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.

The Stückelberg–Kibble model as an example of quantized symplectic reduction

“…This is somehwat reminiscent of the BRST (or, in case of QED, the GuptaBleuler) procedure, with the major difference that with Rieffel induction no negativenorm subspace exists, obviating the need to select a physical subspace of H. Also, certain functional-analytic problems that appear in the BRST as well as in the Dirac method are absent with our present techniques [11,13]. By definition of the inner product on the physcial Hilbert space H phys , calculations of correlation functions of operators in A c (as represented on H phys ) may be performed in L ⊗ H tr , [14].…”

“…This subsection gives a quick 'review by example' of some parts of the theory developed in [11] and [13].…”

“…We refer to [13] for a discussion of the technical points involved. This is somehwat reminiscent of the BRST (or, in case of QED, the GuptaBleuler) procedure, with the major difference that with Rieffel induction no negativenorm subspace exists, obviating the need to select a physical subspace of H. Also, certain functional-analytic problems that appear in the BRST as well as in the Dirac method are absent with our present techniques [11,13].…”

“…A mathematically rigorous treatment of the following material, in the style of [13], is possible, but we leave the details to the interested reader; instead, readability commands us to give somewhat loose formulations.…”

“…The method in question has already been successfully applied to certain finite-dimensional constrained systems [11], as well as to free quantum electrodynamics [13,14].…”