General Relativity & Quantum Cosmology

An ultraviolet complete quantum gravity theory is formulated in which vertex functions in Feynman graphs are entire functions and the propagating graviton is described by a local, causal propagator. A scalar-tensor action describes classical gravity theory. The cosmological constant problem is investigated in the context of the ultraviolet complete quantum gravity. Also investigated are black holes and cosmology.

The Exact Renormalization Group (ERG) is a technique which can be fruitfully applied to systems with local interactions that exhibit a large number of degrees of freedom per correlation length. In the first part of the talk I will give a very general overview of the ERG, focussing on its applications in quantum field theory (QFT) and critical phenomena. In the second part I will discuss how a particular extension of the formalism suggests a new understanding of correlation functions in QFTs, in general, and gauge theories in particular.

Guided by idealized but soluble nonrenormalizable models, a nontraditional proposal for the quantization of covariant scalar field theories is advanced, which achieves a term-by-term, divergence-free perturbation analysis of interacting models expanded about a suitable pseudofree theory [differing from a free theory by an $O(\hbar^2)$ term]. This procedure not only provides acceptable solutions for models for which no acceptable solution currently exists, e.g., $\varphi^4_n$, for spacetime dimension $n\ge4$, but offers a new, divergence-free solution, for less-singular models as well, e.g., $\varphi^4_n$, for $n=2,3$.

Over the last decade there has been strong interest in the theory and phenomenology of particle propagation in quantum spacetime. The main results concern possible Planck-scale modifications of the "dispersion" relation between energy and momentum of a particle. I review results establishing that these modifications can be tested using observations of gamma rays from sources at cosmological distances. And I report recent progress in the understanding of the implications of spacetime expansion for such studies. I also discuss recent preliminary results suggesting that the same Planck-scale modifications of the dispersion relation might have an unexpected role in gravitational collapse.

In my talk I would like to discuss the present status of Doubly Special Relativity. DSR is an extension of Special Relativity aimed at describing kinematics of particles and fields in the regime where (quantum) gravity effects might become relevant. I will discuss an interplay between DSR physics and mathematics of Hopf algebras.

A brief introduction to the notorious "cosmological constant problem" is given. Then, a particular approach is discussed, which has been developed by Volovik and the present speaker over the last years and which goes under the name of q-theory. This approach provides a possible solution of the main cosmological constant problem, why is |Lambda|^(1/4) negligible compared to the energy scales of the electroweak standard model (not to mention the Planck energy)? The next problem is, of course, the small but nonzero value of the actual cosmological constant, responsible for the observed "accelerating universe." This problem can also be addressed in the framework of q-theory. In fact, the observed value Lambda ~ (meV)^4 may correspond to the remnant vacuum energy density of dynamical processes taking place at a cosmic age set by the mass scale M ~ E_ew of ultramassive particles with electroweak interactions. A first estimate of the required value of the energy scale E_ew ranges from 3 to 9 TeV. If correct, this estimate implies the existence of new TeV-scale physics beyond the standard model.

Combining the principles of general relativity and quantum theory still remains as elusive as ever. Recent work, that concentrated on one of the points of contact (and conflict) between quantum theory and general relativity, suggests a new perspective on gravity. It appears that the gravitational dynamics in a wide class of theories - including, but not limited to, standard Einstein's theory - can be given a purely thermodynamic interpretation. In this approach gravity appears as an emergent phenomenon, like e.g., gas or fluid dynamics. I will describe the necessary background, key results and their implications as suggested by my recent work in this area.

Attempts to go beyond the framework of local quantum field theory include scenarios in which the action of external symmetries on the quantum fields Hilbert space is deformed. A common feature of these models is that the quantum group symmetry of their Hilbert spaces induces additional structure in the multiparticle states which in turns reflects a non-trivial momentum-dependent statistics. In certain particular models which might be relevant for quantum gravity the richer structure of the deformed Fock space allows for the possibility of entanglement between the field modes and certain ''planckian'' degrees of freedom invisible to an observer that cannot probe the Planck scale.

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. This construction reveals a number of insights into spin foams including the nature of the continuum limit.

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.