This series consists of talks in the area of Quantum Gravity.
Loop quantum gravity and spin foams are two closely related theories of quantum gravity. There is an expectation that the sum over histories or path integral formulation of LQG will take the form of a spin foam, although a rigorous connection between the two is available only in 2+1 gravity. Understanding the relation between them will resolve many open questions of both theories. We probe the connection through an exactly soluble model of loop quantum cosmology. Beginning from the canonical theory we construct a spin foam like expansion of LQC.
I will describe a very special (infinite-parameter) family of gravity theories that all describe, exactly like General Relativity, just two propagating degrees of freedom. The theories are obtained by generalizing Plebanski's self-dual (chiral) formulation of GR. I will argue that this class of gravity theories provides a potentially powerful new framework for testing the asymptotic safety conjecture in quantum gravity.
A quantum theory of gravity implies a quantum theory of geometries. To
this end we will introduce different phases spaces and choices for the
space of discretized geometries. These are derived through a canonical
analysis of simplicity constraints - which are central for spin foam
models - and gluing constraints. We will discuss implications for
spin foam models and map out how to obtain a path integral
quantization starting from a canonical quantization.
Diffeomorphism symmetry is the underlying symmetry of general
relativity and deeply intertwined with its dynamics. The notion of
diffeomorphism symmetry is however obscured in discrete gravity, which
underlies most of the current quantum gravity models. We will propose
a notion of diffeomorphism symmetry in discrete models and find that
such a symmetry is weakly broken in many models. This is connected to
the problem of finding a consistent canonical dynamics for discrete
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.