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:
 * $$ \left ( \epsilon^{\mu \kappa \rho \nu} \gamma_5 \gamma_\kappa \partial_\rho - i m \sigma^{\mu \nu} \right)\psi_\nu = 0 $$

where $$\epsilon^{\mu\kappa\rho\nu}$$ is the Levi-Civita symbol, $$\gamma_\kappa$$ are Dirac matrices (with $$\kappa=0,1,2,3$$) and $$\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3$$, $$m$$ is the mass, $$\sigma^{\mu\nu} \equiv \frac{i}{2} [\gamma^\mu,\gamma^\nu] $$, and $$\psi_\nu$$ 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:
 * $$\mathcal{L}=-\tfrac{1}{2}\;\bar{\psi}_\mu \left( \epsilon^{\mu \kappa \rho \nu} \gamma_5 \gamma_\kappa \partial_\rho - i m \sigma^{\mu \nu} \right)\psi_\nu$$

where the bar above $$\psi_\mu$$ denotes the Dirac adjoint.

This equation controls the propagation of the wave function of composite objects such as the delta baryons 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 $$\psi_\mu \rightarrow \psi_\mu + \partial_\mu \epsilon$$, where $$\epsilon\equiv \epsilon_\alpha$$ 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.

Equations of motion in the massless case
Consider a massless Rarita–Schwinger field described by the Lagrangian density
 * $$ \mathcal L_{RS} = \bar \psi_\mu \gamma^{\mu\nu\rho} \partial_\nu \psi_\rho,$$

where the sum over spin indices is implicit, $$\psi_\mu$$ are Majorana spinors, and
 * $$ \gamma^{\mu\nu\rho} \equiv \frac{1}{3!} \gamma^{[\mu}\gamma^\nu \gamma^{\rho]}. $$

To obtain the equations of motion we vary the Lagrangian with respect to the fields $$\psi_\mu$$, obtaining:
 * $$ \delta \mathcal L_{RS} =

\delta \bar \psi_\mu \gamma^{\mu\nu\rho} \partial_\nu \psi_\rho + \bar \psi_\mu \gamma^{\mu\nu\rho} \partial_\nu \delta \psi_\rho = \delta \bar \psi_\mu \gamma^{\mu\nu\rho} \partial_\nu \psi_\rho - \partial_\nu \bar \psi_\mu \gamma^{\mu\nu\rho} \delta \psi_\rho + \text{ boundary terms} $$ using the Majorana flip properties we see that the second and first terms on the RHS are equal, concluding that
 * $$ \delta \mathcal L_{RS} = 2 \delta \bar \psi_\mu \gamma^{\mu\nu\rho} \partial_\nu \psi_\rho, $$

plus unimportant boundary terms. Imposing $$ \delta \mathcal L_{RS} = 0$$ we thus see that the equation of motion for a massless Majorana Rarita–Schwinger spinor reads:
 * $$ \gamma^{\mu\nu\rho} \partial_\nu \psi_\rho = 0. $$

The gauge symmetry of the massless Rarita-Schwinger equation allows the choice of the gauge $$\gamma^\mu \psi_\mu = 0$$, reducing the equations to:

\gamma^\nu{\partial}_\nu \psi_\mu = 0, \quad \partial^\mu \psi_\mu = 0, \quad \gamma^\mu \psi_\mu = 0. $$ A solution with spins 1/2 and 3/2 is given by:

\psi_0=\kappa, \quad \psi_i = \psi^{TT}_i + \frac{\gamma^j\partial_j}{\nabla^2}\gamma_0\partial_i\kappa, $$ where $$\nabla^2 $$ is the spatial Laplacian, $$ \psi^{TT}_i$$ is doubly transverse, carrying spin 3/2, and $$\kappa$$ satisfies the massless Dirac equation, therefore carrying spin 1/2.

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:
 * $$\partial_\mu \rightarrow D_\mu = \partial_\mu - i e A_\mu $$.

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.