Abstract:
We consider gravitational wave modes in the FRW metrics in a de Sitter phase and show that the state space splits into many unitarily inequivalent representations of the canonical commutation relations. Non-unitary time evolution is described as a trajectory in the space of the representations. The generator of time evolution is related to the entropy operator. The thermodynamic arrow of time is shown to point in the same direction of the cosmological arrow of time. The vacuum is a two-mode SU(1,1) squeezed state of thermo field dynamics. The link between expanding geometry, squeezing and thermal properties is exhibited.

Abstract:
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and total energies are demonstrated. This material provides background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approximation and other methods.

Abstract:
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and total energies are demonstrated. This material provides background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approximation and other methods.

Abstract:
We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential.

Abstract:
As a ramification of a motivational discussion for previous joint work, in which equations of motion for the finite spectral action of the Standard Model were derived, we provide a new analysis of the results of the calculations herein, switching from the perspective of Spectral triple to that of Fredholm module and thus from the analogy with Riemannian geometry to the pre-metrical structure of the Noncommutative geometry. Using a suggested Noncommutative version of Morse theory together with algebraic $K$-theory to analyse the vacuum solutions, the first two summands of the algebra for the finite triple of the Standard Model arise up to Morita equivalence. We also demonstrate a new vacuum solution whose features are compatible with the physical mass matrix.

Abstract:
In loop quantum gravity, states of the gravitational field turn out to be excitations over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also important to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.

Abstract:
In loop quantum gravity, states of the gravitational field turn out to be excitations over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also important to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.

Abstract:
One of the central difficulties of settling the $L^2$-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface.

Abstract:
The topos theory is a theory which is used for deciding a number of problems of theory of relativity, gravitation and quantum physics. It is known that topos-theoretic geometry can be successfully developed within the framework of Synthetic Differential Geometry of Kock-Lawvere (SDG), the models of which are serving the toposes, i.e. categories possessing many characteristics of traditional Theory of Sets. In the article by using ideas SDG, non-classical spherically symmetric solution of the vacuum Einstein equations is given.

Abstract:
The space of Dirac operators for the Connes-Chamseddine spectral action for the standard model of particle physics coupled to gravity is studied. The model is extended by including right-handed neutrino states, and the S0-reality axiom is not assumed. The possibility of allowing more general fluctuations than the inner fluctuations of the vacuum is proposed. The maximal case of all possible fluctuations is studied by considering the equations of motion for the vacuum. Whilst there are interesting non-trivial vacua with Majorana-like mass terms for the leptons, the conclusion is that the equations are too restrictive to allow solutions with the standard model mass matrix.