Geometric Quantization and Quantum Mechanics pp 168-197 | Cite as

# Relativistic Dynamics in an Electromagnetic Field

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## 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
[cf. Sec. 2.3]. Assuming that Y is orientable, and following the reasoning of Sec. 7.2 leading to a metaplectic structure on (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)

^{*}Ydθ_{Y}), 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 = D^{C}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, dθ_{Y}). 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).## Preview

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## Copyright information

© Springer-Verlag New York Inc. 1980