Relativistic Dynamics in an Electromagnetic Field

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^*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).