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:

${\displaystyle \left(\epsilon ^{\mu \kappa \rho \nu }\gamma _{5}\gamma _{\kappa }\partial _{\rho }-im\sigma ^{\mu \nu }\right)\psi _{\nu }=0}$

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.

This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:

${\displaystyle {\mathcal {L}}=-{\tfrac {1}{2}}\;{\bar {\psi }}_{\mu }\left(\epsilon ^{\mu \kappa \rho \nu }\gamma _{5}\gamma _{\kappa }\partial _{\rho }-im\sigma ^{\mu \nu }\right)\psi _{\nu }}$

where the bar above ${\displaystyle \psi _{\mu }}$ denotes the Dirac adjoint.

This equation controls the propagation of the wave function of composite objects such as the delta baryons (
Δ
) or for the conjectural gravitino. So far, no elementary particle with spin 3/2 has been found experimentally.

The massless Rarita–Schwinger equation has a fermionic gauge symmetry: is invariant under the gauge transformation ${\displaystyle \psi _{\mu }\rightarrow \psi _{\mu }+\partial _{\mu }\epsilon }$, where ${\displaystyle \epsilon \equiv \epsilon _{\alpha }}$ is an arbitrary spinor field. This is simply the local supersymmetry of supergravity, and the field must be a gravitino.

"Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.