Spacetime Covariant Form of Ashtekar's Constraints

The Lagrangian formulation of classical field theories and in particular general relativity leads to a coordinate-free, fully covariant analysis of these constrained systems. This paper applies multisymplectic techniques to obtain the analysis of Palatini and self-dual gravity theories as constrained systems, which have been studied so far in the Hamiltonian formalism. The constraint equations are derived while paying attention to boundary terms, and the Hamiltonian constraint turns out to be linear in the multimomenta. The equivalence with Ashtekar's formalism is also established. The whole constraint analysis, however, remains covariant in that the multimomentum map is evaluated on {\it any} spacelike hypersurface. This study is motivated by the non-perturbative quantization program of general relativity.Comment: 22 pages, plain Tex, no figures, accepted for publication in Nuovo Cimento

Graviton propagator in loop quantum gravity

We compute some components of the graviton propagator in loop quantum gravity, using the spinfoam formalism, up to some second order terms in the expansion parameter.Comment: 41 pages, 6 figure

Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory

The present paper studies the large-j asymptotics of the Lorentzian EPRL spinfoam amplitude on a 4d simplicial complex with an arbitrary number of simplices. The asymptotics of the spinfoam amplitude is determined by the critical configurations. Here we show that, given a critical configuration in general, there exists a partition of the simplicial complex into three type of regions R_{Nondeg}, R_{Deg-A}, R_{Deg-B}, where the three regions are simplicial sub-complexes with boundaries. The critical configuration implies different types of geometries in different types of regions, i.e. (1) the critical configuration restricted into R_{Nondeg}$implies a nondegenerate discrete Lorentzian geometry, (2) the critical configuration restricted into R_{Deg-A}$ is degenerate of type-A in our definition of degeneracy, but implies a nondegenerate discrete Euclidean geometry on R_{Deg-A}, (3) the critical configuration restricted into R_{Deg-B} is degenerate of type-B, and implies a vector geometry on R_{Deg-B}. With the critical configuration, we further make a subdivision of the regions R_{Nondeg} and R_{Deg-A} into sub-complexes (with boundary) according to their Lorentzian/Euclidean oriented 4-simplex volume V_4(v), such that sgn(V_4(v)) is a constant sign on each sub-complex. Then in the each sub-complex, the spinfoam amplitude at the critical configuration gives the Regge action in Lorentzian or Euclidean signature respectively on R_{Nondeg} or R_{Deg-A}. The Regge action reproduced here contains a sign factor sgn(V_4(v)) of the oriented 4-simplex volume. Therefore the Regge action reproduced here can be viewed a discretized Palatini action with on-shell connection. Finally the asymptotic formula of the spinfoam amplitude is given by a sum of the amplitudes evaluated at all possible critical configurations, which are the products of the amplitudes associated to different type of geometries.Comment: 54 pages, 2 figures, reference adde

Laplacians on discrete and quantum geometries

We extend discrete calculus for arbitrary ($p$-form) fields on embedded lattices to abstract discrete geometries based on combinatorial complexes. We then provide a general definition of discrete Laplacian using both the primal cellular complex and its combinatorial dual. The precise implementation of geometric volume factors is not unique and, comparing the definition with a circumcentric and a barycentric dual, we argue that the latter is, in general, more appropriate because it induces a Laplacian with more desirable properties. We give the expression of the discrete Laplacian in several different sets of geometric variables, suitable for computations in different quantum gravity formalisms. Furthermore, we investigate the possibility of transforming from position to momentum space for scalar fields, thus setting the stage for the calculation of heat kernel and spectral dimension in discrete quantum geometries.Comment: 43 pages, 2 multiple figures. v2: discussion improved, references added, minor typos correcte

Coordinated spatial and temporal expression of Hox genes during embryogenesis in the acoel Convolutriloba longifissura

Background: Hox genes are critical for patterning the bilaterian anterior-posterior axis. The evolution of their clustered genomic arrangement and ancestral function has been debated since their discovery. As acoels appear to represent the sister group to the remaining Bilateria (Nephrozoa), investigating Hox gene expression will provide an insight into the ancestral features of the Hox genes in metazoan evolution. Results: We describe the expression of anterior, central and posterior class Hox genes and the ParaHox ortholog Cdx in the acoel Convolutriloba longifissura. Expression of all three Hox genes begins contemporaneously after gastrulation and then resolves into staggered domains along the anterior-posterior axis, suggesting that the spatial coordination of Hox gene expression was present in the bilaterian ancestor. After early surface ectodermal expression, the anterior and central class genes are expressed in small domains of putative neural precursor cells co-expressing ClSoxB1, suggesting an evolutionary early function of Hox genes in patterning parts of the nervous system. In contrast, the expression of the posterior Hox gene is found in all three germ layers in a much broader posterior region of the embryo. Conclusion: Our results suggest that the ancestral set of Hox genes was involved in the anteriorposterior patterning of the nervous system of the last common bilaterian ancestor and were later co-opted for patterning in diverse tissues in the bilaterian radiation. The lack of temporal colinearity of Hox expression in acoels may be due to a loss of genomic clustering in this clade or, alternatively, temporal colinearity may have arisen in conjunction with the expansion of the Hox cluster in the Nephrozoa