jk22 said:
Spin 2 representations are 5x5 matrices.
This is true if, and only if, the object which carries the spin-2 representation has
non-zero mass. In this case, the (massive)
irreducible spin-2 representation (of the Poincare group) can be represented by a
conserved,
traceless and
symmetric rank-2 tensor field: Consider the symmetric tensor field (i.e., we have
10 components) h^{\mu\nu} = h^{\nu\mu}, take its trace h = \eta_{\rho \sigma}h^{\rho\sigma} then form the following traceless symmetric tensor field G^{\mu\nu} = h^{\mu\nu} - \frac{1}{4} \eta^{\mu\nu} h . Now G^{\mu\nu} has 10 - 1= 9 components because G^{\mu\nu} = G^{\nu\mu} and \eta_{\mu\nu}G^{\mu\nu} = 0. So, to reduce the number of components to
5, we need to impose
4 more conditions: Usually, in field theory G^{\mu\nu} is generated by a
conserved source T^{\mu\nu} \ , \partial_{\mu}T^{\mu\nu} = 0, and satisfies the second-order equation ( \partial^{2} + m^{2} ) G^{\mu\nu} = \alpha T^{\mu\nu} \ . Therefore, the required
4 conditions on G^{\mu\nu} are given by \partial_{\mu}G^{\mu\nu} = 0. So, when a tensor field G^{\mu\nu} satisfies the conditions,G^{\mu\nu} = G^{\nu\mu} \ , \eta_{\mu\nu}G^{\mu\nu} = 0 and \partial_{\mu}G^{\mu\nu} = 0, we say that G^{\mu\nu} carries a
massive irreducible spin-2 representation of the Poincare group.
In the
massless case, we can show that the appropriate tensor is given by G^{\mu\nu} = h^{\mu\nu} - \frac{1}{2} \eta^{\mu\nu} h \ . Notice, in this case, that G^{\mu\nu} is still
symmetric, but
not traceless. So this tensor has
10 components. Again, we require G^{\mu\nu} to be
identically conserved, i.e., \partial_{\mu}G^{\mu\nu} = 0 (because, in this case the equation of motion has the form \partial^{2}G^{\mu\nu} = \beta T^{\mu\nu}). Therefore, the number of components of G^{\mu\nu} has been reduced to 10 - 4 = 6. However, we can show that theory is
invariant under the following “gauge” transformations h^{\mu\nu} \to h^{\mu\nu} - \partial^{\mu} \chi^{\nu} - \partial^{\nu} \chi^{\mu} \ ,where \chi^{\mu} is an arbitrary 4-vector field. This allows us to fix
4 out of the above
6 components of G^{\mu\nu}. Thus, there are
2 and only
2 independent components left in G^{\mu\nu} as it should be for
massless fields. So, the massless “spin”-2
irreducible representation of the Poincare group is carried by a
symmetric rank-2 tensor with only 2 independent components.
