Except for a few special cases, quantum energy inequalities proved to date have constrained stress-energy tensors 'normal-ordered' with respect to a reference state, sometimes called 'difference quantum inequalities'. The reliance on a reference state is problematic, because one does not have access to, e.g., the two-point function of a suitable reference state in a generic curved spacetime background. In this talk, based on work with Calvin Smith (York), I will present a general derivation of an absolute quantum energy inequality, which does not appeal to a reference state at any stage, and is manifestly locally covariant. I will also describe other connections between quantum energy inequalities and local covariance, based on work with Mitch Pfenning (West Point).

A quantum field theory describing both geometry and matter must be background independent as prescribed by the Einstein equations of General Relativity according to which geometry and matter determine each other in tandem dynamically. There is no a priori distinguished background metric which makes the framework of ordinary QFT inapplicable as it is heavily based on the locality principle which only makes sense when the causality structure of an externally prescribed background metric is known. One of the background independent approaches to quantum gravity is Loop Quantum Gravity (LQG) whose principles and status we describe in this talk.

Quantum inequality (QI) bounds have been used to place severe restrictions on "designer spacetimes", such as traversable wormholes and warp drives. Most of the QI bounds which have been proven to date involve averages of the energy density in some observer's frame, over his worldline. In the current work, a QI bound on the expectation value of the null-contracted stress tensor, averaged over a timelike worldline, is used to obtain constraints on the geometries of traversable wormholes. In the present context, this type of QI bound has certain advantages over the previous QIs. Particular attention is given to the wormhole models of Visser, Kar, and Dadhich (VKD) and to those of Kuhfittig. These are models which use arbitrarily small amounts of exotic matter for wormhole maintenance. We show that macroscopic VKD models are either ruled out or severely constrained by the QI bound. A recent model of Kuhfittig is shown to be, despite claims to the contrary, non-traversable.

I will briefly review the proposed mechanism of generation of inhomogeneities from quantum fluctuations during inflation. Then I will argue that that picture might not be complete and a new physical mechanism may be needed to accomplish the transition to inhomogeneity. I will give a phenomenological description of such a mechanism and speculate about the underlying physics. The work I report on was carried out in collaboration with A. Perez and D. Sudarsky.

One of the basic steps when quantizing a classical theory is the choice of an appropriate kinematical Hilbert space which the quantized operators are represented on. In quantum mechanics this selection is unique due to the celebrated Stone-von Neumann theorem. Recently, it has been investigated whether similar results may fix both Hilbert space and representation in loop quantum gravity as well. It has turned out that the diffeomorphism invariance of general relativity indeed provides us with strong uniqueness results, as will be discussed in this talk.

In the talk I will mainly deal with a scalar quantum field propagating on given, static spacetimes. In is known, that for static spacetimes with horizons, such as the Schwarzschild spacetime, the ground states exhibit intriguing behavior near the horizon. For instance, the expectation value of the energy density diverges to minus infinity at the Schwarzschild horizon. On the other hand, it is also known that for static spacetimes without horizons no such divergency should be expected (the ground states are Hadamard states). However, it is not known, whether the ground states for spacetimes which only merely avoid collapse do exhibit large negative energy densities anywhere. In my talk I will show that Quantum Weak Energy Inequalities can be employed to yield an answer to this question. This general method, which avoids cumbersome direct calculations, provides physically tight bounds if the spacetimes are isometric to Schwarzschild up to a small distance from the horizon. I will discuss some recent results of mine, together with the problems currently under attack and some possible future generalizations.

Sphalerons are static but unstable finite-energy solutions of the classical Yang-Mills-Higgs field equations. Via the phenomenon of spectral flow, sphalerons are linked to the possible existence of anomalies. After illustrating this interconnection for the well-known example of the chiral $U(1)$ anomaly, I will discuss a possible sphaleron solution related to the non-Abelian (Bardeen) anomaly in SU(3) gauge theory.

A string in a flat background is an integrable system allowing for the construction of an algebra of diffeomorphism-invariant functionals on the worldsheet. The approach where this algebra is used as a starting point for quantization is reviewed and it is shown that it is inequivalent to quantization methods based on Fock space -- even in the critical dimension and regardless of the ordering prescription.

Quantum field theories based on interactions which contain the Moyal star product suffer, in the general case when time does not commute with space, from several diseases: quantum equation of motions contain unusual terms, conserved currents can not be defined and the residual spacetime symmetry is not maintained. All these problems have the same origin: time ordering does not commute with taking the star product. Here we show that these difficulties can be circumvented by a new definition of time ordering: namely with respect to a light-cone variable. In particular the original spacetime symmetries SO(1,1) x SO(2) and translation invariance turn out to be respected. Unitarity is guaranteed as well.

Scientific Organizers

Klaus Sibold

Universität Leipzig

Administrative Contact

Katja Bieling

Max Planck Institute for Mathematics in the Sciences
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