겔만 행렬(Gell-Mann matrices)

개요

• a family of traceless Hermitian -matrices, orthonormalized\[\left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right),\left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{array} \right),\left( \begin{array}{ccc} \frac{1}{\sqrt{3}} & 0 & 0 \\ 0 & \frac{1}{\sqrt{3}} & 0 \\ 0 & 0 & -\frac{2}{\sqrt{3}} \end{array} \right)\]

• 리대수 \(\mathfrak{su}(3)\) 의 기저
• 쿼크를 다루기 위해 도입됨
• SU(3) 대칭성이 등장하는 게이지 이론 에서 사용된다

성질

• \([g_i, g_j] = if^{ijk} g_k\)
• \(f^{123} = 1 \ , \quad f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \frac{1}{2} \ , \quad f^{458} = f^{678} = \frac{\sqrt{3}}{2}\)
• \(\mathrm{tr}(\lambda_i \lambda_j) = 2\delta_{ij}\)

메모

• http://mathoverflow.net/questions/89331/why-the-gell-mann-matrices-in-the-su3-model-need-to-be-trace-orthogonal
• Math Overflow http://mathoverflow.net/search?q=

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxak43OUd5QTRNVGs/edit

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Gell-Mann_matrices
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The World of Mathematical Equations

리뷰논문, 에세이, 강의노트

• http://arxiv.org/pdf/hep-ph/0109241.pdf

관련도서

• M. Gell-Mann, Y. Ne'eman, "The eightfold way" , Benjamin (1964)