Rarita-Schwinger action

In theoretical physics, the Rarita–Schwinger equation is the relativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinger in 1941.

In modern notation it can be written as:

where ${\displaystyle \epsilon ^{\mu \kappa \rho \nu }}$ is the Levi-Civita symbol, ${\displaystyle \gamma _{5}}$ and ${\displaystyle \gamma _{\nu }}$ are Dirac matrices, ${\displaystyle m}$ is the mass, ${\displaystyle \sigma ^{\mu \nu }\equiv {\frac {i}{2}}[\gamma ^{\mu },\gamma ^{\nu }]}$, and ${\displaystyle \psi _{\nu }}$ is a vector-valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the (1/2, 1/2) ⊗ ((1/2, 0) ⊕ (0, 1/2)) representation of the Lorentz group, or rather, its (1, 1/2) ⊕ (1/2, 1) part.

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