**Unformatted text preview: **atar: No higher-spin
conserved charges The Coleman-Mandula theorem (Coleman and Mandula,
1967) and its generalization to the case of supersymmetric
theories with or without massless particles given by Haag,
Lopuszanski, and Sohnius (1975) strongly restrict the symmetries of the S matrix of an interacting relativistic ﬁeld
theory in four-dimensional Minkowski spacetime.11 More
precisely, (i) if the elastic two-body scattering amplitudes
are generically nonvanishing (at almost all energies and
angles), and (ii) if there is only a ﬁnite number of particle
11 For an extended pedagogical review, see Weinberg (2000),
Chapter 24. 992 Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . species on and below any given mass shell, then the maximal
´
possible extension of the Poincare algebra is the (semi)direct
sum of a superalgebra (a superconformal algebra in the
massless case) and an internal symmetry algebra spanned
by elements that commute with the generators of the
´
Poincare algebra.
In particular, this theorem rules out higher symmetry generators (equivalently, conserved charges) that could have
come from higher-spin symmetries surviving at large distances. The argument goes as follows: the gauge symmetries
associated with massless particles may survive at spatial
inﬁnity as nontrivial rigid symmetries. In turn, such symmetries should lead to the conservation of some asymptotic
charges. Under the hypotheses of the generalized ColemanMandula theorem, nontrivial conserved charges associated
with asymptotic higher-spin symmetries cannot exist.
This corollary of the generalized Coleman-Mandula theorem partially overlaps with the Weinberg low-energy theorem
because the conservation law (1) precisely corresponds to the
existence of a conserved charge Q1 ÁÁÁsÀ1 which is a symmetric tensor of rank s À 1 that commutes with the translations, but does not commute with the Lorentz generators.
D. Generalized Weinberg-Witten theorem The Weinberg-Witten theorem (Weinberg and Witten,
1980) states that a massless particle of spin strictly greater
than 1 cannot possess an energy-momentum tensor T
which is both Lorentz covariant and gauge invariant.12 Of
course, this no-go theorem does not preclude gravitational
interactions. In the spin-two case, it implies that there cannot
exist any gauge-invariant energy-momentum tensor for the
graviton. This proves that the energy of the gravitational ﬁeld
cannot be localized, but it obviously does not prevent the
graviton from interacting with matter or with itself.
Recently, a reﬁnement of the Weinberg-Witten theorem
was presented (Porrati, 2008) that genuinely prevents massless particles of spin strictly greater than 2 from coupling
minimally to the graviton in ﬂat background. The minimality
condition is stated according to the Weinberg equivalence
principle, namely, it refers to Lorentz minimal spin-two
coupling (see Sec. II.B.2). In the Lagrangian appr...

View
Full
Document