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.
Contents
In modern notation it can be written as:
where
This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:
where the bar above
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
"Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
Equations of motion in the massless case
Consider a massless Rarita-Schwinger field described by the Lagrangian density
where the sum over spin indices is implicit,
To obtain the equations of motion we vary the Lagrangian with respect to the fields
using the Majorana flip properties we see that the second and first terms on the RHS are equal, concluding that
plus unimportant boundary terms. Imposing
Drawbacks of the equation
The current description of massive, higher spin fields through either Rarita–Schwinger or Fierz–Pauli formalisms is afflicted with several maladies.
Superluminal propagation
As in the case of the Dirac equation, electromagnetic interaction can be added by promoting the partial derivative to gauge covariant derivative:
In 1969, Velo and Zwanziger showed that the Rarita–Schwinger Lagrangian coupled to electromagnetism leads to equation with solutions representing wavefronts, some of which propagate faster than light. In other words, the field then suffers from acausal, superluminal propagation; consequently, the quantization in interaction with electromagnetism is essentially flawed. In extended supergravity, though, Das and Freedman have shown that local supersymmetry solves this problem.