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 2(2j + 1)-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 angularmomentum 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 characterised 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.MSC 2010: 81R20, 81R25.

On the Notions of Energy Tensors in Tetrad-Affine Gravity

“…For matter fields of either integer or semi-integer spin greater than one-half one may wish to consider an appropriate specialized setting, leading to possible generalizations of the Dirac equation [45,46,47]. However, issues about the Lagrangian treatment of such setting suggest that we provisionally confine ourselves to the Lagrangian (8) for all matter fields of spin different from one-half.…”

Gauge field theory without groups

“…Besides the aforementioned book, details about this subject can be found in a dedicated paper[21] 17. In an even more extended theory, the fermion bundle can be written as (ER ⊗ U ) ⊕ (EL ⊗ U ⋆ ) 18.…”

“…See also the dedicated paper[19] for the details, that lie ouside the scope of this presentation 20. We need not be concerned with duality in the most general acceptation 21. This setting suffices if one is not concerned with completions, namely with infinite sums: our multi-particle states only contain finitely many particles, though their number can be arbitrarily large.…”