Abstract
On-shell methods are particularly suited for exploring the scattering of electrically and magnetically charged objects, for which there is no local and Lorentz invariant Lagrangian description. In this paper we show how to construct a Lorentz-invariant S-matrix for the scattering of electrically and magnetically charged particles, without ever having to refer to a Dirac string. A key ingredient is a revision of our fundamental understanding of multi-particle representations of the Poincaré group. Surprisingly, the asymptotic states for electric-magnetic scattering transform with an additional little group phase, associated with pairs of electrically and magnetically charged particles. The corresponding “pairwise helicity” is identified with the quantized “cross product” of charges, e1g2− e2g1, for every charge-monopole pair, and represents the extra angular momentum stored in the asymptotic electromagnetic field. We define a new kind of pairwise spinor-helicity variable, which serves as an additional building block for electric-magnetic scattering amplitudes. We then construct the most general 3-point S-matrix elements, as well as the full partial wave decomposition for the 2 → 2 fermion-monopole S-matrix. In particular, we derive the famous helicity flip in the lowest partial wave as a simple consequence of a generalized spin-helicity selection rule, as well as the full angular dependence for the higher partial waves. Our construction provides a significant new achievement for the on-shell program, succeeding where the Lagrangian description has so far failed.
Highlights
Unitary representations of the Poincaré group, classified by Wigner [1] in the 1930s, provide the foundation of the quantum mechanical (QM) description of particle physics and quantum field theory
In the first part of this paper we address the general construction of multi-particle states and introduce the concept of the pairwise little group (LG), which is necessary to fully classify the multi-particle representations of the Poincaré group
As we argued in the previous section, in the case of the electric-magnetic S-matrix, the transformation rule involves an additional pairwise LG phase associated with the angular momentum in the EM field, as can be seen in eq (2.26)
Summary
Unitary representations of the Poincaré group, classified by Wigner [1] in the 1930s, provide the foundation of the quantum mechanical (QM) description of particle physics and quantum field theory. In a beautiful, under-appreciated paper in 1972 Zwanziger [3] found that quantum states with both electric and magnetic charges transform in non-trivial multiparticle representations of the Poincaré group. These new spinor-helicity variables, together with the standard spinors for massless and massive particles, serve as a complete set of building blocks for the magnetic (and non-magnetic) S-matrix.
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