Fernando Lledo'
Massless relativistic wave equations and quantum field theory
(205K, Latex2e)
ABSTRACT. We give a simple and direct construction of a massless quantum field
with arbitrary discrete helicity that satisfies
Wightman axioms and the corresponding relativistic wave equation
in the distributional sense. We underline the
mathematical differences to massive models.
The construction is based on the notion of massless free net
(cf. Section 3) and the detailed analysis of covariant and massless
canonical (Wigner) representations of the Poincare'
group. A characteristic feature of massless models with nontrivial
helicity is the fact that the fibre degrees of freedom
of the covariant and canonical representations do not coincide.
We use massless relativistic wave equations
as constraint equations reducing the fibre degrees of freedom of
the covariant representation. They are characterized by invariant
(and in contrast with the massive case non reducing) one-dimensional
projections. The definition of one-particle Hilbert space structure
that specifies the quantum field uses distinguished elements of the
intertwiner space between E(2) (the two-fold cover of the
2-dimensional Euclidean group) and its conjugate.
We conclude with a brief comparison between the free nets
constructed in Section 3 and a recent alternative
construction that uses the notion of modular localization.