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# Rarita-Schwinger equation

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:

$\epsilon^{\mu \nu \rho \sigma} \gamma^5 \gamma_\nu \partial_\rho \psi_\sigma + m\psi^\mu = 0$

where εμνρσ is the Levi-Civita symbol, γ5 and γν are Dirac matrices, m is the mass and ψμ is a vector-valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the $\left(\frac{1}{2},\frac{1}{2}\right)\otimes \left(\left(\frac{1}{2},0\right)\oplus \left(0,\frac{1}{2}\right)\right)$ representation of the Lorentz group, or rather, its $\left(1,\frac{1}{2}\right) \oplus \left(\frac{1}{2},1 \right)$ part. This field equation can be derived from the following Lagrangian:

$\mathcal{L}=\frac{1}{2} \epsilon^{\mu \nu \rho \sigma} \bar{\psi}_\mu \gamma^5 \gamma_\nu \partial_\rho \psi_\sigma - m \bar{\psi}_\mu \psi^\mu$

where the bar above ψμ denotes the Dirac adjoint. As in the case of the Dirac equation, electromagnetic interaction can be added simply by promoting the partial derivative to gauge covariant derivative:

$\partial_\mu \rightarrow D_\mu = \partial_\mu - i e A_\mu$

The massless Rarita-Schwinger equation has a gauge symmetry, under the gauge transformation of $\psi_\mu \rightarrow \psi_\mu + \partial_\mu \epsilon$, where $\mathcal{\epsilon}$ is an arbitrary spinor field.

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

This equation is useful for the wave function of composite objects like Delta (Δ) baryons or for proposed elementary fields like the gravitino. So far, no fundamental particle with spin 3/2 has been found experimentally.

## Drawbacks of the formalism

The current description of massive, higher spin fields through either Rarita-Schwinger or Fierz–Pauli formalisms is afflicted with several maladies. Upon gauging, high spin fields suffer from acausal, superluminal propagation; besides, the quantization of these systems in interaction with electromagnetism is essentially flawed. Also, algebraic inconsistencies appear upon gauging which can only be avoided by requiring that all equations involving derivatives be obtainable from a Lagrangian, a procedure that becomes involved because of the need of introducing auxiliary fields in order to obtain all constraints from the Lagrangian.

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. There are some Lorentz frames that allow the consistent formulation of quantum mechanics and quantum field theory; there are some others that don’t.

## References

• W. Rarita and J. Schwinger, On a Theory of Particles with Half-Integral Spin Phys. Rev. 60, 61 (1941).
• Collins P.D.B., Martin A.D., Squires E.J., Particle physics and cosmology (1989) Wiley, Section 1.6.
• G. Velo, D. Zwanziger, Phys. Rev. 186, 1337 (1969).
• G. Velo, D. Zwanziger, Phys. Rev. 188, 2218 (1969).
• M. Kobayashi, A. Shamaly, Phys. Rev. D 17,8, 2179 (1978).