Abstract
Spinning particle models can be used to describe higher spin fields in first quantization. In this paper we discuss how spinning particles with gauged O(N) supersymmetries on the worldline can be consistently coupled to conformally flat spacetimes, both at the classical and at the quantum level. In particular, we consider canonical quantization on flat and on (A)dS backgrounds, and discuss in detail how the constraints due to the worldline gauge symmetries produce geometrical equations for higher spin fields, i.e. equations written in terms of generalized curvatures. On flat space the algebra of constraints is linear, and one can integrate part of the constraints by introducing gauge potentials. This way the equivalence of the geometrical formulation with the standard formulation in terms of gauge potentials is made manifest. On (A)dS backgrounds the algebra of constraints becomes quadratic, nevertheless one can use it to extend much of the previous analysis to this case. In particular, we derive general formulas for expressing the curvatures in terms of gauge potentials and discuss explicitly the cases of spin 2, 3 and 4.
Highlights
We consider canonical quantization on flat and on (A)dS backgrounds, and discuss in detail how the constraints due to the worldline gauge symmetries produce geometrical equations for higher spin fields, i.e. equations written in terms of generalized curvatures
After a review of the model, we have shown how these spinning particles can be coupled to conformally flat spaces, both classically and quantum mechanically, extending the result of [8], where the coupling to (A)dS spaces was obtained at the classical level
We have shown that in flat space these equations reproduce the so-called geometrical equations for higher spin curvatures
Summary
(i) it is symmetric under exchanges of the s blocks, antisymmetric in the d indices of each block, traceless, and satisfies the algebraic Bianchi identities (J constraints); this part is summarized by saying that the tensor R is an irreducible representation of the Lorentz group specified by the Young tableau with d rows and s columns. The curvature R that solves these constraints has “s” symmetric blocks of “d” antisymmetric indices each, and satisfies the algebraic Bianchi identities. This proves symmetry under exchange of the block relative to the fermions ψI with the block relative to the fermion ψJ As these transformations are connected to the identity, they are obtained by exponentiating the infinitesimal generators used in (4.15), so that this symmetry must be a consequence of (4.15), i.e. of the algebraic Bianchi identities.
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