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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 equationHere, 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...

The twistor equation states thatwhere the parentheses denote symmetrization, 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 connection, which is equivalent to the Levi-Civita connection on Minkowski space. The zero rest mass equation can be solved by twistor functions. The solution uses ideas from complex variable theory and cohomology.

In particle physics, a spinor field of order describes a particle of spin , where is an integer or half-integer. Therefore, a spinor of order contains as much information as a tensor of order . As a result of this, particles of integer spin (bosons) can be described equally well by tensor fields or spinor fields, whereas particles of half-integer spin (fermions) can be described only by spinor fields. Spinor fields describing particles of zero rest mass satisfy the zero rest mass equation.

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