狄拉克矩陣

在理論物理學中，狄拉克矩陣$\{ \gamma^0, \gamma^1, \gamma^2, \gamma^3 \}$，又稱γ矩陣，是狄拉克方程式中所引入的四個矩陣，它們是包立矩陣的推廣，滿足反對易關係：

$\displaystyle\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} I_4$，其中的上標，依據愛因斯坦求和約定，其為偽標量（不是冪次方的意思）。

狄拉克表象四個矩陣：

$\gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix},\quad \gamma^1 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}$

$\gamma^2 = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix},\quad \gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}.$

參考資料

• Halzen, Francis; Martin, Alan. Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. 1984. ISBN 0-471-88741-2. 
• A. Zee, Quantum Field Theory in a Nutshell (2003), Princeton University Press: Princeton, New Jersey. ISBN 0-691-01019-6. See chapter II.1.
• M. Peskin, D. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0-201-50397-2] See chapter 3.2.
• W. Pauli. Contributions mathématiques à la théorie des matrices de Dirac. Ann. Inst. Henri Poincaré. 1936, 6: 109 [2012-05-06].