Difference between revisions of "Dirac matrices"
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| − | Four Hermitian matrices | + | Four Hermitian matrices, denoted by $ \alpha_{1} $, $ \alpha_{2} $, $ \alpha_{3} $ and $ \beta $, of dimension $ 4 \times 4 $ that satisfy the relations |
| + | \begin{gather} | ||
| + | \alpha_{k} \alpha_{j} + \alpha_{j} \alpha_{k} = 2 \delta_{k j} \mathsf{I}_{4}, \\ | ||
| + | \alpha_{k} \beta + \beta \alpha_{k} = \mathbf{0}_{4}, \\ | ||
| + | \alpha_{k} \alpha_{k} = \beta^{2} = \mathsf{I}_{4}, | ||
| + | \end{gather} | ||
| + | where $ \mathsf{I}_{4} $ is the $ (4 \times 4) $ identity matrix. The matrices $ \alpha_{1} $, $ \alpha_{2} $, $ \alpha_{3} $ and $ \beta $ may also be replaced by the Hermitian matrices $ \gamma^{k} = - i \beta \alpha_{k} $, where $ k \in \{ 1,2,3 \} $, and by the anti-Hermitian matrix $ \gamma^{0} = i \beta $. These then satisfy the relation | ||
| + | $$ | ||
| + | \gamma^{\alpha} \gamma^{\beta} + \gamma^{\beta} \gamma^{\alpha} = - 2 \eta^{\alpha \beta} \mathsf{I}_{4}, \qquad \forall \alpha,\beta \in \{ 0,1,2,3 \}. | ||
| + | $$ | ||
| + | Here, $ \eta^{\alpha \beta} \stackrel{\text{df}}{=} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} $. It is therefore possible to write the [[Dirac equation|Dirac equation]] in a form that is covariant with respect to the Lorentz group of transformations. The matrices $ \alpha_{k} $, $ \beta $ and $ \gamma^{k} $, where $ k \in \{ 0,1,2,3 \} $, are defined up to an arbitrary unitary transformation and may be represented in various ways. One such representation is | ||
| + | $$ | ||
| + | \gamma^{0} = - i \begin{bmatrix} \mathsf{I}_{2} & \mathbf{0}_{2} \\ \mathbf{0}_{2} & - \mathsf{I}_{2} \end{bmatrix}; \qquad | ||
| + | \gamma^{k} = - i \begin{bmatrix} \mathbf{0}_{2} & \boldsymbol{\sigma}_{k} \\ - \boldsymbol{\sigma}_{k} & \mathbf{0}_{2} \end{bmatrix}, | ||
| + | $$ | ||
| + | where the $ \boldsymbol{\sigma}_{k} $’s are the $ (2 \times 2) $ [[Pauli matrices|Pauli matrices]], while $ \mathsf{I}_{2} $ and $ \mathbf{0}_{2} $ are the $ (2 \times 2) $ identity and zero matrices respectively. Dirac matrices may be used to factorize the [[Klein–Gordon equation|Klein–Gordon equation]] in the following manner: | ||
| + | $$ | ||
| + | (\Box - m^{2}) E \psi | ||
| + | = \left( \sum_{k = 0}^{3} \gamma^{k} \frac{\partial}{\partial x^{k}} - m E \right) \! \left( \sum_{l = 0}^{3} \gamma^{l} \frac{\partial}{\partial x^{l}} + m E \right) \psi | ||
| + | = 0, | ||
| + | $$ | ||
| + | where $ \Box $ denotes the [[D'Alembert operator|d’Alembert operator]]. | ||
| − | + | The Dirac matrices were Introduced by P. Dirac in 1928, in his derivation of the Dirac equation. | |
| − | + | ====Comments==== | |
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| − | + | For references, see the article on the [[Dirac equation|Dirac equation]]. | |
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Latest revision as of 16:06, 15 December 2016
Four Hermitian matrices, denoted by $ \alpha_{1} $, $ \alpha_{2} $, $ \alpha_{3} $ and $ \beta $, of dimension $ 4 \times 4 $ that satisfy the relations \begin{gather} \alpha_{k} \alpha_{j} + \alpha_{j} \alpha_{k} = 2 \delta_{k j} \mathsf{I}_{4}, \\ \alpha_{k} \beta + \beta \alpha_{k} = \mathbf{0}_{4}, \\ \alpha_{k} \alpha_{k} = \beta^{2} = \mathsf{I}_{4}, \end{gather} where $ \mathsf{I}_{4} $ is the $ (4 \times 4) $ identity matrix. The matrices $ \alpha_{1} $, $ \alpha_{2} $, $ \alpha_{3} $ and $ \beta $ may also be replaced by the Hermitian matrices $ \gamma^{k} = - i \beta \alpha_{k} $, where $ k \in \{ 1,2,3 \} $, and by the anti-Hermitian matrix $ \gamma^{0} = i \beta $. These then satisfy the relation $$ \gamma^{\alpha} \gamma^{\beta} + \gamma^{\beta} \gamma^{\alpha} = - 2 \eta^{\alpha \beta} \mathsf{I}_{4}, \qquad \forall \alpha,\beta \in \{ 0,1,2,3 \}. $$ Here, $ \eta^{\alpha \beta} \stackrel{\text{df}}{=} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} $. It is therefore possible to write the Dirac equation in a form that is covariant with respect to the Lorentz group of transformations. The matrices $ \alpha_{k} $, $ \beta $ and $ \gamma^{k} $, where $ k \in \{ 0,1,2,3 \} $, are defined up to an arbitrary unitary transformation and may be represented in various ways. One such representation is $$ \gamma^{0} = - i \begin{bmatrix} \mathsf{I}_{2} & \mathbf{0}_{2} \\ \mathbf{0}_{2} & - \mathsf{I}_{2} \end{bmatrix}; \qquad \gamma^{k} = - i \begin{bmatrix} \mathbf{0}_{2} & \boldsymbol{\sigma}_{k} \\ - \boldsymbol{\sigma}_{k} & \mathbf{0}_{2} \end{bmatrix}, $$ where the $ \boldsymbol{\sigma}_{k} $’s are the $ (2 \times 2) $ Pauli matrices, while $ \mathsf{I}_{2} $ and $ \mathbf{0}_{2} $ are the $ (2 \times 2) $ identity and zero matrices respectively. Dirac matrices may be used to factorize the Klein–Gordon equation in the following manner: $$ (\Box - m^{2}) E \psi = \left( \sum_{k = 0}^{3} \gamma^{k} \frac{\partial}{\partial x^{k}} - m E \right) \! \left( \sum_{l = 0}^{3} \gamma^{l} \frac{\partial}{\partial x^{l}} + m E \right) \psi = 0, $$ where $ \Box $ denotes the d’Alembert operator.
The Dirac matrices were Introduced by P. Dirac in 1928, in his derivation of the Dirac equation.
Comments
For references, see the article on the Dirac equation.
Dirac matrices. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirac_matrices&oldid=40015