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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 Dirac matrix ： ウィキペディア英語版 Gamma matrices In mathematical physics, the gamma matrices, $\backslash$, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra ''C''ℓ1,3(R). It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin-½ particles. In Dirac representation, the four contravariant gamma matrices are :$\backslash gamma^0\; =\; \backslash begin$ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end,\quad \gamma^1 = \begin 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end :$\backslash gamma^2\; =\; \backslash begin$ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end,\quad \gamma^3 = \begin 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end. Analogous sets of gamma matrices can be defined in any dimension and signature of the metric. For example, the Pauli matrices are a set of "gamma" matrices in dimension 3 with metric of Euclidean signature (3,0). In five spacetime dimensions, the four gammas above together with the fifth gamma matrix to be presented below generate the Clifford algebra. ==Mathematical structure== The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation :$\backslash displaystyle\backslash \; =\; \backslash gamma^\backslash mu\; \backslash gamma^\backslash nu\; +\; \backslash gamma^\backslash nu\; \backslash gamma^\backslash mu\; =\; 2\; \backslash eta^\; I\_4$ where $\backslash$ is the anticommutator, $\backslash eta^$ is the Minkowski metric with signature and $I\_4$ is the identity matrix. This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. Covariant gamma matrices are defined by :$\backslash displaystyle\; \backslash gamma\_\backslash mu\; =\; \backslash eta\_\; \backslash gamma^\backslash nu\; =\; \backslash left\backslash ,$ and Einstein notation is assumed. Note that the other sign convention for the metric, necessitates either a change in the defining equation: :$\backslash displaystyle\backslash \; =\; -2\; \backslash eta^\; I\_4$ or a multiplication of all gamma matrices by $i$, which of course changes their hermiticity properties detailed below. Under the alternative sign convention for the metric the covariant gamma matrices are then defined by :$\backslash displaystyle\; \backslash gamma\_\backslash mu\; =\; \backslash eta\_\; \backslash gamma^\backslash nu\; =\; \backslash left\backslash$. 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「Gamma matrices」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.