# Rarita-Schwinger equation

In

theoretical physics , the**Rarita-Schwinger equation**is the

relativisticfield equation of spin-3/2fermion s. It is similar to theDirac equation for spin-1/2 fermions. This equation was first introduced byWilliam Rarita andJulian Schwinger in1941 . In modern notation it can be written as::$epsilon^\{mu\; u\; ho\; sigma\}\; gamma^5\; gamma\_\; u\; partial\_\; ho\; psi\_sigma\; +\; mpsi^mu\; =\; 0$where $epsilon^\{mu\; u\; ho\; sigma\}$ is theLevi-Civita symbol , $gamma^5$ and $gamma\_\; u$ areDirac matrices , $m$ is the mass and $psi\_mu$ is a vector-valuedspinor 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 followingLagrangian ::$mathcal\{L\}=\; frac\{1\}\{2\}\; epsilon^\{mu\; u\; ho\; sigma\}\; ar\{psi\}\_mu\; gamma^5\; gamma\_\; u\; partial\_\; ho\; psi\_sigma\; -\; m\; ar\{psi\}\_mu\; gamma^\{mu\; u\}psi\_\; u$where the bar above $psi\_mu$ denotes theDirac adjoint .As in the case of the Dirac equation, electromagnetic interaction can be added simply by promoting the partial derivative togauge 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)baryon s or for proposed elementary fields like thegravitino . So far, no fundamental particle with spin 3/2 has been foundexperiment ally.**Drawbacks of the formalism**The current description of massive, higher spin fields through either

Rarita-Schwinger orFierz–Pauli formalisms is afflicted with several maladies. Upon gauging, high spin fields suffer from acausal, superluminal propagation; besides, thequantization 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 someLorentz frames that allow the consistent formulation ofquantum mechanics andquantum 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).

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