Dual variables and a connection picture for the Euclidean Barrett-Crane model

The partition function of the SO(4)- or Spin(4)-symmetric Euclidean Barrett-Crane model can be understood as a sum over all quantized geometries of a given triangulation of a four-manifold. In the original formulation, the variables of the model are balanced representations of SO(4) which describe the quantized areas of the triangles. We present an exact duality transformation for the full quantum theory and reformulate the model in terms of new variables which can be understood as variables conjugate to the quantized areas. The new variables are pairs of S^3-values associated to the tetrahedra. These S^3-variables parameterize the hyperplanes spanned by the tetrahedra (locally embedded in R^4), and the fact that there is a pair of variables for each tetrahedron can be viewed as a consequence of an SO(4)-valued parallel transport along the edges dual to the tetrahedra. We reconstruct the parallel transport of which only the action of SO(4) on S^3 is physically relevant and rewrite the Barrett-Crane model as an SO(4) lattice BF-theory living on the 2-complex dual to the triangulation subject to suitable constraints whose form we derive at the quantum level. Our reformulation of the Barrett-Crane model in terms of continuous variables is suitable for the application of various analytical and numerical techniques familiar from Statistical Mechanics.Comment: 33 pages, LaTeX, combined PiCTeX/postscript figures, v2: note added, TeX error correcte

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

Properties of Quantum Hall Skyrmions from Anomalies

It is well known that the Fractional Quantum Hall Effect (FQHE) may be effectively represented by a Chern-Simons theory. In order to incorporate QH Skyrmions, we couple this theory to the topological spin current, and include the Hopf term. The cancellation of anomalies for chiral edge states, and the proviso that Skyrmions may be created and destroyed at the edge, fixes the coefficients of these new terms. Consequently, the charge and the spin of the Skyrmion are uniquely determined. For those two quantities we find the values $e\nu N_{Sky}$ and $\nu N_{Sky}/2$, respectively, where $e$ is electron charge, $\nu$ is the filling fraction and $N_{Sky}$ is the Skyrmion winding number. We also add terms to the action so that the classical spin fluctuations in the bulk satisfy the standard equations of a ferromagnet, with spin waves that propagate with the classical drift velocity of the electron.Comment: 8 pages, LaTeX file; Some remarks are included to clarify the physical results obtained, and the role of the Landau-Lifshitz equation is emphasized. Some references adde

Reality conditions inducing transforms for quantum gauge field theory and quantum gravity

For various theories, in particular gauge field theories, the algebraic form of the Hamiltonian simplifies considerably if one writes it in terms of certain complex variables. Also general relativity when written in the new canonical variables introduced by Ashtekar belongs to that category, the Hamiltonian being replaced by the so-called scalar (or Wheeler-DeWitt) constraint. In order to ensure that one is dealing with the correct physical theory one has to impose certain reality conditions on the classical phase space which generally are algebraically quite complicated and render the task of finding an appropriate inner product into a difficult one. This article shows, for a general theory, that if we prescribe first a {\em canonical} complexification and second a $^*$ representation of the canonical commutation relations in which the real connection is diagonal, then there is only one choice of a holomorphic representation which incorporates the correct reality conditions {\em and} keeps the Hamiltonian (constraint) algebraically simple ! We derive a canonical algorithm to obtain this holomorphic representation and in particular explicitly compute it for quantum gravity in terms of a {\em Wick rotation transform}.Comment: Latex, 23 page

Liquefaction Risk Mitigation — Manchester Airport

Densification of loose sandy soil by Vibroflotation was designed and constructed to mitigate the risk of seismically-induced liquefaction for the proposed 15,000 square meter terminal building. The analyses of the geotechnical data and the design of the densification based upon specified parameters is reported. Field installation methods and post-densification results are discussed

The Hidden Quantum Groups Symmetry of Super-renormalizable Gravity

In this paper we consider the relation between the super-renormalizable theories of quantum gravity (SRQG) studied in [arXiv:1110.5249v2, arXiv:1202.0008] and an underlying non-commutativity of spacetime. For one particular super-renormalizable theory we show that at linear level (quadratic in the Lagrangian) the propagator of the theory is the same we obtain starting from a theory of gravity endowed with {\theta}-Poincar\'e quantum groups of symmetry. Such a theory is over the so called {\theta}-Minkowski non-commuative spacetime. We shed new light on this link and show that among the theories considered in [arXiv:1110.5249v2, arXiv:1202.0008], there exist only one non-local and Lorentz invariant super-renormalizable theory of quantum gravity that can be described in terms of a quantum group symmetry structure. We also emphasize contact with pre-existent works in the literature and discuss preservation of the equivalence principle in our framework.Comment: 10 page

AQFT from n-functorial QFT

There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to "extended" functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra "of observables", the latter uses n-functors which assign to each patch a "propagator of states". In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor ("parallel surface transport") naturally yields a local net. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schroedinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo-Riemannian structure and higher dimensions.Comment: 39 pages; further examples added: Hopf spin chains and asymptotic inclusion of subfactors; references adde

Quantum Interaction ϕ44\phi^4_4: the Construction of Quantum Field defined as a Bilinear Form

We construct the solution $\phi(t,{\bf x})$ of the quantum wave equation $\Box\phi + m^2\phi + \lambda:\!\!\phi^3\!\!: = 0$ as a bilinear form which can be expanded over Wick polynomials of the free $in$-field, and where $:\!\phi^3(t,{\bf x})\!:$ is defined as the normal ordered product with respect to the free $in$-field. The constructed solution is correctly defined as a bilinear form on $D_{\theta}\times D_{\theta}$, where $D_{\theta}$ is a dense linear subspace in the Fock space of the free $in$-field. On $D_{\theta}\times D_{\theta}$ the diagonal Wick symbol of this bilinear form satisfies the nonlinear classical wave equation.Comment: 32 pages, LaTe