In a former paper we proposed a model for the quantization of gravity by working in a bundle $E$ where we realized the Hamilton constraint as the Wheeler-DeWitt equation. However, the corresponding operator only acts in the fibers and not in the base space. Therefore, we now discard the Wheeler-DeWitt equation and express the Hamilton constraint differently, either with the help of the Hamilton equations or by employing a geometric evolution equation. There are two possible modifications possible which both are equivalent to the Hamilton constraint and which lead to two new models. In the first model we obtain a hyperbolic operator that acts in the fibers as well as in the base space and we can construct a symplectic vector space and a Weyl system. In the second model the resulting equation is a wave equation in $\so \times (0,\infty)$ valid in points $(x,t,\xi)$ in $E$ and we look for solutions for each fixed $\xi$. This set of equations contains as a special case the equation of a quantized cosmological Friedman universe without matter but with a cosmological constant, when we look for solutions which only depend on $t$. Moreover, in case $\so$ is compact we prove a spectral resolution of the equation.

In this paper, we establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh--Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and the Cauchy formula (1882) for the magnetic field. This enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and on the other hand, it allows us to define a special good unknown of the magnetic field to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh--Taylor sign condition required by Morando, Tarkhinin and Trebeschi (\emph{J. Differential Equations} \textbf{258} (2015), no. 7, 2531--2571; \emph{Arch. Ration. Mech. Anal.} \textbf{228} (2018), no. 2, 697--742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is {\it no loss of derivatives} in our well-posedness theory. The solution is constructed as {\it the inviscid limit} of solutions to well-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD.

The ``principle of the fermionic projector'' provides a new mathematical framework for the formulation of physical theories and is a promising approach for physics beyond the standard model. The book begins with a brief review of relativity, relativistic quantum mechanics and classical gauge theories, with the emphasis on the basic physical concepts and the mathematical foundations. The external field problem and Klein's paradox are discussed and then resolved by introducing the so-called fermionic projector, a global object in space-time which generalizes the notion of the Dirac sea. The mathematical core of the book is to give a precise definition of the fermionic projector and to employ methods of hyperbolic differential equations for its detailed analysis. The fermionic projector makes it possible to formulate a new type of variational principles in space-time. The mathematical tools for the analysis of the corresponding Euler-Lagrange equations are developed. A particular variational principle is proposed which gives rise to an effective interaction showing many similarities to the interactions of the standard model. The main chapters of the book are easily accessible for beginning graduate students in mathematics or physics. Several appendices provide supplementary material which will be useful to the experienced researcher.

Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

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