Multi-particle Representations of the Poincar\'e Group

TL;DR

This paper extends the concept of multi-particle states in quantum field theory by introducing pairwise helicities, which account for additional angular momentum sources in scattering processes involving electric and magnetic charges.

Contribution

It introduces pairwise helicities as new quantum numbers for multi-particle states, generalizing the asymptotic state construction beyond direct product states.

Findings

Reproduces standard multi-particle states when pairwise helicities vanish.

Recovers Zwanziger's electric-magnetic multi-particle states in special cases.

Identifies pairwise helicity as a label for extra little group phase.

Abstract

In this work we extend the definition of asymptotic multi-particle states of the $S$-matrix, beyond the direct products of one-particle states. We identify new quantum numbers which we call pairwise helicities, or $q_{ij}$, associated with asymptotically separated pairs of particles. These signal the appearance of a new source of angular momentum, beyond the orbital and spin contributions. The essence of our construction is to first treat all single particles as well as all particle pairs independently, ultimately projecting onto the physical states. The resulting representations reproduce the usual direct product states for vanishing $q_{ij}$, while for vanishing spins they reproduce Zwanziger's electric-magnetic multi-particle states. Pairwise helicity then appears as a label for the extra little group phase for our quantum states, in addition to their standard little group transformation. Our newly defined multi-particle states are the correct asymptotic states for the scattering of electric and magnetic charges, with pairwise helicity identified as $q_{ij}=e_i g_j-e_j g_i$.