Rarita–Schwinger equation
Field equation for spin-3/2 fermions / From Wikipedia, the free encyclopedia
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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:[1]
where is the Levi-Civita symbol, are Dirac matrices (with ) and , is the mass, , and 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.[2]
This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:[3]
where the bar above 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 , where 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.