Massless particles, electromagnetism, and Rieffel induction

TL;DR

This paper explores the relationship between different representations of the Poincaré group for massless particles, using symplectic reduction and Rieffel induction to construct the quantum electromagnetic field algebra.

Contribution

It introduces a novel application of Rieffel induction to quantum electromagnetism, connecting classical symplectic reduction with quantum operator algebra techniques.

Findings

Massless particle representations relate via gauge symmetries and symplectic reduction.

Constructs the quantum electromagnetic algebra directly in the vacuum state.

Develops a positive semi-definite sesquilinear form using Gaussian measures on the gauge group.

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 Poincar\'{e} 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 electro- magnetism. 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 electro- magnetism (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.