Influence of quantum matter fluctuations on geodesic deviation

We study the passive influence of quantum matter fluctuations on the expansion parameter of a congruence of timelike geodesics in a semiclassical regime. In particular, we show that, the perturbations of this parameter can be considered to be elements of the algebra of matter fields at all perturbative order. Hence, once a quantum state for matter is chosen, it is possible to explicitly evaluate the behavior of geometric fluctuations. After introducing the formalism necessary to treat similar problems, in the last part of the paper, we estimate the approximated probability of having a geodesic collapse in a flat spacetime due to those fluctuations.Comment: 21 pages, published version, J. Phys. A: Math. Theor. 47 (2014) 37520

A new class of Fermionic Projectors: M{\o}ller operators and mass oscillation properties

Recently, a new functional analytic construction of quasi-free states for a self-dual CAR algebra has been presented in \cite{Felix2}. This method relies on the so-called strong mass oscillation property. We provide an example where this requirement is not satisfied, due to the nonvanishing trace of the solutions of the Dirac equation on the horizon of Rindler space, and we propose a modification of the construction in order to weaken this condition. Finally, a connection between the two approaches is built.Comment: 21 pages, accepted for publication in Letters in Mathematical Physics ( 998

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by $E$, a second order elliptic partial differential operator of metric type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant observables in terms of equivariant sections of a bundle of smooth, regular polynomial functionals over the affine space of the parametrices associated to $E$. Subsequently, adapting to the case in hand a strategy first introduced by Hollands and Wald in a Lorentzian setting, we prove the existence of Wick powers of the underlying field, extending the procedure to smooth, local and polynomial functionals and discussing in the process the regularization ambiguities of such procedure. Subsequently we endow the space of Wick powers with an algebra structure, dubbed E-product, which plays in a Riemannian setting the same role of the time ordered product for field theories on globally hyperbolic spacetimes. In particular we prove the existence of the E-product and we discuss both its properties and the renormalization ambiguities in the underlying procedure. As last step we extend the whole analysis to observables admitting derivatives of the field configurations and we discuss the quantum M{\o}ller operator which is used to investigate interacting models at a perturbative level.Comment: 35 page

Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati and Moretti for vector valued free fields. As a by-product of such classification, we prove that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.Comment: 24 page

Classical KMS Functionals and Phase Transitions in Poisson Geometry

In this paper we study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinstein's seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of $b$-Poisson manifolds, where we provide a complete characterization of the convex cone of KMS measures.Comment: 47 page

Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary

We consider the wave operator on static, Lorentzian manifolds with timelike boundary and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus we prove that answering this question is equivalent to studying the self-adjoint extensions of an associated elliptic operator on a Riemannian manifold with boundary $(M,g)$. The latter is diffeomorphic to any, constant time hypersurface of the underlying background. In turn, assuming that $(M,g)$ is of bounded geometry, this problem can be tackled within the framework of boundary triples. These consist of the assignment of two surjective, trace operators from the domain of the adjoint of the elliptic operator into an auxiliary Hilbert space $\mathsf{h}$, which is the third datum of the triple. Self-adjoint extensions of the underlying elliptic operator are in one-to-one correspondence with self-adjoint operators $\Theta$ on $\mathsf{h}$. On the one hand, we show that, for a natural choice of boundary triple, each $\Theta$ can be interpreted as the assignment of a boundary condition for the original wave operator. On the other hand, we prove that, for each such $\Theta$, there exists a unique advanced and retarded fundamental solution. In addition, we prove that these share the same structural property of the counterparts associated to the wave operator on a globally hyperbolic spacetime.Comment: 25 pages, typos corrected, to appear in Lett. Math. Phy

Global wave parametrices on globally hyperbolic spacetimes

In a recent work the first named author, Levitin and Vassiliev have constructed the wave propagator on a closed Riemannian manifold $M$ as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. In this paper, first we give a natural reinterpretation of the underlying algorithmic construction in the language of ultrastatic Lorentzian manifolds. Subsequently we show that the construction carries over to the case of static backgrounds thanks to a suitable reduction to the ultrastatic scenario. Finally we prove that the overall procedure can be generalised to any globally hyperbolic spacetime with compact Cauchy surfaces. As an application, we discuss how, from our procedure, one can recover the local Hadamard expansion which plays a key role in all applications in quantum field theory on curved backgrounds.Comment: 28 pages, final version accepted for publicatio