Spinor field

In particle physics, a spinor field of order 2s describes a particle of spin s, where s is an integer or half-integer. Therefore, a spinor of order 4s contains as much information as a tensor of order 2s. 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.

Zero rest mass equation

Spinor fields describing particles of zero rest mass satisfy the so-called zero rest mass equation. 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 $phi^\{AB...E\}$ is the spinor field describing a particle of spin s (where upper case Latin indices are spinor indices which can take the values 0 and 1), then it is symmetric and has 2s indices. If the particle is also of zero rest mass, then $phi^\{AB...E\}$ satisfies the zero rest mass equation

:$abla\_\{A^\text{'}\}^Aphi^\{AB...E\}=0$

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

$phi$ 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.

References

* Said, Salem. "Spinor Field." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/SpinorField.html
* Said, Salem. "Zero Rest Mass Equation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/ZeroRestMassEquation.html