Quantum Gravity

The collection of all Dirac operators for a given fermion space defines its space of geometries. 
Formally integrating over this space of geometries can be used to define a path integral and a thus a theory of quantum gravity. 
In general this expression is complicated, however for fuzzy spaces a simple expression for the general form of the Dirac operator exists. This simple expression allows us to explore the space of geometries using Markov Chain Monte Carlo methods and thus examine the path integral in a manner similar to that used in CDT. 
In doing this we find a phase transition and indications that the geometries close to this phase transition might be manifold like.

In this talk I will sketch the relation between unitary representations of the BMS3 group and three-dimensional, asymptotically flat gravity. More precisely, after giving an exact definition of the BMS group in three dimensions, I will argue that its unitary representations are classified by orbits of CFT stress tensors under conformal transformations. These stress tensors, in turn, can be interpreted as Bondi mass aspects for asymptotically flat metrics. I will also show how one can compute characters of the BMS3 group, which coincide with suitable gravitational one-loop partition functions

I will present a generalization of the spinor approach of Euclidean loop quantum gravity to the 3D Lorentzian case, where the gauge group is the noncompact SU(1,1). The key tool of this generalization is the recoupling theory between unitary infinite-dimensional representations and non-unitary finite-dimensional ones, needed to generalize the Wigner-Eckart theorem to tensor operators for SU(1,1).
SU(1,1) tensor operators are used to build observables and a quantum Hamiltonian constraint, analogous to the one introduced by Bonzom and Livine in the Euclidean case. I will show that the Lorentzian Ponzano-Regge amplitude is a solution of the Hamiltonian constraint, by making use of the Biedenharn-Elliott relations (generalized to the case where unitary and non-unitary SU(1,1) representations are coupled to each other).

In quantum gravity, observables must be diffeomorphism-invariant. Such observables are nonlocal, in contrast with the standard assumption of locality in flat spacetime quantum field theory. I will show how to construct 'gravitationally dressed' observables in linearized gravity that become local in the weak gravity limit, and whose corrections to exact locality are characterized by the Newtonian potential. One can attempt to make these observables more local by concentrating gravitational field lines into a smaller solid angle. In AdS_3 gravity I will show that nonperturbatively there are sharp limits to how much the gravitational dressing can be concentrated.

Based on arXiv:1507.07921 with Steve Giddings, and arXiv:1510.00672 with Don Marolf and Eric Mintun.

In this talk, I will review the Refined Gribov-Zwanziger framework designed to deal with the so-called Gribov copies in Yang-Mills theories and its standard BRST soft breaking. I will show that, within this scenario, the BRST transformations are modified in the non-perturbative regime in order to be a symmetry of the model. This fact has been supported by recent lattice simulations and opens a new avenue for the investigation of non-perturbative effects in Yang-Mills theories.

In the summer of 2015, the speaker led a team at Microsoft Research, comprised primarily of research interns from seven universities, to demonstrate a social, ambulatory, Mixed Reality system for the first time.  Each intern developed a preliminary, domain-specific exploration of how such a system could be used.  One of the interns, Andrzej Banburski of the Perimeter Institute, demonstrated an interface to Mathematica.  The very concept and terminology of Virtual Reality had been first articulated by the speaker almost four decades earlier precisely in the hopes of experiencing this type of interface.  Mathematical expressions floating in the air is not just more vivid that writing them on a blackboard; a different workflow and mental framework becomes possible.  For instance, functions appear as lenses that can be compounded as they are lined up.  One can peer through a stack of such lenses at different data or starting conditions.  Instead of erasing and replacing portions of expressions on a blackboard, a stack of lenses can fork and merge, so that a range of variations can be explored at once as aspects of a sculptural form.   The purpose is not just to make math more accessible to those who might find this interface more inviting, but to free math from pre-computational, notation-bound conventions.  The speaker will argue that this type of interface, while still only barely explored, could eventually have a significant impact both on computer science and other disciplines, including theoretical physics. 

We analyse different classical formulations of General Relativity in the Batalin (Frad-
kin) Vilkovisky framework with boundary, as a first step in the program of CMR [1] quantisation. Success and failure in satisfying the axioms will allow us to discriminate among the different descriptions, suggesting that some are more suitable than others in view of perturbative quantisation. Based on a joint work with A. Cattaneo [2, 3] we will present the details of the application of the BV-BFV formalism to the Einstein-Hilbert and Palatini-Holst formulations of General Relativity. We show that the two descriptions are no longer equivalent from this point of view, and we discuss possible interpretations of this result.

[1] A. S. Cattaneo, P. Mnëv, N. Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332, 2, 535-603 (2014); A.S. Cattaneo, P. Mnëv, N. Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, arXiv:1507.01221
[2] A. S. Cattaneo, M. Schiavina, BV-BFV analysis of General Relativity. Part I: Einstein Hilbert action, arXiv:1509.05762 (2015).
[3] A. S. Cattaneo, M. Schiavina, BV-BFV analysis of General Relativity. Part II: Palatini Holst action, in preparation; A. S. Cattaneo, M. Schiavina, On time, in preparation.

The Hartle-Hawking (HH) no-boundary proposal provides a Euclidean path integral prescription for a measure on the space of all possible initial conditions. Apart from  saddle point  and  minisuper-space calculations, it is hard to obtain results using the unregulated path integral. A promising choice of spacetime  regularisation comes from the causal set (CST) approach to quantum gravity.  Using analytic results as well as Markov Chain Monte Carlo and numerical integration methods we obtain  the HH wave function in a theory of non-perturbative 2d CST. We find that the  wave function is sharply peaked with the peak geometry changing discretely with "temperature". In the low temperature regime the peak corresponds to causal sets which have no continuum counterpart but exhibit physically interesting behaviour. They show a rapid spatial expansion with respect to the discrete proper time as well as  a high degree of spatial homogeneity due to extensive overlap of the causal past. While our results are limited to 2 dimensions they provide a concrete example of how quantum gravity could explain the initial conditions for our observable universe.

The kinematical framework of canonical loop quantum gravity has mostly been studied in the context of compact Cauchy slices. However many key physical notions such as total energy and momentum require the use of asymptotically flat boundary conditions (and hence non-compact slices). We present a quantum kinematics, based on the Koslowski-Sahlmann representation, that successfully incorporates such asymptotically flat boundary conditions. Based on joint work with Madhavan Varadarajan.

I will describe a proposal for a generalization of the BMS group in which the conformal isometries of the sphere (Lorentz group) are replaced by arbitrary sphere diffeomorphisms. I describe the computation of canonical charges and show that the associated Ward identities are equivalent to the Cachazo-Strominger subleading soft graviton formula. Based on joint work with Alok Laddha.

An algebraic view allows one to compute properties of physical states without having to introduce an explicit Hilbert space. In relational systems without absolute time, a state cannot be completely positive once a choice for a (possibly local) internal time has been made. A more general class of `almost-positive' states is introduced in this talk, together with a discussion of their constrained dynamics.

We evaluate the one-graviton loop contribution to the vacuum polarization on de Sitter background in a 1-parameter family of exact, de Sitter invariant gauges. Our result is computed using dimensional regularization and fully renormalized with BPHZ counterterms, which must include a noninvariant owing to the time-ordered interactions.

Because the graviton propagator engenders a physical breaking of de Sitter invariance two structure functions are needed to express the result. In addition to its relevance for the gauge issue this is the first time a covariant gauge graviton propagator has been used to compute a noncoincident loop.