Quantum Gravity

In describing condensed matter, some well established paradigms have allowed much progress to be made in understanding and using materials.  But in the last 15 - 20 years, new materials, such as heavy fermions, high temperature superconductors, and now charge density wave-supporting materials,  have been shown to require new paradigms in describing them.  While much progress has been achieved in that time, we still do not have  a widely accepted theoretical description of the nature of their electronic excitations.

We  will  briefly review  the  issue  of    "information loss"    during  the Hawking   evaporation of a  black hole,  and  argue that    the  quantum dynamical  reduction  theories, which  have  been  developed to  address the   measurement problem  in quantum mechanics,  possess the  elements to  diffuse the ``paradox”  at the qualitative  and  at   the  quantitative  level, leading to what  seems  to be an overall  coherent  picture.  

Although the inflationary predictions for the primordial power spectrum of    density inhomogeneities   seem  very successful, there is an obscure part in our understanding of the emergence of the seeds of cosmic   structure:  How  does  a universe which at one pint  in time is described by a  state that is fully  homogeneous and isotropic,  evolve into a state that is not,   given that the dynamics  does not  contain any  source  for the  undoing of  such symmetry?   We will  discuss  an   approach that  offers  a resolution  of the  issue  in the context of  proposals to deal  with the measurement  problem in quantum  mechanics. We will  see that the   approach  naturally  resolves the difficulties  raised  by the  lack of  observation of tensor modes.

Path integrals are at the heart of quantum field theory. In spite of their covariance and seeming simplicity, they are hard to define and evaluate. In contrast, functional differentiation, as it is used, for example, in variational problems, is relatively straightforward. This has motivated the development of new techniques that allow one to express functional integration in terms of functional differentiation. In fact, the new techniques allow one to express integrals in general through differentiation. These techniques therefore add to the general toolbox for integration and integral transforms such as the Fourier and Laplace transforms. Here, we review some of these results, we give simpler proofs and we add new results, for example, on expressing the Laplace transform and its inverse in terms of derivatives, results that may be of use in quantum field theory, e.g., in the context of heat traces.

In order to introduce the cosmological constant in a simplicial geometry, constant curvature should be introduced on simplex faces. This yields a compactification of the phase space and the finiteness of the Hilbert for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete, pointing to discreetness of time, in addition to space.

Quantum effects render black holes unstable.  Besides Hawking radiation there is another, genuinely quantum gravitational, source of instability: the Hajicek-Kiefer explosion via tunnelling to a white hole.  A recent result in classical general relativity makes this decay channel plausible: there is an exact external solution of the Einstein equations locally (but not globally) isometric to extended Schwarzschild, which describes an object collapsing into a black hole and then exploding out of a white hole. The tunnelling time can in principle be computed using Loop Quantum Gravity. If it is sufficiently short, present explosions of primordial black hole could be observables, opening a new observational window on quantum gravity. I discuss the possibility that these explosions could be related to the recently observed and mysterious Fast Radio Bursts.