gamma matrices

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1 Convention
2 Algebra
3 Associated matrices
- 3.1 4 dimensions
- 3.2 d dimensions
4 Representations
- 4.1 4 dimensions
5 Identities
- 5.1 D (even) dimensions
6 See also
7 References

Convention

Note on convention: $\eta_{\mu\nu} = \operatorname{diag}(1,-1,-1,-1,...)\,$ , while $d\,$ labels the number of spacetime dimensions.

Algebra

$\left\{ \gamma^\mu, \gamma^\nu \right\} \equiv \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu\nu} I\,$ .

Associated matrices

4 dimensions

$\gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3 = -\frac{i}{4!} \epsilon^{\mu\nu\rho\sigma} \gamma_\mu \gamma_\nu \gamma_\rho \gamma_\sigma\,$ .

d dimensions

$\gamma^{d+1} = i \gamma^0 \gamma^1 \gamma^2 ... \gamma^{d-1}\,$ .

$\sigma^{\mu\nu} \equiv \frac{i}{2} [\gamma^\mu, \gamma^\nu]\,$ .

$S^{\mu\nu} \equiv \frac{i}{4} [\gamma^\mu, \gamma^\nu]\,$ .

$\Sigma^{\mu\nu} \equiv \frac{1}{4} [\gamma^\mu, \gamma^\nu]\,$ .

Representations

The $\sigma^i\,$ are the Pauli matrices.

4 dimensions

Dirac basis:

$\gamma^0 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix}.$

Weyl or Chiral basis:

$\gamma^0 = \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix},\quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix},\quad \gamma^5 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}.$

i.e.,

$\gamma^\mu = \begin{pmatrix} 0 & \sigma^\mu \\ \bar\sigma^\mu & 0 \end{pmatrix}\,$ .

Majorana basis:

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

$\gamma^2 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \quad \gamma^3 = \begin{pmatrix} 0 & -i\sigma^1 \\ -i\sigma^1 & 0 \end{pmatrix}, \quad \gamma^5 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$

Identities

D (even) dimensions

$\left\{ \gamma^{D+1}, \gamma^\mu \right\} = 0\,$ .

$\left[ \gamma^{D+1}, S^{\mu\nu} \right] = 0\,$ .

$\operatorname{tr}\left(\gamma^{\mu_1}\gamma^{\mu_2}\gamma^{\mu_3}...\gamma^{\mu_p}\gamma^{D+1}\right) = \begin{cases} 0, & \mbox{if }p < D \\ (2i)^\frac{D}{2} \epsilon^{\mu_1 \mu_2 \mu_3...\mu_p}, & \mbox{if }p = D \end{cases}\,$ .