Abstract
Lagrangians for massive, unconstrained, higher-spin bosons and fermions are proposed. The idea is to modify the geometric, gauge invariant Lagrangians describing the corresponding massless theories by the addition of suitable quadratic polynomials. These polynomials provide generalisations of the Fierz–Pauli mass term containing all possible traces of the basic field. No auxiliary fields are needed.
Highlights
The central object in the theories of massless spin-1 and spin-2 fields is the curvature
Μs, to which we shall restrict our attention in this work, the conditions to be met by these fields in order to describe the free propagation of massive, irreducible representations of the Poincare group are contained in the Fierz systems [16]: ( − m2) φ μ1 ... μs = 0, ∂ αφ α μ2 ... μs = 0, φ α α μ3 ... μs = 0
To the best of our knowledge, this is the first description of massive higher-spin theories which does not involve any auxiliary fields
Summary
The central object in the theories of massless spin-1 and spin-2 fields is the curvature. Μs−3, introduced in that context in order to eliminate the non-localities of the irreducible formulation Since this field represents a pure-gauge contribution to the equations of motion, needed to guarantee invariance under a wider symmetry than the constrained one of (1.16), the corresponding higher-derivative terms, they could in general be the source of problems, both at the classical and at the quantum level, should not interfere in this case with the physical content of the theory itself. As a matter of principle, all of the infinitely many geometric Lagrangians describing the same free dynamics, being built from divergenceless Einstein tensors, are equivalently amenable to the quadratic deformation by means of the generalised Fierz-Pauli mass terms given in (2.99) and (3.66) In this sense, an issue of uniqueness is present at the massive level as well, as discussed, and the analysis of the current exchange in this case is not sufficient to provide a selection principle, as showed in the same Section by the example of spin 4.
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