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**Unformatted text preview: **the basic assumptions of canonical second quantization that led up to the notion of free ﬁelds to begin with.
Thus a satisfactory resolution seems certainly much more
demanding than even that of quantizing ordinary general
relativity (though the prolongation of the Einstein-Cartan
reformulation of general relativity as a soldered Yang-Mills
theory for the spacetime isometry algebra soon leads to
inﬁnite-dimensional algebras as well) which actually leaves
2
These constraints on massless particle scattering appear only in
spacetimes of dimension D ! 4 to which we restrict our attention in
this paper. Indeed, in dimension D 3 massless ﬁelds of helicity
s ! 2 have no local propagating degrees of freedom. Pure massless
higher-spin gravities in lower dimensions are of Chern-Simons type
which do not share most of the exotic features of their higherdimensional cousins discussed here.
3
The precise link between, on the one hand, the Fradkin-Vasiliev
cubic action and, on the other hand, the fully interacting Vasiliev
equations, remains to be found.
4
We thus leave out many other interesting features of the Vasiliev
system, such as its unfolded, or Cartan integrable, formulation, and
the link between its ﬁrst-quantization, deformed Wigner oscillators,
singletons, and compositeness of massless particles in anti–de Sitter
spacetime. Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 room for a naive optimism: the quantization of higher-spin
gauge theories could lead to a radically new view on quantum
ﬁeld theory altogether, and, in particular, on the formidable
spin-two barrier set up by the requirement of power-counting
renormalizability.
Indeed, at the classical level, there exist the aforementioned higher-spin gravities (Vasiliev, 1990, 1992, 1996,
2003; Sezgin and Sundell, 1998, 2001a,2001b, 2002a): these
are special instances of interacting higher-spin gauge theories
constituting what one may think of as the simplest possible
higher-spin extensions of general relativity. Their minimal
bosonic versions (in D ! 4 ordinary spacetime dimensions)
consist of a propagating scalar, metric, and tower of massless
ﬁelds of even spins, s ¼ 4; 6; . . . (these models can then be
extended by various forms of ‘‘matter’’ and suitable higherspin counterparts, in a supersymmetric setup in case fermions
are included).
As mentioned, a key feature of higher-spin gravity is its
double perturbative expansion: besides the expansion in
numbers of ﬁelds, weighted by a dimensionless coupling g,
there is a parallel albeit strongly coupled expansion in numbers of pairs of derivatives, weighted by a dimensionful
parameter, the cosmological constant Ã, thus serving as
both infrared and ultraviolet cutoff. Hence classical higherspin gravity prefers a nonvanishing cosmological constant,
unlike string theory in ﬂat spacetime which also has a double
perturbative expansion but with a strictly massless sector
accessible at low energies in a weakly coupled der...

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