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 Fermionic Signature Operator and Quantum States in Rindler Space-Time

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling-Rindler vacuum. Moreover, the fermionic signature operator gives a covariant construction of general thermal states, in particular of the Unruh state. The fermionic signature operator is shown to be well-defined in asymptotically Rindler space-times. In four-dimensional Rindler space-time, our construction gives rise to new quantum states.Comment: 27 pages, LaTeX, more details on self-adjoint extension (published version

On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided. Keywords: symmetric hyperbolic systems, symmetric positive systems, admissible boundary conditions, Dirac operator, normally hyperbolic operator, Klein-Gordon operator, heat operator, reaction-diffusion operator, globally hyperbolic manifolds with timelike boundary

On the uniqueness of invariant states

Given an abelian group G endowed with a T-pre-symplectic form, we assign to it a symplectic twisted group *-algebra W_G and then we provide criteria for the uniqueness of states invariant under the ergodic action of the symplectic group of automorphism. As an application, we discuss the notion of natural states in quantum abelian Chern-Simons theory.Comment: 29 pages -- accepted in Advances in Mathematic

Quantization of linearized gravity by Wick rotation in Gaussian time

We consider the problem of quantizing linearized Einstein equations. We provide a rigorous construction applicable to a large class of Einstein metrics. More precisely, we prove the existence of Hadamard states in the harmonic gauge on analytic backgrounds of bounded geometry. This includes examples such as Minkowski, de Sitter, Kerr-Kruskal and Schwarzschild-de Sitter spacetimes. The states are constructed from Calder\'on projectors for an elliptic boundary problem obtained by Wick rotation near a Cauchy surface. The key ingredients of the proof are gauge transformations in the Wick-rotated setting and various techniques from elliptic theory and microlocal analysis.Comment: 62 page

Invariant States on Noncommutative Tori

For any number h such that \u210f:=h/2\u3c0 is irrational and any skew-symmetric, non-degenerate bilinear form \u3c3:\u21242g 7\u21242g\u2192\u2124, let be \ue22dhg,\u3c3 be the twisted group 17-algebra \u2102[\u21242g] and consider the ergodic group of 17-automorphisms of \ue22dhg,\u3c3 induced by the action of the symplectic group Sp(\u21242g,\u3c3). We show that the only Sp(\u21242g,\u3c3)-invariant state on \ue22dhg,\u3c3 is the trace state \u3c4

The quantization of Proca fields on globally hyperbolic spacetimes: Hadamard states and M{\o}ller operators

This paper deals with several issues concerning the algebraic quantization of the real Proca field in a globally hyperbolic spacetime and the definition and existence of Hadamard states for that field. In particular, extending previous work, we construct the so-called M\o ller $*$-isomorphism between the algebras of Proca observables on paracausally related spacetimes, proving that the pullback of these isomorphisms preserves the Hadamard property of corresponding quasifree states defined on the two spacetimes. Then, we pull-back a natural Hadamard state constructed on ultrastatic spacetimes of bounded geometry, along this $*$-isomorphism, to obtain a Hadamard state on a general globally hyperbolic spacetime. We conclude the paper, by comparing the definition of a Hadamard state, here given in terms of wavefront set, with the one proposed by Fewster and Pfenning, which makes use of a supplementary Klein-Gordon Hadamard form. We establish an (almost) complete equivalence of the two definitions.Comment: 45 pages --- accepted in Annales Henri Poincar\'

On the Cauchy problem for the Fadaray tensor on globally hyperbolic manifolds with timelike boundary

We study the well-posedness of the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary. The existence of Green operators for the operator $\mathrm{d}+\delta$ and a suitable pre-symplectic structure on the space of solutions are discussed.Comment: 19 pages -- accepted in Rendiconti Lincei Matematica e Applicazion

Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states

We discuss the quantization of linearized gravity on globally hyperbolic, asymptotically flat, vacuum spacetimes and the construction of distinguished states which are both of Hadamard form and invariant under the action of all bulk isometries. The procedure, we follow, consists of looking for a realization of the observables of the theory as a sub-algebra of an auxiliary, non-dynamical algebra constructed on future null infinity $\Im^+$. The applicability of this scheme is tantamount to proving that a solution of the equations of motion for linearized gravity can be extended smoothly to $\Im^+$. This has been claimed to be possible provided that a suitable gauge fixing condition, first written by Geroch and Xanthopoulos, is imposed. We review its definition critically showing that there exists a previously unnoticed obstruction in its implementation leading us to introducing the concept of radiative observables. These constitute an algebra for which a Hadamard state induced from null infinity and invariant under the action of all spacetime isometries exists and it is explicitly constructed.Comment: 31 pages, added reference