On the propagation of polarization sets for systems of real principal type

“…Moreover, both authors investigate also the polarization set of the two-point functions of Hadamard states. The polarization set is a generalization of the wavefront set for vector-bundle distributions introduced by Dencker [7]. In components of a local frame for a vector-bundle, a vector-bundle distribution u is locally represented as an element (u 1 , .…”

Section: Introductionmentioning

confidence: 99%

Microlocal Spectrum Condition and Hadamard Form for Vector-Valued Quantum Fields in Curved Spacetime

2001

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Abstract.Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed 'wavefront set spectrum condition'), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher.In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance scaling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.

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“…Both systems (2-1) and (2-2) are real, time reversal invariant, and their solutions satisfy reciprocity. We describe how the system (2-2) can be decoupled by transforming it with appropriate pseudodifferential operators; see Taylor [98], Ivrii [67] and Dencker [39]. The goal is to transform the operator P il by conjugation with a matrixvalued pseudodifferential operator…”

Section: Propagation Of Elastic Waves In Smoothly Varying Mediamentioning

confidence: 99%

“…Under the assumptions made by Beylkin [10], there exists microlocally an invertible map, transforming seismic data to a reflectivity function r(x, e), of which the singular part should not depend on e. We consider such a transformation in a general framework that allows the presence of caustics and in anisotropic elastic rather than isotropic acoustic media. The treatment of elastic waves is based upon the decoupling of the hyperbolic system into n scalar equations (see Taylor [98], Ivrii [67], Dencker [39]) after which Fourier integral operator techniques are invoked. Each scalar equation governs the propagation of a particular mode, such as qP and S 1,...,n−1 .…”

Section: Introductionmentioning

confidence: 99%

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

Stolk¹,

Hoop²

2001

Comm Pure Appl Math

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Abstract. We review applications of microlocal analysis (MA) to reflection seismology. In this inverse method one attempts to estimate the index of refraction of waves in the earth from seismic data measured at the Earth's surface. Seismic imaging creates images of the Earth's upper crust using seismic waves generated by artificial sources and recorded into extensive arrays of sensors (geophones or hydrophones). The technology is based on a complex, and rapidly evolving, mathematical theory that employs advanced solutions to a wave equation as tools to solve approximately the general seismic inverse problem, with complications introduced by the heterogeneity and anisotropy of the Earth's crust. We describe several important developments using MA to generate these wave-solutions by manipulating the wavefields directly on their phase space. We also consider some recent applications of MA to global seismology.

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On the propagation of polarization sets for systems of real principal type

Cited by 98 publications

References 3 publications

Propagation of polarization in elastodynamics with residual stress and travel times

Propagation of polarization in elastodynamics with residual stress and travel times

Microlocal Spectrum Condition and Hadamard Form for Vector-Valued Quantum Fields in Curved Spacetime

Microlocal analysis of seismic inverse scattering in anisotropic elastic media

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