Zero Rest Mass Equation

Spinor fields describing particles of zero rest mass satisfy the so-called zero rest mass equations. Examples of zero rest mass particles include the neutrino (a fermion) and the gauge bosons (as long as gauge symmetry is not violated) such as the photon.

If  is the spinor field describing a particle of spin  (where upper case Latin indices are spinor indices which can take the values 0 and 1), then it is symmetric and has  indices. If the particle is also of zero rest mass, then  satisfies the zero rest mass equation

Here, in a Lorentz transformation, primed spinors transform under the conjugate of the transformation for unprimed ones, Einstein summation is used throughout, and  denotes the spinor, which is equivalent to the Levi-Civita connection on Minkowski space.

 has one index for the neutrino, two for the photon, and four for the graviton. For the photon, the equation obtained states the vanishing of the divergence of the field strength tensor. For the graviton, it gives the Bianchi identity for a linearized Weyl tensor.