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Register a new userHuygens principle for the generalized Dirac operator in curved spacetime
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Authors
Karen Yagdjian
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In this article we give sufficient conditions for the generalized Dirac operator to obey the incomplete Huygens principle, as well as necessary and sufficient conditions to obey the Huygens principle by the Dirac operator in the curved spacetime of the Friedmann-Lema^itre-Robertson-Walker models of cosmology.
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The operator associated to the angular part of the Dirac equation in the Kerr-Newman background metric is a block operator matrix with bounded diagonal and unbounded off-diagonal entries. The aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above. In the last section, these analytic bounds are compared to numerical values from the literature.
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We present a variational approach which shows that the wave functions belonging to quantum systems in different potential landscapes, are pairwise linked to each other through a generalized continuity equation. This equation contains a source term proportional to the potential difference. In case the potential landscapes are related by a linear symmetry transformation in a finite domain of the embedding space, the derived continuity equation leads to generalized currents which are divergence free within this spatial domain. In a single spatial dimension these generalized currents are invariant. In contrast to the standard continuity equation, originating from the abelian $U(1)$-phase symmetry of the standard Lagrangian, the generalized continuity equations derived here, are based on a non-abelian $SU(2)$-transformation of a Super-Lagrangian. Our approach not only provides a rigorous theoretical framework to study quantum mechanical systems in potential landscapes possessing local symmetries, but it also reveals a general duality between quantum states corresponding to different Schr{o}dinger problems.
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