Scattering on Magnetic Monopoles

The time-dependent scattering theory of charged particles on magnetic monopoles is investigated within a mathematical frame-work, which duely pays attention to the fact that the wave-functions of the scattered particles are sections in a non-trivial complex line-bundle. It is found that Möller operators have to be defined in a way which takes into account the peculiar long-range behaviour of the monopole field. Formulas for the scattering matrix and the differential cross-section are derived, and, as a by-product, a momentum space picture for particles, which are described by sections in the underlying complex line-bundle, is presented.

The additive and multiplicative orders of a complex line-bundle

Let X be a finite CW-complex, and ξ: Eξ → X a complex line-bundle on X. This bundle determines an element c1(ξ) in H2(X:Z) and a class ; let m(ξ), a(ξ) be the orders (possibly infinite) of these elements in their respective groups. How are m(ξ) and a(ξ) related? The answer is contained in the following three propositions.