Quantum Gravity

In the sixties, Roger Penrose came up with a radical new idea for a quantum geometry which would be entirely background independent, combinatorial, discrete (countable number of degrees of freedom), and involve only integers and fractions, not complex or real numbers. The basic structures are spin-networks. One reason we might believe that space or space-time might be discrete is that current physique tells us that matter is discrete and that matter and geometry are related through gravity. Once a discrete theory is decided on, it seems awkward that the dynamics would retain "continuous elements" in the form of real numbers (used for the probabilities for example). The great achievement of Penrose's theory is that there is a well defined procedure which gives the semi-classical limit geometry (always of the same dimension) without any input on topology (the fundamental theory does not contain a manifold).

In the late 80s it became clear through notable work by Witten and others that there is a deep connection between (2+1) gravity and Chern-Simons theory making it possible to quantize. In the case where the cosmological constant is negative, spacetime has a boundary and classically there are black holes. We will study the features of this theory as an arena for seeing the simple realization of the Holographic Principle and the duality between quantum gravity theories in asymptotically Anti-deSitter space and conformal field theories. Extensions and open problems with the theory will also be discussed.

It is an open question why gravity is so much weaker than the other three interactions we know. One possible answer which has been suggested is that this mismatch is only apparently so, and a feature we observe on large distances. The strength of gravity on small distances could grow faster than an extrapolation of Newton\'s law would imply, such that it becomes comparable to the other interactions at distances that will be testable in the soon future. The concrete scenario for this is that our world could have additional compactified extra dimensions. If that was the case, quantum gravitational effects could become observable at the Large Hadron Collider. The most prominent features in these models are the production of mini black holes, and graviton emission.

In the first part of the talk, a brief introduction to general relativity and quantum theory is given. Their independent successes are discussed, as well as the desire and difficulty in merging them, to obtain a unique language to describe the universe. Then I focus on Loop quantum gravity, a particular approach towards this objective, in which a discrete microscopic structure of spacetime is envisaged.

An ingredient in recent discussions of curvature singularity avoidance in quantum gravity is the "inverse scale factor" operator and its generalizations. I describe a general lattice origin of this idea, and show how it applies to the Coulomb singularity in quantum mechanics, and more generally to lattice formulations of quantum gravity. The example also demonstrates that a lattice discretized Schrodinger or Wheeler-DeWitt equation is computationally equivalent to the so called "polymer" quantization derived from loop quantum gravity.

This talk discusses Einstein\'s special and general theories of relativity in a simple manner. Some simple mathematical arguments are included. This leads to a brief description of Loop Quantum Gravity - one of the most popular way of trying to describe gravity using the ideas of quantum mechanics. In addition, the Standard model of particle physics, which describes all known particles and their (non-gravitational) interactions is explained. We conclude with a description of some recent research which may bring the Standard Model and Loop Quantum Gravity together into a single theory.

We introduce a framework that allows to calculate cosmological perturbations in a gauge invariant manner to any order. The two main features of this framework are to take physical observables as basic objects and to treat the variables describing the background geometry as fully dynamical. Backreaction effects can therefore naturally adressed. At the end I will mention applications to Loop Quantum Cosmology.

With a cosmic flight simulator, we'll take a scenic journey through space and time. After exploring our local Galactic neighborhood, we'll travel back 13.7 billion years to explore the Big Bang itself and how state-of-the-art measurements are transforming our understanding of our cosmic origin and ultimate fate. We then turn to the question of whether this can all be described purely mathematically, and discuss implications ranging from standard physics topics like symmetries, irreducible representations, units, free parameters and initial conditions to broader issues like parallel universes, simulations and and Goedel incompleteness.

Hints from quantum gravity suggest that a preferred frame may actually exist. One way to accommodate such a frame in general relativity without sacrificing diffeomorphism invariance is to couple the metric to a dynamical, time like, unit-norm vector field--the "aether". I will discuss properties and observational tests of a class of such theories, including post-Newtonian effects and radiation from binary pulsar systems.