Higher-dimensional gamma matrices

In mathematical physics, higher-dimensional gamma matrices are the matrices which satisfy the Clifford algebra

$\{ \Gamma_a ~,~ \Gamma_b \} = 2 \eta_{a b} I_N$

with the metric given by

$\eta = \parallel \eta_{a b} \parallel = \text{diag}(%2B1,-1, \dots, -1)$

where $a,b = 0,1, \dots, d-1$ and $I_N$ the identity matrix in $N= 2^{[d/2]}$ dimensions.

They have the following property under hermitian conjugation

$\Gamma_0^\dagger= %2B\Gamma_0 ~,~ \Gamma_i^\dagger= -\Gamma_i ~(i=1,\dots,d-1)$

1 Charge conjugation
2 Symmetry properties
3 Example of an explicit construction in chiral base

Charge conjugation

Since the groups generated by $\ \Gamma_a$ , $-\Gamma_a^T$ , $\Gamma_a^T$ are the same we deduce from Schur's lemma that there must exist a similarity transformation which connect them. This transformation is generated by the charge conjugation matrix. Explicitly we can introduce the following matrices

$C_{(%2B)} \Gamma_a C_{(%2B)}^{-1} = %2B \Gamma_a^T$

$C_{(-)} \Gamma_a C_{(-)}^{-1} = - \Gamma_a^T$

They can be constructed as real matrices in various dimensions as the following table shows

D	$C^*_{(%2B)}= C_{(%2B)}$	$C^*_{(-)}= C_{(-)}$
$2$	$C^T_{(%2B)}=C_{(%2B)};~~~C^2_{(%2B)}=1$	$C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1$
$3$		$C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1$
$4$	$C^T_{(%2B)}=-C_{(%2B)};~~~C^2_{(%2B)}=-1$	$C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1$
$5$	$C^T_{(%2B)}=-C_{(%2B)};~~~C^2_{(%2B)}=-1$
$6$	$C^T_{(%2B)}=-C_{(%2B)};~~~C^2_{(%2B)}=-1$	$C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=1$
$7$		$C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=1$
$8$	$C^T_{(%2B)}=C_{(%2B)};~~~C^2_{(%2B)}=1$	$C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=1$
$9$	$C^T_{(%2B)}=C_{(%2B)};~~~C^2_{(%2B)}=1$
$10$	$C^T_{(%2B)}=C_{(%2B)};~~~C^2_{(%2B)}=1$	$C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1$
$11$		$C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=-1$

Symmetry properties

A $\Gamma$ matrix is called symmetric if

$( C \Gamma_{a_1 \dots a_n} )^T = %2B ( C \Gamma_{a_1 \dots a_n} )$

otherwise it is called antisymmetric. In the previous expression $C$ can be either $C_{(%2B)}$ or $C_{(-)}$ . In odd dimension there is not ambiguity but in even dimension it is better to choose whichever one of $C_{(%2B)}$ or $C_{(-)}$ which allows for Majorana spinors. In $D=6$ there is not such criterion and therefore we consider both.

D	C	Symmetric	Antisymmetric
$3$	$C_{(-)}$	$\gamma_{a}$	$I_2$
$4$	$C_{(-)}$	$\gamma_{a} ~,~ \gamma_{a_1 a_2}$	$I_4 ~,~ \gamma_\text{chir} ~,~ \gamma_\text{chir} \gamma_a$
$5$	$C_{(%2B)}$	$\Gamma_{a_1 a_2}$	$I_4 ~,~ \Gamma_a$
$6$	$C_{(-)}$	$I_8 ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3}$	$\Gamma_a ~,~ \Gamma_\text{chir}~,~ \Gamma_\text{chir} \Gamma_a ~,~ \Gamma_{a_1 a_2}$
$7$	$C_{(-)}$	$I_8 ~,~ \Gamma_{a_1 a_2 a_3}$	$\Gamma_a ~,~ \Gamma_{a_1 a_2}$
$8$	$C_{(%2B)}$	$I_{16} ~,~ \Gamma_{a} ~,~ \Gamma_\text{chir} ~,~ \Gamma_\text{chir}\Gamma_{a_1 a_2 a_3} ~,~ \Gamma_{a_1 \dots a_4}$	$\Gamma_\text{chir} \Gamma_a ~,~ \Gamma_{a_1 a_2} ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3}$
$9$	$C_{(%2B)}$	$I_{16} ~,~ \Gamma_{a} ~,~ \Gamma_{a_1 \dots a_4} ~,~ \Gamma_{a_1 \dots a_5}$	$\Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3}$
$10$	$C_{(-)}$	$\Gamma_{a} ~,~ \Gamma_\text{chir} ~,~ \Gamma_\text{chir} \Gamma_a ~,~ \Gamma_{a_1 a_2} ~,~ \Gamma_\text{chir} \Gamma_{a_1 \dots a_4} ~,~ \Gamma_{a_1 \dots a_5}$	$I_{32} ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3} ~,~ \Gamma_{a_1 \dots a_4} ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2 a_3}$
$11$	$C_{(-)}$	$\Gamma_a ~,~ \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 \dots a_5}$	$I_{32} ~,~ \Gamma_{a_1 a_2 a_3} ~,~ \Gamma_{a_1 \dots a_4}$

Example of an explicit construction in chiral base

We construct the $\Gamma$ matrices in a recursive way, first in all even dimensions and then in odd ones.

d = 2

We take

$\gamma_0= \sigma_1 ~,~ \gamma_1= -i \sigma_2$

and we can easily check that the charge conjugation matrices are

$C_{(%2B)}= \sigma_1 = C_{(%2B)}^* = s_{(2,%2B)} C_{(%2B)}^T = s_{(2,%2B)} C_{(%2B)}^{-1} ~~~~ s_{(2,%2B)}=%2B1$

$C_{(-)}= i \sigma_2 = C_{(-)}^* = s_{(2,-)} C_{(-)}^T = s_{(2,-)} C_{(-)}^{-1} ~~~~ s_{(2,-)}=-1$

We can also define the hermitian chiral $\gamma_\text{chir}$ to be

$\gamma_\text{chir}= \gamma_0 \gamma_1 = \sigma_3 = \gamma_\text{chir}^\dagger$

generic even d = 2k

We now construct the $\Gamma_a$ ( $a=0,\dots d%2B1$ ) matrices and the charge conjugations $C_{(\pm)}$ in $d%2B2$ dimensions starting from the $\gamma_{a'}$ ( $a'=0, \dots, d-1$ ) and $c_{(\pm)}$ matrices in $d$ dimensions. Explicitly we have

$\Gamma_{a'} = \gamma_{a'} \otimes \sigma_3 ~(a'=0, \dots, d-1) ~~,~~ \Gamma_{d} = I \otimes (i \sigma_1),~~ \Gamma_{d%2B1}= I \otimes (i \sigma_2)$

Then we can construct the charge conjugation matrices

$C_{(%2B)} = c_{(-)} \otimes \sigma_1 ~~~~,~~~~ C_{(-)} = c_{(%2B)} \otimes (i \sigma_2)$

with the following properties

$C_{(%2B)}= C_{(%2B)}^* = s_{(d%2B2,%2B)} C_{(%2B)}^T = s_{(d%2B2,%2B)} C_{(%2B)}^{-1} ~~~~ s_{(d%2B2,%2B)}= s_{(d,-)}$

$C_{(-)}= C_{(-)}^* = s_{(d%2B2,-)} C_{(-)}^T = s_{(d%2B2,-)} C_{(-)}^{-1} ~~~~ s_{(d%2B2,-)}=-s_{(d,%2B)}$

Starting from the values for $d=2$ , $s_{(2,%2B)}=%2B1,~~~ s_{(2,-)}=-1$ we can compute all the signs $s_{(d,\pm)}$ which have a periodicity of 8, explicitly we find

	$d=8 k$	$d=8 k%2B2$	$d=8 k%2B4$	$d=8 k%2B6$
$s_{(d,%2B)}$	+1	+1	−1	−1
$s_{(d,-)}$	+1	−1	−1	+1

Again we can define the hermitian chiral matrix in $d%2B2$ dimensions as

$\Gamma_\text{chir}= \alpha_{d%2B2} \Gamma_0 \Gamma_1 \dots \Gamma_{d-1} = \gamma_\text{chir} \otimes \sigma_3 ~~~~ \alpha_d= i^{d/2-1}$

which is diagonal by construction and transforms under charge conjugation as

$C_{(\pm)} \Gamma_\text{chir} C_{(\pm)}^{-1} = \beta_{d%2B2} \Gamma_\text{chir}^T ~~~~ \beta_d= (-)^{d(d-1)/2}$

generic odd d = 2k + 1

We consider the previous construction for $d-1$ (which is even) and then we simply take all $\Gamma_{a}$ ( $a=0, \dots, d-2$ ) matrices to which we add $\Gamma_{d-1}= i \Gamma_\text{chir}$ ( the $i$ is there in order to have an antihermitian matrix).

Finally we can compute the charge conjugation matrix: we have to choose between $C_{(%2B)}$ and $C_{(-)}$ in such a way that $\Gamma_{d-1}$ transforms as all the others $\Gamma$ matrices. Explicitly we require

$C_{(s)} \Gamma_\text{chir} C_{(s)}^{-1} = \beta_{d} \Gamma_\text{chir}^T = s \Gamma_\text{chir}^T$