Theorem using the relativistic normalization for one

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Unformatted text preview: get lim hþs; p þ qjT j þ s; pi ¼ p p : q! 0 (B2) This is tantamount to saying that, at low energy, the only possible coupling between gravity and everything else is done Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 via the minimal coupling procedure, bringing no more than two derivatives (or one if the spin is half-integer) in the interaction. More precisely, among all possible interaction terms there must always be that coming from minimal coupling @ ! @ þ ÀðhÞ, with the nonvanishing coefficient  related to Newton’s constant. Since q is spacelike (off-shell soft graviton), one goes in the frame in which q ¼ ð0; ÀqÞ, p ¼ ðjqj=2; q=2Þ, and p þ q ¼ ðjqj=2; Àq=2Þ (the massless spin-s particle is on shell), and deduces that a rotation RðÞ by an angle  around the q direction acts on the one-particle states as RðÞjp; þsi ¼ expðÆisÞjp; þsi, RðÞjp þ q; þsi ¼ expðÇisÞjp þ q; þsi since RðÞ is a rotation of  around p but of À around p þ q ¼ Àp. Decomposing T under space rotations in terms of spherical tensors as the complex spin zero tensor T0;0 plus the real components fT1;m g1 ¼À1 and m fT2;m g2 ¼À2 , one can write the following relation: m eÆ2is hþs; p þ qjTj;m j þ s; pi ¼ hþs; p þ qjRy Tj;m Rj þ s; pi ¼ eim hþs; p þ qjTj;m j þ s; pi (B3) which admits, for s > 1, the only solution hþs; p þ qjT j þ s; pi ¼ 0. If T is a tensor under Lorentz transformations then this implies that hþs; p þ qjT j þ s; pi ¼ 0 in all frames, in contradiction with the equivalence principle (B2). This seems to kill gravity itself, but of course in that case as usually happens in gauge theories, T is not a Lorentz tensor (which is the same as saying that T is not gauge invariant). One can define matrix elements for T that transform as Lorentz tensors only at the price of introducing nonphysical, pure-gauge states. This is what Porrati (2008) did in order to accommodate the Weinberg-Witten argument to gauge theories for spin-s fields, s > 1 and prove that massless higherspin particles cannot exist around a flat background if their tensor T appearing in hþs; p þ qjT j þ s; pi complies with the equivalence principle (B2). Denoting by v all one-particle spin-s states, whether or not spurious (pure gauge), the matrix element under consideration is denoted hv0 ; p þ qjT jv; pi. The method used by Porrati (2008) in order to derive the S matrix is to perform the standard perturbative expansion of the effective action (where g ¼  þ h ) A¼ 1 Z 4 pffiffiffiffiffiffiffiffi 1 Z d4 q ~à h ðqÞ d x ÀgR þ 16G 2 ð2Þ4   ðhv0 ; p þ qjT  jv; pi þ T  Þ þ Oðh2 Þ: (B4) The linear interaction terms include the matrix element and another effective tensor T  which summarizes the effect of any other matter field but that we omit from now on without loss of generality. To linear order, Einstein’s equations become L  h ðqÞ ¼ 16...
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This document was uploaded on 09/28/2013.

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