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Relativistic Dynamics in an Electromagnetic Field

  • Jȩdrzej Śniatycki
Chapter
  • 383 Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 30)

Abstract

The relativistic dynamics of a particle with charge e in an external electromagnetic field f can be described in terms of the phase space (T*Y,ω e ), where Y is the space-time manifold, Л: T*Y → Y is the cotangent bundle projection, and
$$ {\omega_e} = d{\theta_Y} + e\pi *f $$
(10.1)
[cf. Sec. 2.3]. Assuming that Y is orientable, and following the reasoning of Sec. 7.2 leading to a metaplectic structure on (T*YdθY), we obtain a metaplectic structure on (T*Y,ω e ). The vertical distribution D on T*Y tangent to the fibres of Л is Lagrangian with respect to the symplectic form ω e , so that of F = DC is a polarization of (T*Y,ω e ). The metalinear structure of F induced by the metaplectic structure on (T*Y, we isomorphic to that induced by the metaplectic structure on (T*Y, dθY). Hence, we can apply the results of Sec. 7.2. We denote by \( \tilde{D}F \) the metalinear frame bundle of F induced by the metaplectic structure and by √∧4 F the associated line bundle corresponding to the character χ of ML (4,C).

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Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • Jȩdrzej Śniatycki
    • 1
  1. 1.The University of CalgaryCalgaryCanada

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