A first-order Lagrangian theory of fields with arbitrary spin

Abstract

The bundles suitable for a description of higher-spin fields can be built in terms of a 2-spinor bundle as the basic “building block”. This allows a clear, direct view of geometric constructions aimed at a theory of such fields on a curved spacetime. In particular, one recovers the Bargmann–Wigner equations and the [Formula: see text]-dimensional representation of the angular-momentum algebra needed for the Joos–Weinberg equations. Looking for a first-order Lagrangian field theory we argue, through considerations related to the 2-spinor description of the Dirac map, that the needed bundle must be a fibered direct sum of a symmetric “main sector” — carrying an irreducible representation of the angular-momentum algebra — and an induced sequence of “ghost sectors”. Then one indeed gets a Lagrangian field theory that, at least formally, can be expressed in a way similar to the Dirac theory. In flat spacetime, one gets plane-wave solutions that are characterized by their values in the main sector. Besides symmetric spinors, the above procedures can be adapted to anti-symmetric spinors and to Hermitian spinors (the latter describing integer-spin fields). Through natural decompositions, the case of a spin-2 field describing a possible deformation of the spacetime metric can be treated in terms of the previous results.