This series consists of talks in the area of Quantum Gravity.
The space of convex polyhedra can be given a dynamical structure. Exploiting this dynamics we have performed a Bohr-Sommerfeld quantization of the volume of a tetrahedral grain of space, which is in excellent agreement with loop gravity. Here we present investigations of the volume of a 5-faced convex polyhedron. We give for the first time a constructive method for finding these polyhedra given their face areas and normals to the faces and find an explicit formula for the volume. This results
In spacetime physics any set C of events—a causal set—is taken to be partially ordered by the relation £ of possible causation: for p, q Î C, p £ q means that q is in p’s future light cone. Fotini Markopoulou has proposed that the causal structure of spacetime itself be represented by “sets evolving over C” —that is, in essence, by the topos Set C of presheaves on Cop.
Recent progress in the quantization of nonrenormalizable scalar fields has found that a suitable non-classical modification of the ground state wave function leads to a result that eliminates term-by-term divergences that arise in a conventional perturbation analysis. After a brief review of both the scalar field story and the affine quantum gravity program, examination of the procedures used in the latter surprisingly shows an analogous formulation which already implies that affine quantum gravity is not plagued by divergences that arise in a standard perturbation study.
We propose a new method of unifying gravity and the Yang-Mills fields by introducing a spin-foam model. We realize a unification between an SU(2) Yang-Mills interaction and 3D general relativity by considering a constrained Spin(4) ~SO(4) Plebanski action.
I will review the construction of lattice theories which maintain one or more exact supersymmetries for non zero lattice spacing concentrating in particular on the case of N=4 super Yang-Mills. Such lattice theories may be studied using Monte Carlo techniques borrowed from lattice QCD and can be used to explore issues in holography. In three dimensions the same constructions can be used to formulate a topological theory of gravity which we argue is equivalent to Witten's Chern Simons theory.
We introduce an exactly solvable model to test various proposals for the imposition of the spin foam simplicity constraints. This model is a three-dimensional Holst-Plebanski action for the gauge group SO(4), in which the simplicity constraints mimic the situation of the four-dimensional theory. In particular, the canonical analysis reveals the presence of secondary second class constraints conjugated to the primary ones.
During the last couple of years Dupuis, Freidel, Livine, Speziale and Tambornino developed a twistorial formulation for loop quantum gravity.
Constructed from Ashtekar--Barbero variables, the formalism is restricted to SU(2) gauge transformations.
In this talk, I perform the generalisation to the full Lorentzian case, that is the group SL(2,C).
The emergence of fractal features in the microscopic structure of space-time is a common theme in many approaches to quantum gravity. In particular the spectral dimension, which measures the return probability of a fictitious diffusion process on space-time, provides a valuable probe which is easily accessible both in the continuum functional renormalization group and discrete Monte Carlo simulations of the gravitational action.
We relate the discrete classical phase space of loop gravity to the continuous phase space of general relativity. Our construction shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. We resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise-flat geometry, showing that both geometries belong to the same equivalence class. We also establish a clear relationship between Regge geometries and the piecewise-flat spin foam geometries.