Physics of roller coasters

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

$\{{\Gamma }_a,{\Gamma }_b\}=2{\eta }_{ab}{I}_N$

with the metric given by

$\eta =\parallel {\eta }_{ab}\parallel =\text{diag}(+1,-1,\dots ,-1)$

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

${\Gamma }_0^{†}=+{\Gamma }_0,{\Gamma }_i^{†}=-{\Gamma }_i(i=1,\dots ,d-1)$

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 connects them. This transformation is generated by the charge conjugation matrix. Explicitly we can introduce the following matrices

$C_{(+)}{\Gamma }_a{C}_{(+)}^{-1}=+{\Gamma }_a^T$

$C_{(-)}{\Gamma }_a{C}_{(-)}^{-1}=-{\Gamma }_a^T$

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

D	$C_{(+)}^{\ast }=C_{(+)}$	$C_{(-)}^{\ast }=C_{(-)}$
$2$	$C_{(+)}^T=C_{(+)};C_{(+)}^2=1$	$C_{(-)}^T=-C_{(-)};C_{(-)}^2=-1$
$3$		$C_{(-)}^T=-C_{(-)};C_{(-)}^2=-1$
$4$	$C_{(+)}^T=-C_{(+)};C_{(+)}^2=-1$	$C_{(-)}^T=-C_{(-)};C_{(-)}^2=-1$
$5$	$C_{(+)}^T=-C_{(+)};C_{(+)}^2=-1$
$6$	$C_{(+)}^T=-C_{(+)};C_{(+)}^2=-1$	$C_{(-)}^T=C_{(-)};C_{(-)}^2=1$
$7$		$C_{(-)}^T=C_{(-)};C_{(-)}^2=1$
$8$	$C_{(+)}^T=C_{(+)};C_{(+)}^2=1$	$C_{(-)}^T=C_{(-)};C_{(-)}^2=1$
$9$	$C_{(+)}^T=C_{(+)};C_{(+)}^2=1$
$10$	$C_{(+)}^T=C_{(+)};C_{(+)}^2=1$	$C_{(-)}^T=-C_{(-)};C_{(-)}^2=-1$
$11$		$C_{(-)}^T=C_{(-)};C_{(-)}^2=-1$

Symmetry properties

A $\Gamma$ matrix is called symmetric if

$(C{\Gamma }_{a_1\dots a_n}{)}^T=+(C{\Gamma }_{a_1\dots a_n})$

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

D	C	Symmetric	Antisymmetric
$3$	$C_{(-)}$	${\gamma }_a$	$I_2$
$4$	$C_{(-)}$	${\gamma }_a,{\gamma }_{a_1{a}_2}$	$I_4,{\gamma }_{\text{chir}},{\gamma }_{\text{chir}}{\gamma }_a$
$5$	$C_{(+)}$	${\Gamma }_{a_1{a}_2}$	$I_4,{\Gamma }_a$
$6$	$C_{(-)}$	$I_8,{\Gamma }_{\text{chir}}{\Gamma }_{a_1{a}_2},{\Gamma }_{a_1{a}_2{a}_3}$	${\Gamma }_a,{\Gamma }_{\text{chir}},{\Gamma }_{\text{chir}}{\Gamma }_a,{\Gamma }_{a_1{a}_2}$
$7$	$C_{(-)}$	$I_8,{\Gamma }_{a_1{a}_2{a}_3}$	${\Gamma }_a,{\Gamma }_{a_1{a}_2}$
$8$	$C_{(+)}$	$I_{16},{\Gamma }_a,{\Gamma }_{\text{chir}},{\Gamma }_{\text{chir}}{\Gamma }_{a_1{a}_2{a}_3},{\Gamma }_{a_1\dots a_4}$	${\Gamma }_{\text{chir}}{\Gamma }_a,{\Gamma }_{a_1{a}_2},{\Gamma }_{\text{chir}}{\Gamma }_{a_1{a}_2},{\Gamma }_{a_1{a}_2{a}_3}$
$9$	$C_{(+)}$	$I_{16},{\Gamma }_a,{\Gamma }_{a_1\dots a_4},{\Gamma }_{a_1\dots a_5}$	${\Gamma }_{a_1{a}_2},{\Gamma }_{a_1{a}_2{a}_3}$
$10$	$C_{(-)}$	${\Gamma }_a,{\Gamma }_{\text{chir}},{\Gamma }_{\text{chir}}{\Gamma }_a,{\Gamma }_{a_1{a}_2},{\Gamma }_{\text{chir}}{\Gamma }_{a_1\dots a_4},{\Gamma }_{a_1\dots a_5}$	$I_{32},{\Gamma }_{\text{chir}}{\Gamma }_{a_1{a}_2},{\Gamma }_{a_1{a}_2{a}_3},{\Gamma }_{a_1\dots a_4},{\Gamma }_{\text{chir}}{\Gamma }_{a_1{a}_2{a}_3}$
$11$	$C_{(-)}$	${\Gamma }_a,{\Gamma }_{a_1{a}_2},{\Gamma }_{a_1\dots a_5}$	$I_{32},{\Gamma }_{a_1{a}_2{a}_3},{\Gamma }_{a_1\dots a_4}$

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

$C_{(+)}={\sigma }_1=C_{(+)}^{\ast }=s_{(2,+)}{C}_{(+)}^T=s_{(2,+)}{C}_{(+)}^{-1}{s}_{(2,+)}=+1$

$C_{(-)}=i{\sigma }_2=C_{(-)}^{\ast }=s_{(2,-)}{C}_{(-)}^T=s_{(2,-)}{C}_{(-)}^{-1}{s}_{(2,-)}=-1$

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

${\gamma }_{\text{chir}}={\gamma }_0{\gamma }_1={\sigma }_3={\gamma }_{\text{chir}}^{†}$

generic even d = 2k

We now construct the ${\Gamma }_a$ ( $a=0,\dots d+1$) matrices and the charge conjugations $C_{(\pm )}$ in $d+2$ dimensions starting from the ${\gamma }_{a^{\prime }}$ ($a^{\prime }=0,\dots ,d-1$) and $c_{(\pm )}$ matrices in $d$ dimensions. Explicitly we have

${\Gamma }_{a^{\prime }}={\gamma }_{a^{\prime }}\otimes {\sigma }_3(a^{\prime }=0,\dots ,d-1),{\Gamma }_d=I\otimes (i{\sigma }_1),{\Gamma }_{d+1}=I\otimes (i{\sigma }_2)$

Then we can construct the charge conjugation matrices

$C_{(+)}=c_{(-)}\otimes {\sigma }_1,C_{(-)}=c_{(+)}\otimes (i{\sigma }_2)$

with the following properties

$C_{(+)}=C_{(+)}^{\ast }=s_{(d+2,+)}{C}_{(+)}^T=s_{(d+2,+)}{C}_{(+)}^{-1}{s}_{(d+2,+)}=s_{(d,-)}$

$C_{(-)}=C_{(-)}^{\ast }=s_{(d+2,-)}{C}_{(-)}^T=s_{(d+2,-)}{C}_{(-)}^{-1}{s}_{(d+2,-)}=-s_{(d,+)}$

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

	$d=8k$	$d=8k+2$	$d=8k+4$	$d=8k+6$
$s_{(d,+)}$	+1	+1	−1	−1
$s_{(d,-)}$	+1	−1	−1	+1

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

${\Gamma }_{\text{chir}}={\alpha }_{d+2}{\Gamma }_0{\Gamma }_1\dots {\Gamma }_{d-1}={\gamma }_{\text{chir}}\otimes {\sigma }_3{\alpha }_d=i^{d/2-1}$

which is diagonal by construction and transforms under charge conjugation as

$C_{(\pm )}{\Gamma }_{\text{chir}}{C}_{(\pm )}^{-1}={\beta }_{d+2}{\Gamma }_{\text{chir}}^T{\beta }_d=(-{)}^{d(d-1)/2}$

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 $C_{(+)}$ and $C_{(-)}$ in such a way that ${\Gamma }_{d-1}$ transforms as all the others $\Gamma$ matrices. Explicitly we require

$C_{(s)}{\Gamma }_{\text{chir}}{C}_{(s)}^{-1}={\beta }_d{\Gamma }_{\text{chir}}^T=s{\Gamma }_{\text{chir}}^T$