1,720,974 research outputs found
Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds
No full textIn this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action of the intertwining operators in the context of algebraic quantum field theory. In particular, we provide a new geometric proof for the existence of the so-called Hadamard states on globally hyperbolic manifolds
The quantization of Maxwell theory in the Cauchy radiation gauge: Hodge decomposition and Hadamard states
No full textThe aim of this paper is to prove the existence of Hadamard states for the Maxwell equations on any globally hyperbolic spacetime. This will be achieved by introducing a new gauge fixing condition, the Cauchy radiation gauge, that will allow to suppress all the unphysical degrees of freedom. The key ingredient for achieving this gauge is a new Hodge decomposition for differential k-forms in Sobolev spaces on complete (possibly noncompact) Riemannian manifolds
The well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with timelike boundary
No full textWe consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial boundary value problem coupled toMIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions
ON THE CAUCHY PROBLEM FOR FRIEDRICHS SYSTEMS ON GLOBALLY HYPERBOLIC MANIFOLDS WITH TIMELIKE BOUNDARY
No full textIn 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
The fermionic signature operator and quantum states in Rindler space-time
No full textThe 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
On the uniqueness of invariant states
No full textGiven an abelian group G endowed with a T=R/Z-pre-symplectic form, we assign to it a symplectic twisted group ⁎-algebra WG 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
Wick Rotation of Linearized Gravity in Gaussian Time and Calderón Projectors
No full textMotivated by the quantization of linearized gravity, we consider gauge-fixed linearized Einstein equations and their Wick rotation near a Cauchy surface. We show that Calderón projectors for the Wick-rotated equations induce Hadamard bi-solutions on the Lorentzian level. On the other hand, we find smoothing obstructions to gauge-invariance and positivity conditions needed in quantization. These obstructions are primarily due to boundary terms arising in the Wick-rotated theory and depend on the boundary conditions
A new class of Fermionic Projectors: Møller operators and mass oscillation properties
No full textRecently, a new functional analytic construction of quasi-free states for a self-dual CAR algebra has been presented in Finster and Reintjes (Adv Theor Math Phys 20:1007, 2016). 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
Injective Tensor Products in Strict Deformation Quantization
No full textThe aim of this paper is twofold. Firstly we provide necessary and sufficient criteria for the existence of a strict deformation quantization of algebraic tensor products of Poisson algebras, and secondly we discuss the existence of products of KMS states. As an application, we discuss the correspondence between quantum and classical Hamiltonians in spin systems and we provide a relation between the resolvent of Schödinger operators for non-interacting many particle systems and quantization maps
Invariant States on Noncommutative Tori
No full textFor any number h such that h := h/2 pi is irrational and any skew-symmetric, non-degenerate bilinear form sigma : Z(2g) x Z(2g) -> Z, let be A(g,sigma)(h) be the twisted group *-algebra C[Z(2g)] and consider the ergodic group of *-automorphisms of A(g,sigma)(h) induced by the action of the symplectic group Sp (Z(2g), sigma). We show that the only Sp (Z(2g),sigma)-invariant state on A(g,sigma)(h) is the trace state tau
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