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Two singularity theorems can be proven if one attempts to let a Lorentzian cobordism interpolate between two topologically distinct manifolds. On the other hand, Cartier and DeWitt-Morette have given a rigorous definition for quantum field theories (qfts) by means of path integrals. This article uses their results to study whether qfts can be made compatible with topology changes. We show that path integrals over metrics need a finite norm for the latter and for degenerate metrics, this problem can sometimes be resolved with tetrads. We prove that already in the neighborhood of some cuspidal singularities, difficulties can arise to define certain qfts. On the other hand, we show that simple qfts can be defined around conical singularities that result from a topology change in a simple setup. We argue that the ground state of many theories of quantum gravity will imply a small cosmological constant and, during the expansion of the universe, will cause frequent topology changes. Unfortunately, it is difficult to describe the transition amplitudes consistently due to the aforementioned problems. We argue that one needs to describe qfts by stochastic differential equations, and in the case of gravity, by Regge calculus in order to resolve this problem., Comment: 85 pages, to appear in rmp. The article is now in the production stage

We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma$ that we construct are either Birkhoff sections, meaning that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partial\Sigma$ as limit sets of the orbits of the geodesic flow that do not return to $\Sigma$. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow., Comment: 33 pages, 12 figures; version 2: minor modifications. To appear in Ergodic Theory and Dynamical Systems

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a $C^2$ open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are $C^2$ stably ergodic if and only if they are Anosov., Comment: 8 pages

Hawking's spacetime foam model predicts that due to quantum fluctuations, spacetime is filled with black hole like objects. We argue that Hawking's model implies a cosmological constant of the observed order and that it can also be used to solve the problem of time in quantum gravity., Comment: 12 pages

This is a review article on quantum gravity. In section 1, the Penrose singularity theorem is proven. In section 2, the covariant quantization approach of gravity is reviewed. In section 3, an article by Hawking is reviewed that shows the gravitational path integral at one loop level to be dominated by contributions from some kind of virtual gravitational instantons. In section 4, the canonical, non-perturbative quantization approach is reviewed. In section 5, arguments from Hawking are mentioned which show the gravitational path integral to be an approximate solution of the Wheeler deWitt equation. In section 6, the black hole entropy is derived in various ways. Section 6.1 uses the gravitational path integral for this calculation. Section 6.2 shows how the black hole entropy can be derived from canonical quantum gravity. In section 7.1, arguments from Dvali and Gomez who claim that gravity can be quantized in a way which would be in some sense self-complete are critically assessed. In section 7.2 a model from Dvali and Gomez for the description of quantum mechanical black holes is critically assessed and compared with the standard quantization methods of gravity., Comment: 100 pages

This article contains a review of Nelson's analysis of Bell's theorem. It shows that Bell's inequalities can be violated with a theory of local random variables if one accepts that the outcomes of these variables are not predetermined prior to measurement. The article describes the relation between Bell's theorem and the Strong Free Will theorem of Conway and Kochen. Then, the original articles of Bell are analyzed in detail. Following an article of Faris, it is explained that Bell's work on the hidden variable question in fact describes two separate theorems. Bell's first theorem says that there can be no model for the singlet state where an outcome does not depend locally on the settings of the detector where the outcome was measured. Bell's second theorem shows that Bell's inequalities can be violated by a theory that is either not deterministic, or violates causality in the sense of relativity or the free will assumption of the experimenters. It is shown in detail where Bell implicitly makes the various locality assumptions that Nelson has shown to be necessary for deriving Bell's inequality. The article closes by relating the various assumptions needed to derive Bell's theorem with the reality criterion of EPR, Comment: 19 pages, section added where Bell's original articles are analyzed and compared in detail with Nelson's work

This is a reply to the comment of Schmelzer on my earlier paper. In contrast to Schmelzer's claims, it is shown that passive locality does not correspond to the completeness condition of EPR. Nevertheless, it is argued that Schmelzer has found several errors in my article which, fortunately, can be corrected. In spite of Schmelzer's arguments, the Fritsche-Haugk theory can be used to define a local hidden variable theory for the singlet state., Comment: The correct parts of this article are now in a separate article at arxiv.org. The stochastic model in this article must be described in more detail before it is suitable for publication

In 1985, Edward Nelson, who formulated the theory of stochastic mechanics, made an interesting remark on Bell's theorem. Nelson analysed the latter in the light of classical fields that behave randomly. He found that if a stochastic hidden variable theory fulfils certain conditions, the inequality of Bell can be violated. Moreover, Nelson was able to prove that this may happen without any instantaneous communication between the two spatially separated measurement stations. Since Nelson's article got almost overlooked by physicists, we try to review his comments on Bell's theorem. We argue that a modification of stochastic mechanics published recently by Fritsche and Haugk can be extended to a theory which fulfils the requirements of Nelson's analysis. The article proceeds to derive the quantum mechanical formalism of spinning particles and the Pauli equation from this version of stochastic mechanics. Then, we investigate Bohm's version of the EPR experiment. Additionally, other setups, like entanglement swapping or time and position correlations, are shortly explained from the viewpoint of our local hidden-variable model. Finally, we mention that this theory could perhaps be relativistically extended., Comment: The stochastic model presented in this article was defect. The parts of this work that are correct are now in a separate article on arxiv.org