Quantum Gravity

By explicit construction, I will show that one can in a simple way introduce and measure gravitational holonomies and Wilson loops in lattice formulations of nonperturbative quantum gravity based on (Causal) Dynamical Triangulations.

I propose a quantum gravity model in which the fundamental degrees of freedom are pure information bits for both discrete space-time points and links connecting them. The Hamiltonian is a very simple network model consisting of a ferromagnetic Ising model for space-time vertices and an antiferromagnetic Ising model for the links. As a result of the frustration arising between these two terms, the ground state self-organizes as a new type of low-clustering graph with finite Hausdorff dimension. The model has three quantum phases: a mean field phase in which the spectral and Hausdorff dimensions coincide and are larger then 4. A fluctuations-dominated phase in which the Hausdorff dimension can only be 4 and the spectral dimension is lower than the Hausdorff dimension and a disordered phase in which there is no space-time interpretation. The large-scale dimension 4 of the universe is related to the upper critical dimension 4 of the Ising model. An ultraviolet fixed point at the lower critical dimension of the Ising model implies the absence of space-time at very small scales. At finite temperatures the universe emerges without big bang and without singularities from a ferromagnetic phase transition in which space-time itself forms out of a hot soup of information bits. When the temperature is lowered the universe unfolds and expands by lowering its connectivity, a mechanism I have called topological expansion. The model admits also macroscopic black hole configurations corresponding to graphs containing holes with no space time inside and around which there are "Schwarzschild-like horizons" with a lower spectral dimension and an entropy proportional to their surface. 

I will discuss a new duality between strongly coupled and weakly coupled condensed matter systems. It can be obtained by combining the gauge-gravity duality with analog gravity. In my talk I will explain how one arrives at the new duality, what it can be good for, and what questions this finding raises.

In an approach to quantum gravity where space-time arises from coarse graining of fundamentally discrete structures, black hole formation and subsequent evaporation could be described by a unitary evolution without the problems encountered by standard remnant scenarios or the schemes where information is assumed to come out with the radiation while semiclassical evaporation (firewalls and complementarity). I point out the possibility that the final state is purified by correlations with the fundamental pre-geometric structures (in the sense of Wheeler) which are available in such approaches, and, like defects in the underlying space-time weave, can carry zero energy.

We present a new description of discrete space-time in 1+1 dimensions in terms of a set of elementary geometrical units that represent its independent classical degrees of freedom. This is achieved by means of a binary encoding that is ergodic in the class of space-time manifolds respecting coordinate invariance of general relativity. Space-time fluctuations can be represented in a classical lattice gas model whose Boltzmann weights are constructed with the discretized form of the Einstein-Hilbert action. Within this framework, it is possible to compute basic quantities such as the Ricci curvature tensor and the Einstein equations, and to evaluate the path integral of discrete gravity. The description as a lattice gas model also provides a novel way of quantization and, at the same time, to quantum simulation of fluctuating space-time.

A renormalization group transformation implements a scale transformation while resetting the UV cut-off, so that the theories before and after the RG transformation contain the same degrees of freedom, but with a modified action. In Wilson's original proposal, the cut-off is reset by integrating out the degrees of freedom between the old and new cut-off. In recent years we have learned that, alternatively, one can decouple these degrees of freedom by means of local unitary transformations (disentanglers). The result is a reversible renormalization group flow, and an efficient description of many-body ground states. In this talk, I will show you pretty drawings and will use fancy names, such as "multi-scale entanglement renormalization ansatz" (MERA). MERA is currently studied as a possible lattice realization of the AdS/CFT correspondence. I will keep it simple.

We study a contracting universe composed of cold dark matter and radiation, and with a positive cosmological constant. Assuming that loop quantum cosmology captures the correct high-curvature dynamics of the space-time, we calculate the spectrum of scalar and tensor perturbations after the bounce, assuming initial quantum vacuum fluctuations. We find that the modes that exit the (sound) Hubble radius during matter-domination when the effective equation of state is slightly negative due to the cosmological constant will be nearly scale-invariant with a slight red tilt, in agreement with observations. The tensor perturbations are also nearly scale-invariant, and the predicted tensor-to-scalar ratio is small.  Finally, as this scenario predicts a positive running of the scalar index, it can be differentiated from inflationary models.

Instead of formulating the state space of a quantum field theory over a single big Hilbert space, it has been proposed by Jerzy Kijowski to describe quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces. I will discuss the physical motivations for this approach and explain how it can be implemented in the context of LQG. While the resulting state space forms a natural extension of the Ashtekar-Lewandowski Hilbert space, it treats position and momentum variables on equal footing. This paves the way for the construction of semi-classical states beyond fixed graph level, and eventually for the derivation of LQC from full LQG.

I offer a personal perspective on the causal dynamical triangulations approach to the construction of quantum theories of gravity. After briefly introducing the approach's formalism and results, I illuminate this perspective with 3+1 vignettes of recent and ongoing research. Specifically, I review attempts to locate an ultraviolet fixed point through renormalization group analyses; I report measurements of the homogeneity of the approach's quantum geometries; I discuss efforts to determine the large scale effective action in the case of 2+1 dimensions; and I suggest a speculative possibility for reviving the Euclidean dynamical triangulations approach.