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| In [[theoretical physics]], the '''Rarita–Schwinger equation''' is the
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| [[theory of relativity|relativistic]] [[field equation]] of [[spin (physics)|spin]]-3/2 [[fermion]]s. 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:<ref>S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335</ref>
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| :<math> \left ( \epsilon^{\mu \nu \rho \sigma} \gamma_5 \gamma_\nu \partial_\rho - i m \sigma^{\mu \sigma} \right)\psi_\sigma = 0</math>
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| where <math> \epsilon^{\mu \nu \rho \sigma}</math> is the [[Levi-Civita symbol]], <math>\gamma_5</math> and <math>\gamma_\nu</math> are [[Dirac matrices]], <math>m</math> is the mass, <math>\sigma^{\mu \nu} \equiv i/2\left [ \gamma^\mu , \gamma^\nu \right ]</math>, and <math>\psi_\sigma</math> is a vector-valued [[spinor]] with additional components compared to the four component spinor in the Dirac equation. It corresponds to the <math>\left(\tfrac{1}{2},\tfrac{1}{2}\right)\otimes \left(\left(\tfrac{1}{2},0\right)\oplus \left(0,\tfrac{1}{2}\right)\right)</math> [[Representations of the Lorentz group|representation of the Lorentz group]], or rather, its <math>\left(1,\tfrac{1}{2}\right) \oplus \left(\tfrac{1}{2},1 \right)</math> part.<ref>S. Weinberg, "The quantum theory of fields", Vol. 1, Cambridge p. 232</ref>
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| This field equation can be derived as the [[Euler–Lagrange equation]] corresponding to the Rarita-Schwinger [[Lagrangian]]:<ref>S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335</ref>
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| :<math>\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</math> | |
| where the bar above <math>\psi_\mu</math> denotes the [[Dirac adjoint]].
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| This equation is useful for the [[wave function]] of composite objects such as the [[delta baryon]]s ({{SubatomicParticle|Delta}}) or for the hypothetical [[gravitino]]. So far, no [[elementary particle]] with spin 3/2 has been found experimentally.
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| The massless Rarita–Schwinger equation has a gauge symmetry, under the gauge transformation of <math>\psi_\mu \rightarrow \psi_\mu + \partial_\mu \epsilon</math>, where <math>\mathcal{\epsilon}</math> is an arbitrary spinor field.
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| "Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist.
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| ==Drawbacks of the equation==
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| The current description of massive, higher spin fields through either Rarita-Schwinger or [[Fierz–Pauli equation|Fierz-Pauli]] formalisms is afflicted with several maladies.
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| ===Superluminal propagation===
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| As in the case of the Dirac equation, electromagnetic interaction can be added by promoting the partial derivative to [[gauge covariant derivative]]:
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| :<math>\partial_\mu \rightarrow D_\mu = \partial_\mu - i e A_\mu </math>. | |
| 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,
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| the field then suffers from acausal, superluminal propagation; consequently, the [[Quantization (physics)|quantization]] in interaction with electromagnetism is essentially flawed.
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| ==References==
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| <references/>
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| ==Notes==
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| * 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).
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| * Collins P.D.B., Martin A.D., Squires E.J., ''Particle physics and cosmology'' (1989) Wiley, ''Section 1.6''.
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| * G. Velo, D. Zwanziger, ''Propagation and Quantization of Rarita-Schwinger Waves in an External Electromagnetic Potential'', Phys. Rev. 186, 1337 (1969).
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| * G. Velo, D. Zwanziger, ''Noncausality and Other Defects of Interaction Lagrangians for Particles with Spin One and Higher'', Phys. Rev. 188, 2218 (1969).
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| * M. Kobayashi, A. Shamaly, ''Minimal Electromagnetic coupling for massive spin-two fields'', Phys. Rev. D 17,8, 2179 (1978).
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| {{DEFAULTSORT:Rarita-Schwinger equation}}
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| [[Category:Quantum field theory]]
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| [[Category:Spinors]]
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| [[Category:Partial differential equations]]
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| [[Category:Fermions]]
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Also visit my webpage :: riley steele (corta.co)