Quantum Gravity

In this talk I revisit the canonical framework for general relativity in its connection-frame field formulation, exploiting its local holographic nature. I will show how we can understand  the  Gauss law, the Bianchi identity and the space diffeomorphism constraints as conservation laws for local surface charges. These charges being respectively the electric flux, the dual magnetic flux  and momentum charges. Quantization of the surface charge algebra can be done in terms of Kac-Moody edge modes. This leads to an enhanced theory upgrading spin networks to tube networks carrying Virasoro representations. Taking a finite dimensional truncation of this quantization yields states of quantum geometry,  dubbed `Poincaré charge networks’, which carry a representation of the 3D diffeomorphism boundary charges on top of the SU(2) fluxes and gauge transformations. This opens the possibility to have for the first time a framework where spatial diffeomorphism are represented at the quantum level. Moreover, our construction leads naturally to the picture that the relevant geometrical degrees of freedom live on boundaries, that their dynamics and the fabric of quantum space itself is encoded into their entanglement, and it is designed to offer a new setting to study the coarse-graining of gravity both at the classical and the quantum levels.

Gauge theories possess nonlocal features that, in the presence of boundaries, inevitably lead to subtleties. In particular their fundamental degrees of freedom are not point-like. This leads to a non-trivial cutting (C) and sewing (S) problem: 
(C) Which gauge invariant degrees of freedom are associated to a region with boundaries? 
(S) Do the gauge invariant degrees of freedom in two complementary regions R and R’ unambiguously comprise *all* the gauge-invariant degrees of freedom in M = R ∪ R’ ? Or, do new “boundary degrees of freedom” need to be introduced at the interface S = R ∩ R’ ?
In this talk, I will address and answer these questions in the context of Yang-Mills theory. The analysis is carried out at the level of the symplectic structure of the theory, i.e. for linear perturbations over arbitrary backgrounds. I will also discuss how the ensuing results translate into a quasilocal derivation of the superselection of the electric flux through the boundary of a region, and into a novel gluing formula which constructively proves that no ambiguity exists in the gluing of regional gauge-fixed configurations.
Time allowing I will also address how the formalism generalizes the “Dirac dressing” of charged matter fields, and how, in the presence of matter, quasi-local “global” charges (as opposed to gauge charges) emerge at special (i.e. reducible) configurations.

This talk is based on arXiv:1910.04222, with H. Gomes (U. of Cambridge, UK).
See also arXiv:1808.02074, with H. Gomes and F. Hopfmüller (Perimeter)

Quantizing 4D geometries leads to discrete area spectra. Such discrete area spectra are also suggested by the holographic principle and entropy counting for black holes.

 Starting with this input of a discrete area spectrum I will construct a path integral for quantum gravity and discuss (quantum) corrections to the GR dynamics that are forced by the discrete area spectra. The resulting model can serve as effective model for the spin foam approach and clarifies  the dynamical principles and underlying key assumptions for spin foams. The considerations also point towards key phenomenological differences to e.g. the ADM quantization scheme, and thus to a way to falsify the key assumption of discrete area spectra. 

In order to solve the problem of time in quantum gravity, various approaches to a relational quantum dynamics have been proposed. In this talk, I will exploit quantum reduction maps to illustrate a previously unknown equivalence between three of the well-known ones: (1) relational observables in the clock-neutral picture of Dirac quantization, (2) Page and Wootters’ (PW) Schrödinger picture formalism, and (3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. To establish the equivalence, a quantization procedure for relational Dirac observables is developed using covariant POVMs which encompass non-ideal clocks. The quantum reduction maps reveal this procedure as the quantum analog of the classical method of gauge-invariantly extending gauge-fixed quantities. The quantum reduction maps also allow one to extend a recent ‘clock-neutral’ approach to changing temporal reference frames, transforming relational observables and states between different clock choices, and demonstrate a clock dependent temporal nonlocality effect. Using the trinity, I will discuss how Kuchar's three fundamental criticisms against the PW formalism, namely that its conditional probabilities would (i) yield the wrong localization probabilities for relativistic particles, (ii) violate the constraints, and (iii) produce incorrect transition probabilities, can be resolved. Given the trinity, these resolutions also apply to approaches (1) and (3) and corroborate the PW formalism, if done correctly, as a viable approach to the problem of time. Time permitting, I will explain, however, why the slogan `time from entanglement' in the PW formalism is misleading.

In this talk, we present a new outlook on canonical quantum gravity and its coupling to matter.

We will show how this fresh perspective combines the critical elements of holography, loop quantum gravity, and relative locality. I will first focus on the consequences of cutting open a portion of space and show that new symmetry charges and new degrees of freedom reveal themselves. 
I will explain the nature of this boundary symmetry algebra in metric gravity and then first-order gravity. We will see that a rich structure appears that explains from the continuum perspective the non-commutativity of geometric flux, the quantization of the area spectra, the nature of the simplicity constraints but also reveals the dual momentum observables and finally allow to reconcile the elements of canonical gravity with Lorentz invariance.
I will discuss the issue of quantization as a challenge of finding a representation of the boundary algebra and will give clues about where we are in this process.

In this talk I will discuss some recent results on boundary degrees of freedom (or edge modes), and their description via an extended phase space structure containing extra boundary fields. Motivated by a slight modification of the covariant phase space formalism, I will show how the use of a boundary Lagrangian enables to include the edge modes in the phase space and to obtain their boundary dynamics. This will be illustrated on the example of Maxwell theory, where in addition the edge modes can be understood as contributing to entanglement entropy. The possible extension of this result to gravity will then be mentioned.

Our earlier findings indicate the violation of the 'volume simplicity' constraint in the current Spinfoam models (EPRL-FK-KKL). This result, and related problems in LQG, promted to revisit the metric/vielbein degrees of freedom in the classical Einstein-Cartan gravity. Notably, I address in detail what constitutes a 'geometry' and its 'group of motions' in such Poincare gauge theory. In a differential geometric scheme that I put forward the local translations are not broken but exact, and their relation to diffeomorphism transformations is clarified. The refined notion of a tensor takes into account the (relative) localization in spacetime, whereas the key concept of 'development' generalizes parallel transport (of vectors and points) to affine spaces. I advocate for this Cartan connection as the fundamental d.o.f. of the gravitational field, and discuss implications for discretization and quantization. Based on [1905.06931].

In this talk, I will describe the framework of large D matrix models, which provides new limits for matrix models where the sum over planar graphs simplifies when D is large. The basic degrees of freedom are a set of D real matrices of size NxN which is invariant under O(D). These matrices can be naturally interpreted as a real tensor of rank three, making a compelling connection with tensor models. Furthermore, they have a natural interpretation in terms of D-brane constructions in string theory. I will present a way to define a large D scaling of the coupling constants such that the sum over Feynman graphs of fixed genus in matrix models admits a well-defined large D expansion. In particular, in the large D limit, the sum over planar graphs truncates to a tractable, yet non-trivial, sum over generalized melonic graphs. This family of graphs has been shown to display very interesting properties, especially in the case of quantum mechanical models such as the SYK model and SYK-like tensor models. If time allows, I will also explain how one can use the large D limit of matrix models to simplify the sum over all genera, which is notoriously divergent. 

According to the Asymptotic Safety conjecture, a (non-perturbatively)
renormalizable quantum field theory of gravity could be constructed
based on the existence of a non-trivial fixed point of the
renormalization group flow.
The existence of this fixed point can be established, e.g., via the
non-perturbative methods of the functional renormalization group (FRG).
In practice, the use of the FRG methods requires to work within
truncations of the gravitational action, and higher-derivative operators
naturally lead to the presence of several poles in the propagator. The
question is whether these poles represent a real problem for the
unitarity of the theory.

Using QED as a working example, in this talk I will discuss some aspects
of non-perturbative unitarity in Quantum Field Theory. I will show that
the inclusion of quantum effects at all scales is of crucial importance
to assess unitarity of field theories. In particular, poles appearing in
truncations of the action could correspond to fake degrees of freedom of
the theory. Possible conditions to determine, within truncations,
whether a pole represents a fake or a genuine degree of freedom of the
theory will also be discussed.

Hopf algebra lattice models are related to certain topological quantum field theories and give rise to topological invariants of oriented surfaces. Examples are the combinatorial quantisation of Chern-Simons theory and the Kitaev model.
Our main result is the construction of a mapping class group action on these models, formulated under weaker assumptions than the associated topological quantum field theories, namely for pivotal Hopf algebras in symmetric monoidal categories. This description also yields new structures for semisimple finite-dimensional Hopf algebras. The action of the mapping class group is defined in terms of simple graph transformations and allows one to compute quantum Dehn twists explicitly.