The Dirac operator on Lorentzian spin manifolds and the Huygens property.pdf

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The Dirac operator on Lorentzian spin manifolds and the Huygens property.pdf

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The Dirac operator on Lorentzian spin manifolds and the Huygens property

The Dirac operator on Lorentzian spin manifolds and theHuygens propertyHelga BaumAugust 9, 1996AbstractWe consider the Dirac operator D of a Lorentzian spin manifold of even dimen-sion n 4. We prove, that the square D2 of the Dirac operator on plane wavemanifolds and the shifted operator D2 K on Lorentzian space forms of constantsectional curvature K are of Huygens type. Furthermore, we study the Huygensproperty for coupled Dirac operators on 4-dimensional Lorentzian spin manifolds.1 IntroductionIt is a familiar phenomenon that waves propagate quite dierent in 2 and 3 dimensions.When a pebble falls into water at a certain point x0, circular waves around x0 areformed. A given point near x0 will be hit by an initial ripple and later by residualwaves. 3-dimensionally, the situation is quite dierent. If we produce a sound localizedat the neighbourhood of a point x0 then someone near x0 will hear the sound during acertain time interval but no longer. There are no residual waves like those present onthe water surface.The mathematical reason for this dierent behaviour is a special property of the fun-damental solution of the wave operator 2m of the Rm in dimension m = 3. Whereas ingeneral the forward fundamental solution of 2m with respect to the point o 2 Rm+1 issupported in the future cone J+(o) = f(x; t) 2 Rm R j jjxjj tg the forward funda-mental solution in dimension m = 3 and each other odd dimension m 3 is supportedeven in the light cone C+(o) = f(x; t) 2 Rm R j jjxjj = tg . This produces a sharpwave propagation. Operators describing a sharp wave propagation such as 23 arecalled operators of Huygens type or shortly Huygens operators. In 1923, in his YaleLectures, J. Hadamard posed the problem of nding all normally hyperbolic operatorsof Huygens type (see [Had23], p.236). In spite of its age this problem is still far frombeing completely solved. For results of the last thirty years and methods developed totreat this problem see [Gun88, Gun91, Wun94, B