Quantum Gravity

In this talk (based on arXiv:1001.0354) we give a quantum statistical interpretation for the Kauffmann bracket polynomial state sum <K> for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting C(K) denote the Hilbert space for this model, there is a natural unitary transformation U from C(K) to itself such that <K> = <F|U|F> where |F> is a sum over basis states for C(K). The quantum algorithm arises directly from this formula via the Hadamard Test. We then show that the framework for our quantum model for the bracket polynomial is a natural setting for Khovanov homology. The Hilbert space C(K) of our model has basis in one-to-one correspondence with the enhanced states of the bracket state summmation and is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. We show that for the Khovanov boundary operator d defined on C(K) we have the relationship dU + Ud = 0. Consequently, the unitary operator U acts on the Khovanov homology, and we therefore obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. The formula for the Jones polynomial as a graded Euler characteristic is now expressed in terms of the eigenvalues of U and the Euler characteristics of the eigenspaces of U in the homology. The quantum algorithm given here is inefficient, and so it remains an open problem to determine better quantum algorithms that involve both the Jones polynomial and the Khovanov homology.

Einstein's theory of General Relativity has taught us that empty space (or, more precisely, spacetime) is in itself a dynamical and wonderfully rich entity for both theoretical physicists and science fiction authors alike. Although it may stretch our imagination, astrophysical observations leave little doubt that spacetime can bend, move and vibrate. If we want to explain these phenomena from an underlying microscopic and more fundamental structure, we need to bring in quantum theory, leading to even more exotic possibilities such as spacetime foam and wormholes. Do they really exist? How would we know? Are they in conflict with known physics? At least some of these questions may already be within the reach of our fundamental physical theories, not just qualitatively, but also quantitatively. In this talk, Professor Loll will share her insights into how much we know and how much we can still hope to learn about quantum gravity - the elusive quantum theory of space and time.

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 gravity. Finally we will discuss methods to construct models with exact symmetries and elaborate on the connection between diffeomorphism symmetry and triangulation independence.

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. This is illustrated in a 2-d model of causal sets, in which typicality replaces quantum probabilities. The second approach is intrinsic, and uses the quantum measure formulation in which dynamics is described in the language of measure theory and observables are measurable sets in an event algebra. Using the example of complex percolation dynamics I will show that naive attempts to carry out this process for the quantum measure may not work. I will end by discussing possible ways to address this question.

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.

We will review the definitions of spin foam models for quantum gravity and the recent advances in this field, such as the "graviton propagator", the definition of coherent states of geometry and the derivation of non-commutative field theories as describing the effective dynamics of matter coupled to quantum gravity. I will insist on the role of group field theories as providing a non-perturbative definition of spinfoams and their intricate relation with non-commutative geometry and matrix models.

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.