Quantum Gravity

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.

Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. We review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantisation and, more specifically, with the Master constraint quantisation for first class constraints. For first class constraints with non trivial structure functions the equivalence can only be established by passing to Abelian(ised) constraints which is always possible locally in phase space. With the above general considerations, we derive concretely the path integral formulations for GR from the canonical theory. We also show that there in principle exists a spin-foam model consistent with the canonical theory of GR.

The asymptotic formula for the Ponzano-Regge model amplitude is given for non-tardis triangulations of handlebodies in the limit of large boundary spins. The formula produces a sum over all possible immersions of the boundary triangulation in three dimensional Euclidean space weighted by the cosine of the Regge action evaluated on these immersions. Furthermore the asymptotic scaling registers the existence of flexible immersions.

The scaling analysis in the large spin limit of Feynman amplitudes for the Bosonic colored group field theory are considered in any dimension starting with dimension 4. By an explicit integration of two colors, we show that the model is positive. This formulation could be useful for the constructive analysis of this type of models.

Spin foam models aim at defining non-perturbative and background independent amplitudes for quantum gravity. In this work, I argue that the dynamics and the geometric properties of spin foam models can be nicely studied using recursion relations. In 3d gravity and in the 4d Ooguri model, the topological invariance leads to recursion relations for the amplitudes. I also derive recursions from the action of holonomy operators on spin network functionals. Their geometric content is discussed in terms of elementary moves on simplices, and is related to the classical constraints of the underlying theories. Interestingly, our recurrence relations apply to any SU(2) invariant symbol. Another method is considered for non-topological objects, and applied to the 10j-symbol which defines the Barrett-Crane spin foam model.

After implementing an effective minimal length, we will present a new class of spacetimes, describing both neutral and charged black holes. As a result, we will improve the conventional Schwarzschild and Reisner-Nordstroem spacetimes, smearing out their singularities at the origin. On the thermodynamic side, we will show how the new black holes admit a maximum temperature, followed by the ``SCRAM phase'', a thermodynamic stable shut down, characterized by a positive black hole heat capacity. As a consequence, also for the neutral solution, in place of the runaway behavior of the temperature, one finds that the evaporation ends up with a zero temperature extremal black hole, i.e. a final configuration entirely governed by the minimal length. For the charged case, both the Hawking and Schwinger pair creation will be discussed in this new scenario. We will also analyze the above solutions in the presence of extra dimensions and the connections with the production of mini black holes, which is foreseen in the extreme energy hadron collisions at the LHC in the next few months. Finally we will discuss further developments and possible connections with other approaches in this field.

The relation between loop quantum gravity (LQG) and ordinary quantum field theory (QFT) on a fixed background spacetime still bears many obstacles. When looking at LQG and ordinary QFT from a mathematical perspective it turns out that the two frameworks are rather different: Although LQG is a true continuum theory its Hilbert space is defined in terms of certain embedded graphs which are labeled by irreducible representations of SU(2). The natural arena for ordinary QFT, on the other hand, is a Fock space which strongly uses the metric properties of the underlying continuum spacetime. In this talk I will review this issue and show how one can use Born--Oppenheimer methods to further progress towards an understanding of (matter) quantum field theories from first principles.

How sure are you that spacetime is continuous? One approach to quantum gravity, causal set theory, models spacetime as a discrete structure: a causal set. This talk begins with a brief introduction to causal sets, then describes a new approach to modelling a quantum scalar field on a causal set. We obtain the Feynman propagator for the field by a novel procedure starting with the Pauli-Jordan commutation function. The candidate Feynman propagator is shown to agree with the continuum result. This model opens the door to physical predictions for scalar matter on a causal set.

We review some recent results on tachyon nonperturbative solutions of the nonlocal, lowest-level, effective action of string field theory. It is shown how nonlocality is encoded in a spacetime diffusion equation and how the latter emerges from the symmteries of the full, background-independent theory.

The idea behind an intersection between loop quantum gravity and noncommutative geometry is to combine elements of unification with a setup of canonical quantum gravity. In my talk I will first review the construction of a semi-finite spectral triple build over an algebra of holonomy loops. Here, the loop algebra is a noncommutative algebra of functions over a configurations space of connections, and the interaction between the Dirac type operator and the loop algebra captures information of the kinematical part of canonical quantum gravity. Next, I will show how certain normalizable, semi-classical states are build which connects the spectral triple construction to the Dirac Hamiltonian in 3+1 dimensions. Thus, these states can be interpreted as one-particle fermion states in an ambient gravitational field. This analysis indicates that the spectral triple construction involves matter degrees of freedom.

I will discuss the contribution to black hole thermodynamics from a variation in the cosmological constant. The description of black hole with a cosmological constant is facilitated by introducing a two-form potential for the static Killing field. The resulting Smarr formula then includes a term proportional to the cosmological constant times an effective volume, which arises as the difference between the Killing potential on the horizon and the boundary at infinity. This volume is shown to be equal to the difference between the (infinite) volume of AdS and the (infinite) volume outside the black hole horizon of AdS containing a black hole--and so can be interpreted as the volume occupied by the black hole. I will outline the derivation for the first law for AdS black holes including a variation in the cosmological constant. This yields a new work term, the change in the cosmological constant times the effective volume. Hence this is analogous to a "volume times change in pressure" work term in classical thermodynamics. This suggests that the usual change in mass term is better interpreted as a change in the enthalpy, the mass plus the pressure times the volume. In the AdS/CFT correspondence a change in the cosmological constant corresponds to a change in the t'Hooft coupling, for example, a change in the number of degrees of freedom. The effective volume multiplier then looks like a chemical potential. Members of the audience will be asked to contribute their own interpretations at this point.