# A first-order Lagrangian theory of fields with arbitrary spin

## TL;DR

This paper develops a first-order Lagrangian framework for higher-spin fields using 2-spinor bundles, providing a geometric and algebraic foundation for describing fields of arbitrary spin on curved spacetime.

## Contribution

It introduces a novel first-order Lagrangian theory for arbitrary spin fields based on 2-spinor bundles, extending geometric and algebraic methods for higher-spin field descriptions.

## Key findings

- Recovering Bargmann-Wigner equations within the framework
- Formulating a Dirac-like Lagrangian for higher-spin fields
- Applying the theory to spin-2 fields and potential spacetime deformations

## 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 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 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.