Discrete Quantum Gravity Group Bahr

Overview

In recent years, Spin Foam Models have emerged as a promising route to defining a theory of quantum gravity.

They intrinsically rely on a discretization of space-time, which is akin to the introduction of a possibly irregular lattice. The lattice itself does not carry any geometric information such as lengths, angles, etc. This defines a notion of scale, which, in this background-independent context, is given by purely combinatorial data, rather than a length- or energy value.

The path integral itself is realized as a state sum model, i.e. a sum over states, consisting of representation-theoretic data to elements in the lattice. Each such state can be interpreted as a discrete 4d metric, giving the lattice a geometry.

The boundary states arising in this formulation of quantum gravity are Penrose's spin networks. The name „Spin Foam“ refers to the fact that histories of spin networks resemble a collection of "soap films" stuck together.

Not much is known about the renormalization of these models, which is one of the major topics of this research group.

The background-independent nature of the theory makes application of usual renormalization group techniques non-straightforward. Since one sums over different metrics, there is no a priori length scale, or any other geometric quantitiy attached to the lattice. Indeed, the path integral over all metrics requires one to sum over all values such a lattice parameter could have.

The methods for computing the RG flow of spin foam models come from tensor network renormalization, which share technical and conceptual similarities. The work of this group is concerned, among others, with the following topics:

List of topics

• Coarse graining: The precise way in which states on coarse and fine lattices should be related is still open. The resulting RG flow depends on the precise choice, so one has to choose one which preserves the geometric and dynamical notions of GR. The properties of the coarse graining maps, as well as the RG flow, are subject to extensive research, not only in this Emmy-Noether group.
• Continuum limit: While it is known that spin foam models on small lattices have discrete GR as asymptotic limit of large quantum numbers, it is not clear yet in what way continuum GR emerges in the (thermodynamic) limit of large lattices. To check this, at least on some symmetry-reduced regimes, is an important research project.
• RG fixed points and diffeomorphism symmetry: Very little is known about the RG flow of Spin Foam models so far, in particular with its interplay with the diffeomorphisms, which form a gauge symmetry group of GR. While it is known that diffeomorphisms are broken in the discrete gravity regime, one has observed in examples that the symmetry is restored at the RG fixed point, in the form of a 'perfect action'. Whether this persists in the full quantum gravity theory or not is a crucial question for the research programme as a whole.
• Although both Riemannian signature and Lorentzian signature models exist, it is not clear how the two are related, if at all. One task is to define a generalized form of Wick-rotation, which works in the background-independent context, possibly on regular lattices.