Rarita–Schwinger equation

In theoretical physics, the Rarita–Schwinger equation is the relativistic field equation of spin-3/2 fermions in a four-dimensional flat spacetime. 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, ${\displaystyle \gamma _{\kappa }}$ are Dirac matrices (with ${\displaystyle \kappa =0,1,2,3}$) and ${\displaystyle \gamma _{5}=i\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$, ${\displaystyle m}$ 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.