MASSLESS PARTICLES, ELECTROMAGNETISM, AND RIEFFEL INDUCTION

Abstract

The connection between space-time covariant representations (obtained by inducing from the Lorentz group) and irreducible unitary representations (induced from Wigner’s little group) of the Poincaré group is re-examined in the massless case. In the situation relevant to physics, it is found that these are related by Marsden-Weinstein reduction with respect to a gauge group. An analogous phenomenon is observed for classical massless relativistic particles. This symplectic reduction procedure can be (‘second’) quantized using a generalization of the Rieffel induction technique in operator algebra theory, which is carried through in detail for electromagnetism. Starting from the so-called Fermi representation of the field algebra generated by the free abelian gauge field, we construct a new (‘rigged’) sesquilinear form on the representation space, which is positive semi-definite, and given in terms of a Gaussian weak distribution (promeasure) on the gauge group (taken to be a Hilbert Lie group). This eventually constructs the algebra of observables of quantum electromagnetism (directly in its vacuum representation) as a representation of the so-called algebra of weak observables induced by the trivial representation of the gauge group.