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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Rarita-Schwinger equation ： ウィキペディア英語版 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:〔S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335〕 :$\backslash left\; (\; \backslash epsilon^\; \backslash gamma\_5\; \backslash gamma\_\backslash kappa\; \backslash partial\_\backslash rho\; -\; i\; m\; \backslash sigma^\; \backslash right)\backslash psi\_\backslash nu\; =\; 0$ where $\backslash epsilon^$ is the Levi-Civita symbol, $\backslash gamma\_5$ and $\backslash gamma\_\backslash nu$ are Dirac matrices, $m$ is the mass, $\backslash sigma^\; \backslash equiv\; \backslash frac\; ()$, and $\backslash psi\_\backslash nu$ is a vector-valued spinor with additional components compared to the four component spinor in the Dirac equation. It corresponds to the representation of the Lorentz group, or rather, its part.〔S. Weinberg, "The quantum theory of fields", Vol. 1, Cambridge p. 232〕 This field equation can be derived as the Euler–Lagrange equation corresponding to the Rarita–Schwinger Lagrangian:〔S. Weinberg, "The quantum theory of fields", Vol. 3, Cambridge p. 335〕 :$\backslash mathcal=-\backslash tfrac\backslash ;\backslash bar\_\backslash mu\; \backslash left\; (\; \backslash epsilon^\; \backslash gamma\_5\; \backslash gamma\_\backslash kappa\; \backslash partial\_\backslash rho\; -\; i\; m\; \backslash sigma^\; \backslash right)\backslash psi\_\backslash nu$ where the bar above denotes the Dirac adjoint. This equation controls the propagation of the wave function of composite objects such as the delta baryons () 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: is invariant under the gauge transformation $\backslash psi\_\backslash mu\; \backslash rightarrow\; \backslash psi\_\backslash mu\; +\; \backslash partial\_\backslash mu\; \backslash epsilon$, where $\backslash epsilon\backslash equiv\; \backslash epsilon\_\backslash alpha$ is an arbitrary spinor field. "Weyl" and "Majorana" versions of the Rarita–Schwinger equation also exist. ==Equations of motion in the massless case== Consider a massless Rarita-Schwinger field described by the Lagrangian density :$\backslash mathcal\; L\_\; =\; \backslash bar\; \backslash psi\_\backslash mu\; \backslash gamma^\; \backslash partial\_\backslash nu\; \backslash psi\_\backslash rho,$ where the sum over spin indices is implicit, $\backslash psi\_\backslash mu$ are Majorana spinors, and :$\backslash gamma^\; \backslash equiv\; \backslash frac\; \backslash gamma^\backslash gamma^\backslash nu"\; TITLE="\backslash mu\}\backslash gamma^\backslash nu">\backslash gamma^.$ To obtain the equations of motion we vary the Lagrangian with respect to the fields $\backslash psi\_\backslash mu$, obtaining: :$\backslash delta\; \backslash mathcal\; L\_\; =$ \delta \bar \psi_\mu \gamma^ \partial_\nu \psi_\rho + \bar \psi_\mu \gamma^ \partial_\nu \delta \psi_\rho = \delta \bar \psi_\mu \gamma^ \partial_\nu \psi_\rho - \partial_\nu \bar \psi_\mu \gamma^ \delta \psi_\rho + \text using the Majorana flip properties〔Pierre Ramond - Field theory, a Modern Primer - p.40〕 we see that the second and first terms on the RHS are equal, concluding that :$\backslash delta\; \backslash mathcal\; L\_\; =\; 2\; \backslash delta\; \backslash bar\; \backslash psi\_\backslash mu\; \backslash gamma^\; \backslash partial\_\backslash nu\; \backslash psi\_\backslash rho,$ plus unimportant boundary terms. Imposing $\backslash delta\; \backslash mathcal\; L\_\; =\; 0$ we thus see that the equation of motion for a massless Majorana Rarita-Schwinger spinor reads: :$\backslash gamma^\; \backslash partial\_\backslash nu\; \backslash psi\_\backslash rho\; =\; 0.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rarita–Schwinger equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.