This series consists of talks in the area of Quantum Gravity.
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.
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.
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.
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?
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.
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 talk will discuss my attempts to define quantum geometry (and hence quantum gravity) using non-commutative geometries and the interesting mathematical structures that emerge.