See sec iib2 in the lagrangian approach the same

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Unformatted text preview: oach, the same result had already been obtained in various particular instances, where it was shown that the Lorentz minimal coupling prescription applied to free higher-spin gauge fields enters in conflict with their Abelian gauge symmetries (Aragone and Deser, 1979; Berends et al., 1979; Aragone and La Roche, 1982; Boulanger and Leclercq, 2006). The complete no-go result ruling out the Lorentz minimal coupling of type 2-s-s in the Lagrangian approach is given in Boulanger, Leclercq, and Sundell (2008). In between the Lagrangian and the S-matrix approaches lies the light-cone approach where all local cubic vertices in dimensions from four to six have been classified [see, e.g., Metsaev (2006) and references therein] and where the same negative conclusions concerning the Lorentz minimal coupling of higher-spin gauge fields to gravity had already been reached and stated in complete generality. 12 For a pedagogical essay, see, e.g., Loebbert (2008). Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 This being said, consistent cubic vertices between spin-two and higher-spin gauge fields do exist, even in Minkowski spacetime (Boulanger and Leclercq, 2006; Metsaev, 2006; Boulanger, Leclercq, and Sundell, 2008). Instead of describing Lorentz’s minimal coupling, they contain more than two derivatives in total. As one can see, the generalized WeinbergWitten theorem does not by itself forbid such type 2-s-s interactions. The crux of the matter is to combine this theorem with the Weinberg equivalence principle. Together, the Weinberg equivalence principle and the generalized Weinberg-Witten theorem prohibit the cross couplings of massless higher-spin particles with low-spin particles in flat spacetime (Porrati, 2008). The argument goes as follows: elementary particles with spin not greater than 2 are known to couple minimally to the graviton at low energy. Therefore (Weinberg’s equivalence principle) all particles interacting with low-spin particles must also couple minimally to the graviton at low energy, but [generalized Weinberg-Witten theorem (Porrati, 2008) and identical results presented in Metsaev (2006) and Boulanger, Leclercq, and Sundell (2008)] massless higher-spin particles cannot couple minimally to gravity around the flat background. Consequently, at low energies massless higher-spin particles must completely decouple from low-spin ones. Hence, if the same Lagrangian can be used to describe both the low-energy phenomenology and the Planck-scale physics, then no higherspin particles can couple to low-spin particles (including spin two) at all. E. Velo-Zwanziger difficulties In this section, we stress that, contrary to widespread prejudice, the Velo-Zwanziger difficulties do not constitute a serious obstruction to the general program of constructing consistent interactions involving higher-spin fields. The observed pathologies are nothing but symptoms of nonintegrability in the sense of Cartan of the differential equations under...
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