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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 Higher-dimensional gamma matrices ： ウィキペディア英語版 Higher-dimensional gamma matrices In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of Dirac, which are a mainstay of relativistic quantum mechanics. They are utilized in relativistically invariant wave equations for fermions (such as spinors) in arbitrary space-time dimensions, notably in string theory and supergravity. Consider a space-time of dimension with the flat Minkowski metric, : $\backslash eta\; =\; \backslash parallel\; \backslash eta\_\; \backslash parallel\; =\; \backslash text(+1,-1,\; \backslash dots,\; -1)\; ~,$ where . Set . The standard Dirac matrices correspond to taking . The higher gamma matrices are a -long sequence of complex matrices $\backslash Gamma\_i,\backslash \; i=0,\backslash ldots,d-1$ which satisfy the anticommutator relation from the Clifford algebra (generating a representation for it), : $\backslash \; =\; \backslash Gamma\_a\backslash Gamma\_b\; +\; \backslash Gamma\_b\backslash Gamma\_a\; =\; 2\; \backslash eta\_\; I\_N\; ~,$ where is the identity matrix in dimensions. (The spinors acted on by these matrices have components in dimensions.) Such a sequence exists for all values of and can be constructed explicitly, as provided below. The gamma matrices have the following property under hermitian conjugation, : $\backslash Gamma\_0^\backslash dagger=\; +\backslash Gamma\_0\; ~,~\; \backslash Gamma\_i^\backslash dagger=\; -\backslash Gamma\_i$ ~(i=1,\dots,d-1) ~. == Charge conjugation == Since the groups generated by are the same, we can look for a similarity transformation which connects them all. This transformation is generated by a respective charge conjugation matrix. Explicitly, we can introduce the following matrices : $C\_\; \backslash Gamma\_a\; C\_^\; =\; +\; \backslash Gamma\_a^T$ : $C\_\; \backslash Gamma\_a\; C\_^\; =\; -\; \backslash Gamma\_a^T\; ~.$ They can be constructed as real matrices in various dimensions, as the following table shows. In even dimension both $C\_\backslash pm$ exist, in odd dimension just one. ! $C^$ *_= C_ |- | $2$ | $C^T\_=C\_;~~~C^2\_=1$ | $C^T\_=-C\_;~~~C^2\_=-1$ |- | $3$ | | $C^T\_=-C\_;~~~C^2\_=-1$ |- | $4$ | $C^T\_=-C\_;~~~C^2\_=-1$ | $C^T\_=-C\_;~~~C^2\_=-1$ |- | $5$ | $C^T\_=-C\_;~~~C^2\_=-1$ | |- | $6$ | $C^T\_=-C\_;~~~C^2\_=-1$ | $C^T\_=C\_;~~~C^2\_=1$ |- | $7$ | | $C^T\_=C\_;~~~C^2\_=1$ |- | $8$ | $C^T\_=C\_;~~~C^2\_=1$ | $C^T\_=C\_;~~~C^2\_=1$ |- | $9$ | $C^T\_=C\_;~~~C^2\_=1$ | |- | $10$ | $C^T\_=C\_;~~~C^2\_=1$ | $C^T\_=-C\_;~~~C^2\_=-1$ |- | $11$ | | $C^T\_=-C\_;~~~C^2\_=-1$ |} 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「Higher-dimensional 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.