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::$$epsilon^{mu u ho sigma} gamma^5 gamma_ u partial_ ho psi_sigma + mpsi^mu = 0where $$epsilon^{mu u ho sigma} is the Levi-Civita symbol, $$gamma^5 and $$gamma_ u are Dirac matrices, $$m is the mass and $$psi_mu is a vector-valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the $$left( frac{1}{2}, frac{1}{2} ight)otimes left(left( frac{1}{2},0 ight)oplus left(0, frac{1}{2} ight) ight) representation of the Lorentz group, or rather, its $$left(1, frac{1}{2} ight) oplus left( frac{1}{2},1 ight) part.This field equation can be derived from the following Lagrangian::$$mathcal{L}= frac{1}{2} epsilon^{mu u ho sigma} ar{psi}_mu gamma^5 gamma_ u partial_ ho psi_sigma - m ar{psi}_mu gamma^{mu u}psi_ uwhere the bar above $$psi_mu 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 ightarrow D_mu = partial_mu - i e A_mu

The massless Rarita-Schwinger equation has a gauge symmetry, under the gauge transformation of $$psi_mu ightarrow psi_mu + partial_mu epsilon, where $$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, " [http://prola.aps.org/abstract/PR/v60/i1/p61_1 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).