5.7.2 The Spin of the Graviton

Typically the carrier space D is organized in orbits, under the action of G, whose little groups are different non-isomorphic subgroups of the G. In the case of the Lorentz group SO(1, m − 1) and its fundamental m-dimensional representation the relevant orbits are three corresponding to the time-like, space-like and null-like vectors. The corresponding little groups are Gtime = SO(m), Gspace = SO(1, m − 1) and Gnull = SO(m − 2). Therefore the degrees of freedom of the graviton correspond to the following irreducible representation of the little group SO(m − 2):

where the hat denotes the traceless nature of the symmetric tensor. The number of

independent components of such an irreducible representation is the number m(m−3)

2

quoted above. Given its relevance in modern multi-dimensional gravity theories and specifically in supergravities that are the low energy limits of superstring models, in Table 5.1 we list the number of graviton degrees of freedom for all space-time dimensions between D = 11 and D = 3.

In D = 3 the graviton disappears. There is no room for its propagation since there is no residual transverse group. Gravity in three space-time dimension allows only for the Newtonian potential and there are no local propagating degrees of freedom. In D = 4 the transverse group is SO(2), whose unitary irreducible representations are all two-dimensional, as we explained in Sect. 1.6.1. For this reason the graviton has the same number of degrees of freedom as the photon or a Yang-Mills gauge boson, namely 2. Yet although all two-dimensional, the irreducible representations of SO(2) are not all equivalent. According to the discussion of Sect. 1.6.1 they are characterized by an half integer number s that we name the spin of the corresponding particle. In the case of the graviton, just as for the photon, s is integer, but it is s = 2 rather than s = 1. Physically this means that we have two states, one of helicity s = 2 and one of helicity s = −2. The two states are pictorially described in Fig. 5.5.