments, the electron density (n_e) plays a critical role in the gas dynamics, chemistry, and collisional excitation of certain molecules. We determine n_e in the prototypical strongly irradiated photodissociation region (PDR), the Orion Bar, from the detection of new millimeter-wave carbon recombination lines (mmCRLs) and existing far-IR [13CII] hyperfine line observations. We detect 12 mmCRLs (including alpha, beta, and gamma transitions) observed with the IRAM 30m telescope, at ~25'' angular resolution, toward the H/H2 dissociation front (DF) of the Bar. We also present a mmCRL emission cut across the PDR. These lines trace the C+/C/CO gas transition layer. As the much lower frequency carbon radio recombination lines, mmCRLs arise from neutral PDR gas and not from ionized gas in the adjacent HII region. This is readily seen from their narrow line profiles (dv=2.6+/-0.4 km/s) and line peak LSR velocities (v_LSR=+10.7+/-0.2 km/s). Optically thin [13CII] hyperfine lines and molecular lines - emitted close to the DF by trace species such as reactive ions CO+ and HOC+ - show the same line profiles. We use non-LTE excitation models of [13CII] and mmCRLs and derive n_e = 60-100 cm^-3 and T_e = 500-600 K toward the DF. The inferred electron densities are high, up to an order of magnitude higher than previously thought. They provide a lower limit to the gas thermal pressure at the PDR edge without using molecular tracers. We obtain P_th > (2-4)x10^8 cm^-3 K assuming that the electron abundance is equal or lower than the gas-phase elemental abundance of carbon. Such elevated thermal pressures leave little room for magnetic pressure support and agree with a scenario in which the PDR photoevaporates.Comment: Accepted for publication in A&A Letters (includes language editor corrections

A Lorentzian Signature Model for Quantum General Relativity

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value.Comment: 22 pages, latex, amsfonts, Xypic. Version 3: improved presentation. Version 2 is a major revision with explicit formulae included for the evaluation of relativistic spin networks and the computation of examples which have finite value

Spin foam model from canonical quantization

We suggest a modification of the Barrett-Crane spin foam model of 4-dimensional Lorentzian general relativity motivated by the canonical quantization. The starting point is Lorentz covariant loop quantum gravity. Its kinematical Hilbert space is found as a space of the so-called projected spin networks. These spin networks are identified with the boundary states of a spin foam model and provide a generalization of the unique Barrette-Crane intertwiner. We propose a way to modify the Barrett-Crane quantization procedure to arrive at this generalization: the B field (bi-vectors) should be promoted not to generators of the gauge algebra, but to their certain projection. The modification is also justified by the canonical analysis of Plebanski formulation. Finally, we compare our construction with other proposals to modify the Barret-Crane model.Comment: 26 pages; presentation improved, important changes concerning the closure constraint and the vertex amplitude; minor correctio