# Rarita–Schwinger equation

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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:[1]

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

where ϵμνρσ is the Levi-Civita symbol, γ5 and γν are Dirac matrices, m is the mass, σμνi/2[γμ, γν], and ψσ 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]

$\mathcal{L}=-\tfrac{1}{2}\;\bar{\psi}_\mu \left ( \epsilon^{\mu \nu \rho \sigma} \gamma_5 \gamma_\nu \partial_\rho - i m \sigma^{\mu \sigma} \right)\psi_\sigma$

where the bar above ψμ 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 sermonic gauge symmetry, under the gauge transformation of ψμψμ + ∂με, where ε is an arbitrary spinor field.

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

## 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[4] have shown that local supersymmetry solves this problem.

## References

1. ^ S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335
2. ^ S. Weinberg, "The quantum theory of fields", Vol. 1, Cambridge p. 232
3. ^ S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335
4. ^ Das, A.; Freedman, D. Z. (1976). "Gauge quantization for spin-3/2 fields". Nuclear Physics B 114 (2): 271. doi:10.1016/0550-3213(76)90589-7.; Freedman, D. Z.; Das, A. (1977). "Gauge internal symmetry in extended supergravity". Nuclear Physics B 120 (2): 221. doi:10.1016/0550-3213(77)90041-4.

## Notes

• 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, Propagation and Quantization of Rarita–Schwinger Waves in an External Electromagnetic Potential, Phys. Rev. 186, 1337 (1969).
• G. Velo, D. Zwanziger, Noncausality and Other Defects of Interaction Lagrangians for Particles with Spin One and Higher, Phys. Rev. 188, 2218 (1969).
• M. Kobayashi, A. Shamaly, Minimal Electromagnetic coupling for massive spin-two fields, Phys. Rev. D 17,8, 2179 (1978).