翻訳と辞書 Words near each other ・ Dira jansei ・ Dira oxylus ・ Dira Paes ・ Dira Sugandi ・ Dira swanepoeli ・ Dirac (disambiguation) ・ Dirac (software) ・ Dirac (video compression format) ・ Dirac adjoint ・ Dirac algebra ・ Dirac bracket ・ Dirac comb ・ Dirac delta function ・ Dirac equation ・ Dirac equation in curved spacetime ・ Dirac equation in the algebra of physical space ・ Dirac fermion ・ Dirac hole theory ・ Dirac large numbers hypothesis ・ Dirac measure ・ Dirac operator ・ Dirac Prize ・ Dirac sea ・ Dirac spectrum ・ Dirac spinor ・ Dirac string ・ Dirac's theorem ・ Dirac, Charente ・ Dirachma ・ Dirac–von Neumann axioms Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dirac equation in the algebra of physical space ： ウィキペディア英語版 Dirac equation in the algebra of physical space The Dirac equation, as the relativistic equation that describes spin 1/2 particles in quantum mechanics can be written in terms of the Algebra of physical space (APS), which is a case of a Clifford algebra or geometric algebra that is based in the use of paravectors. The Dirac equation in APS, including the electromagnetic interaction, reads :$i\; \backslash bar\; \backslash Psi\backslash mathbf\_3\; +\; e\; \backslash bar\; \backslash Psi\; =\; m\; \backslash bar^\backslash dagger$ Another form of the Dirac equation in terms of the Space time algebra was given earlier by David Hestenes. In general, the Dirac equation in the formalism of geometric algebra has the advantage of providing a direct geometric interpretation. ==Relation with the standard form== The spinor can be written in a null basis as :$$ \Psi = \psi_ P_3 - \psi_ P_3 \mathbf_1 + \psi_ \mathbf_1 P_3 + \psi_ \bar_3, such that the representation of the spinor in terms of the Pauli matrices is :$$ \Psi \rightarrow \begin \psi_ & \psi_ \\ \psi_ & \psi_ \end :$$ \bar^\dagger \rightarrow \begin \psi_^ * & -\psi_^ * \\ -\psi_^ * & \psi_^ * \end The standard form of the Dirac equation can be recovered by decomposing the spinor in its right and left-handed spinor components, which are extracted with the help of the projector :$P\_3\; =\; \backslash frac(\; 1\; +\; \backslash mathbf\_3),$ such that :$$ \Psi_L = \bar^\dagger P_3 :$$ \Psi_R = \Psi P_3^ with the following matrix representation :$$ \Psi_L \rightarrow \begin \psi_^ * & 0 \\ -\psi_^ * & 0 \end :$$ \Psi_R \rightarrow \begin \psi_ & 0 \\ \psi_ & 0 \end The Dirac equation can be also written as :$i\; \backslash partial\; \backslash bar^\backslash dagger\; \backslash mathbf\_3\; +\; e\; A\; \backslash bar^\backslash dagger\; =\; m\; \backslash Psi$ Without electromagnetic interaction, the following equation is obtained from the two equivalent forms of the Dirac equation :$$ \begin 0 & i \bar\\ i \partial & 0 \end \begin \bar^\dagger P_3 \\ \Psi P_3 \end = m \begin \bar^\dagger P_3 \\ \Psi P_3 \end so that :$$ \begin 0 & i \partial_0 + i\nabla \\ i \partial_0 - i \nabla & 0 \end \begin \Psi_L \\ \Psi_R \end = m \begin \Psi_L \\ \Psi_R \end or in matrix representation :$$ i \left( \begin 0 & 1 \\ 1 & 0 \end \partial_0 + \begin 0 & \sigma \\ -\sigma & 0 \end \cdot \nabla \right) \begin \psi_L \\ \psi_R \end = m \begin \psi_L \\ \psi_R \end, where the second column of the right and left spinors can be dropped by defining the single column chiral spinors as :$$ \psi_L \rightarrow \begin \psi_^ * \\ -\psi_^ * \end :$$ \psi_R \rightarrow \begin \psi_ \\ \psi_ \end The standard relativistic covariant form of the Dirac equation in the Weyl representation can be easily identified $$ i \gamma^ \partial_ \psi = m \psi, such that :$$ \psi_= \begin \psi_^ * \\ -\psi_^ * \\ \psi_ \\ \psi_ \end Given two spinors $\backslash Psi$ and $\backslash Phi$ in APS and their respective spinors in the standard form as $\backslash psi$ and $\backslash phi$, one can verify the following identity :$$ \phi^\dagger \gamma^0 \psi = \langle \bar\Psi + (\bar\Phi)^\dagger \rangle_S , such that :$$ \psi^\dagger \gamma^0 \psi = 2 \langle \bar\Psi \rangle_ 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「Dirac equation in the algebra of physical space」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.