This series consists of talks in the area of Quantum Gravity.
We will give a short overview of non-perturbative quantum gravity
models and discuss some key common problems for these models. In
particular we will analyze what background independence requires from
a theory of quantum gravity.
In causal set quantum gravity, spacetime is assumed to have a fundamental atomicity or discreteness, and is replaced by a locally finite poset, the causal set. After giving a brief review of causal sets, I will discuss two distinct approaches to constructing a quantum dynamics for causal sets. In the first approach one borrows heavily from the continuum to construct a partition function for causal sets.
The concept of renormalization lies at the heart of fundamental physics. I introduce in this talk the Connes-Kreimer algebraic approach for expressing renormalizability in quantum field theories as well as the extension of these notions to the manifestly non-local framework of noncommutative quantum field theories. Finally, I will present an attempt to further generalize these concepts to quantum gravity models. based on: 0909.5631 [gr-qc], Class. Quant. Grav. (in press)
"Extended" topological field theory generalizes ordinary TQFT to include spacetimes with boundary. Starting from a gauge theory of flat G-connections, and its boundary restriction, I will describe a plan for constructing an extended topological field theory, for any compact Lie group G. This is based on work-in-progress with Jeffrey Morton.
It has recently uncovered that the intertwiner space for LQG carries a natural representation of the U(N) unitary group. I will describe this U(N) action in details and show how it can be used to compute the LQG black hole entropy, to define coherent intertwiner states and to reformulate the LQG dynamics in new terms.
15 years ago Ishibashi, Kawai and collaborators developed non-critical string field theory, starting with the formalism of dynamical triangulations. The same construction can be repeated using causal dynamical triangulations, and in this case one can actually sum explicitly over all genera. The theory can be viewed as stochastic quantization of space, proper (world sheet) time playing the role of stochastic time.
Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. We review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantisation and, more specifically, with the Master constraint quantisation for first class constraints.
The asymptotic formula for the Ponzano-Regge model amplitude is given for non-tardis triangulations of handlebodies in the limit of large boundary spins. The formula produces a sum over all possible immersions of the boundary triangulation in three dimensional Euclidean space weighted by the cosine of the Regge action evaluated on these immersions. Furthermore the asymptotic scaling registers the existence of flexible immersions.
The scaling analysis in the large spin limit of Feynman amplitudes for the Bosonic colored group field theory are considered in any dimension starting with dimension 4. By an explicit integration of two colors, we show that the model is positive. This formulation could be useful for the constructive analysis of this type of models.
Spin foam models aim at defining non-perturbative and background independent amplitudes for quantum gravity. In this work, I argue that the dynamics and the geometric properties of spin foam models can be nicely studied using recursion relations. In 3d gravity and in the 4d Ooguri model, the topological invariance leads to recursion relations for the amplitudes. I also derive recursions from the action of holonomy operators on spin network functionals.