Function of the eld ie the total number of derivatives

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Unformatted text preview: conventions, the Lagrangian is a polynomial in the length parameter ‘). A weaker form of locality is the requirement that the Lagrangian L is perturbatively local in the sense that it admits a power series expansion in the fields and all their derivatives (so, in our conventions, each vertex must admit a power series expansion in the length scale ‘). Strictly speaking, this weak form of locality is rather a mild form of nonlocality because it is obviously not equivalent to the genuine requirement of locality. Nevertheless, it guarantees that somehow the nonlocality (if any) is under control: at each order in the length scale, the theory is local; the bound on the total number of partial derivatives is controlled by the power of ‘. Concretely, this means that there is no strong nonlocality (such as inverse powers of the Laplacian) and that, perturbatively, it can be treated as a local theory. Effective Lagrangians provide standard examples of perturbatively local theories. We note that if at the cubic level one forgoes the assumption of perturbative locality, then the higher-spin gravitational minimal coupling around flat space would become automatically consistent. Remember that, in the early attempts to minimally couple higher-spin particles around flat space (Berends et al., 1979; Aragone and Deser, 1980; Aragone and La Roche, 1982), the problem was that the higher-spin variation of the cubic Lagrangian creates terms " Smin $ R " Á ðW@’ þ @W’Þ proportional to the spin-two linearized Weyl tensor W , where " is the higher-spin gauge parameter. Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . 998 These terms cannot be compensated by an appropriate local gauge transformation for the spin-two field, since the linearized Weyl tensor (or its symmetrized and traceless derivative) does not vanish on shell. However, if one deals with wildly nonlocal operators and inserts the formal object ‘‘h=h’’ in front of the Weyl tensor, one can compensate R 1 1 the terms " Á ðh hW@’ þ @ h hW’Þ by appropriate nonlocal spin-two gauge transformations of the form h $ 12 h @ ð"@’ þ @"’Þ, using the fact that, contrary to the Weyl tensor, the d’Alembertian of the Weyl tensor is proportional to the field equations for the spin-two field. Schematically, hW $ @C where C denotes the (linearized) Cotton tensor which is itself a linear combination of the curl of the (linearized) Einstein tensor. (ii) Higher-spin vertices are higher derivative The higher-derivative property has been observed in all known examples of higher-spin cubic couplings. A summary of the general situation at the cubic level and in flat space is as follows. Cubic interactions (Metsaev, 2006): In flat space, the total number n of derivatives in any consistent local cubic vertex of type s-s0 -s00 (with s s0 s00 ) is bounded by s0 þ s00 À s n s þ s0 þ s00 : Therefore, the ve...
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This document was uploaded on 09/28/2013.

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