General Relativity & Quantum Cosmology

In 3d quantum gravity, Planck's constant, the Planck length and the cosmological constant control the lack of (co)-commutativity of quantities like angular momenta, momenta and postion coordinates. I will explain this statement, using the quantum groups which arise in the 3d quantum gravity but avoiding technical details. The non-commutative structures in 3d quantum gravity are quite different from those in the deformed version of special relativity desribed by the kappa-Poincare group, but can be related to the latter by an operation called semi-dualisation. I will explain this operation, and make comments on its possible physical significance. The talk is based on joint work with Shahn Majid.

I will introduce Kitaev's suface codes as a block quantum error-correcting code. Recovery procedures will be described in the case of imperfect syndrome measurements. More might be covered if time permits.

ABSTRACT: The asymptotic freedom conjecture for gravitation is explored in which strong renormalization effects (as in QCD) may occur at astrophysical distance scales larger than the solar system. Nonperturbative renormalization group trajectories exhibiting such an infrared fixed point describe a theory of gravitation with a running gravitational coupling which grows at large distance. The concept that this extra gravity may provide the answer to the missing mass inherent in the dark matter paradigm is a natural suggestion. We provide the alternative of Modified Gravity to answer the problem of galaxy rotation curves from the smallest dwarf galaxies to the largest giant galaxies and to galaxy clusters including the Bullet Cluster. The theory may also explain the apparent anomalous deceleration of the Pioneer 10 and 11 space probes, within solar system constraints.

The entanglement entropy between quantum fields inside and outside a black hole horizon is a promising candidate for the microscopic origin of black hole entropy. I will explain the motivation behind this interpretation of black hole entropy, and why it requires quantum gravity. I will then apply these ideas to loop quantum gravity and show how to compute the entanglement entropy of spin network states. The result of this calculation agrees asymptotically with results from the isolated horizon framework, and I will give the reason for this agreement. Finally, I will show that the entanglement entropy gives extensive corrections to the area law, suggesting corrections to the gravitational action.

I will discuss recent results in Supersymmetric Large Extra Dimensions (SLED), a scenario which shows promise towards solving both the hierarchy and the cosmological constant problems. One of the issues which arises in this programme is a direct result of the need to use codimension-2 branes, which can only consistently couple to gravity through a tension term in the action. This precludes us from asking certain interesting questions, such as what will happen when a phase transition occurs on the brane. In this talk, I will describe how these codim-2 branes can be modelled as codim-1, thus enabling us to work with more interesting brane actions.

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.