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| In [[mathematical physics]], '''higher-dimensional gamma matrices''' are the matrices which satisfy the [[Clifford algebra]]
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| : <math> \{ \Gamma_a ~,~ \Gamma_b \} = 2 \eta_{a b} I_N </math>
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| with the metric given by
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| : <math> \eta = \parallel \eta_{a b} \parallel = \text{diag}(+1,-1, \dots, -1)
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| </math>
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| where <math> a,b = 0,1, \dots, d-1 </math> and <math> I_N </math> the identity matrix in <math> N= 2^{[d/2]} </math> dimensions.
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| They have the following property under hermitian conjugation
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| : <math> \Gamma_0^\dagger= +\Gamma_0 ~,~ \Gamma_i^\dagger= -\Gamma_i
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| ~(i=1,\dots,d-1)
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| </math>
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| == Charge conjugation ==
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| Since the groups generated by <math>\ \Gamma_a </math>,
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| <math> -\Gamma_a^T </math>,
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| <math> \Gamma_a^T </math> are the same we deduce from [[Schur's lemma]]
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| that there must exist a [[matrix similarity|similarity transformation]] which connects them.
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| This transformation is generated by the [[charge conjugation]] matrix.
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| Explicitly we can introduce the following matrices
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| : <math> C_{(+)} \Gamma_a C_{(+)}^{-1} = + \Gamma_a^T </math>
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| : <math> C_{(-)} \Gamma_a C_{(-)}^{-1} = - \Gamma_a^T </math>
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| They can be constructed as real matrices in various dimensions as the following table shows
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| {| class="wikitable"
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| |-
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| ! D | |
| ! <math> C^*_{(+)}= C_{(+)} </math>
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| ! <math> C^*_{(-)}= C_{(-)} </math>
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| |-
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| | <math> 2 </math>
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| | <math> C^T_{(+)}=C_{(+)};~~~C^2_{(+)}=1 </math>
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| | <math> C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1 </math>
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| |-
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| | <math> 3 </math>
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| | <math> C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1 </math>
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| |-
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| | <math> 4 </math>
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| | <math> C^T_{(+)}=-C_{(+)};~~~C^2_{(+)}=-1 </math>
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| | <math> C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1 </math>
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| |-
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| | <math> 5 </math>
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| | <math> C^T_{(+)}=-C_{(+)};~~~C^2_{(+)}=-1 </math>
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| |-
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| | <math> 6 </math>
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| | <math> C^T_{(+)}=-C_{(+)};~~~C^2_{(+)}=-1 </math>
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| | <math> C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=1 </math>
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| |-
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| | <math> 7 </math>
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| | <math> C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=1 </math>
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| |-
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| | <math> 8 </math>
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| | <math> C^T_{(+)}=C_{(+)};~~~C^2_{(+)}=1 </math>
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| | <math> C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=1 </math>
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| |-
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| | <math> 9 </math>
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| | <math> C^T_{(+)}=C_{(+)};~~~C^2_{(+)}=1 </math>
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| |-
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| | <math> 10 </math>
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| | <math> C^T_{(+)}=C_{(+)};~~~C^2_{(+)}=1 </math>
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| | <math> C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1 </math>
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| |-
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| | <math> 11 </math>
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| | <math> C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=-1 </math>
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| |}
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| == Symmetry properties ==
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| A <math> \Gamma </math> matrix is called symmetric if
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| : <math> ( C \Gamma_{a_1 \dots a_n} )^T = + ( C \Gamma_{a_1 \dots a_n} ) </math>
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| otherwise it is called antisymmetric.
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| In the previous expression <math> C </math> can be either <math> C_{(+)} </math>
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| or <math> C_{(-)} </math>. In odd dimension there is not ambiguity but
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| in even dimension it is better to choose whichever one of <math> C_{(+)} </math>
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| or <math> C_{(-)} </math> which allows
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| for Majorana spinors. In <math> D=6 </math> there is not such
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| criterion and therefore we consider both.
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| {| class="wikitable"
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| |-
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| ! D
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| ! C
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| ! Symmetric
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| ! Antisymmetric
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| |-
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| | <math> 3 </math>
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| | <math> C_{(-)} </math>
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| | <math> \gamma_{a} </math>
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| | <math> I_2 </math>
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| |-
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| | <math> 4 </math>
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| | <math> C_{(-)} </math>
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| | <math> \gamma_{a} ~,~ \gamma_{a_1 a_2} </math>
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| | <math> I_4 ~,~ \gamma_\text{chir} ~,~ \gamma_\text{chir} \gamma_a </math>
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| |-
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| | <math> 5 </math>
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| | <math> C_{(+)} </math>
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| | <math> \Gamma_{a_1 a_2} </math>
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| | <math> I_4 ~,~ \Gamma_a </math>
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| |-
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| | <math> 6 </math>
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| | <math> C_{(-)} </math>
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| | <math> I_8 ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3} </math>
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| | <math> \Gamma_a ~,~ \Gamma_\text{chir}~,~ \Gamma_\text{chir} \Gamma_a ~,~ \Gamma_{a_1 a_2}</math>
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| |-
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| | <math> 7 </math>
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| | <math> C_{(-)} </math>
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| | <math> I_8 ~,~ \Gamma_{a_1 a_2 a_3} </math>
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| | <math> \Gamma_a ~,~ \Gamma_{a_1 a_2}</math>
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| |-
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| | <math> 8 </math>
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| | <math> C_{(+)} </math>
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| | <math> I_{16} ~,~ \Gamma_{a} ~,~ \Gamma_\text{chir} ~,~ \Gamma_\text{chir}\Gamma_{a_1 a_2 a_3} ~,~ \Gamma_{a_1 \dots a_4} </math>
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| | <math> \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} </math>
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| |-
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| | <math> 9 </math>
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| | <math> C_{(+)} </math>
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| | <math> I_{16} ~,~ \Gamma_{a} ~,~ \Gamma_{a_1 \dots a_4} ~,~ \Gamma_{a_1 \dots a_5} </math>
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| | <math> \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3}</math>
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| |-
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| | <math> 10 </math>
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| | <math> C_{(-)} </math>
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| | <math> \Gamma_{a} ~,~ \Gamma_\text{chir} ~,~ \Gamma_\text{chir} \Gamma_a ~,~ \Gamma_{a_1 a_2}
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| ~,~ \Gamma_\text{chir} \Gamma_{a_1 \dots a_4} ~,~ \Gamma_{a_1 \dots a_5}</math>
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| | <math> I_{32} ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3}
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| ~,~ \Gamma_{a_1 \dots a_4} ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2 a_3} </math>
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| |-
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| | <math> 11 </math>
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| | <math> C_{(-)} </math>
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| | <math> \Gamma_a ~,~ \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 \dots a_5} </math>
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| | <math> I_{32} ~,~ \Gamma_{a_1 a_2 a_3} ~,~ \Gamma_{a_1 \dots a_4}</math>
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| |}
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| == Example of an explicit construction in chiral base ==
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| We construct the <math> \Gamma </math> matrices in a recursive way, first in all even dimensions and then in odd ones.
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| === ''d'' = 2 ===
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| We take
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| : <math> \gamma_0= \sigma_1 ~,~ \gamma_1= -i \sigma_2 </math>
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| and we can easily check that the charge conjugation matrices are
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| : <math> C_{(+)}= \sigma_1 = C_{(+)}^* = s_{(2,+)} C_{(+)}^T = s_{(2,+)} C_{(+)}^{-1} ~~~~ s_{(2,+)}=+1 </math>
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| : <math> C_{(-)}= i \sigma_2 = C_{(-)}^* = s_{(2,-)} C_{(-)}^T = s_{(2,-)} C_{(-)}^{-1} ~~~~ s_{(2,-)}=-1 </math>
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| We can also define the hermitian chiral <math> \gamma_\text{chir} </math> to be
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| : <math> \gamma_\text{chir}= \gamma_0 \gamma_1 = \sigma_3 = \gamma_\text{chir}^\dagger </math> | |
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| === generic even ''d'' = 2''k'' ===
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| We now construct the <math> \Gamma_a </math> ( <math> a=0,\dots d+1 </math>) matrices and the charge conjugations <math> C_{(\pm)} </math> in <math> d+2 </math> dimensions starting from the <math> \gamma_{a'} </math> (<math> a'=0, \dots, d-1 </math>) and <math> c_{(\pm)} </math> matrices in <math> d </math> dimensions.
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| Explicitly we have
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| : <math> \Gamma_{a'} = \gamma_{a'} \otimes \sigma_3 ~(a'=0, \dots, d-1) ~~,~~ \Gamma_{d} = I \otimes (i \sigma_1),~~ \Gamma_{d+1}= I \otimes (i \sigma_2) </math>
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| Then we can construct the charge conjugation matrices
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| : <math> C_{(+)} = c_{(-)} \otimes \sigma_1 ~~~~,~~~~ C_{(-)} = c_{(+)} \otimes (i \sigma_2) </math>
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| with the following properties
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| : <math> C_{(+)}= C_{(+)}^* = s_{(d+2,+)} C_{(+)}^T = s_{(d+2,+)} C_{(+)}^{-1} ~~~~ s_{(d+2,+)}= s_{(d,-)} </math> | |
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| : <math> C_{(-)}= C_{(-)}^* = s_{(d+2,-)} C_{(-)}^T = s_{(d+2,-)} C_{(-)}^{-1} ~~~~ s_{(d+2,-)}=-s_{(d,+)} </math> | |
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| Starting from the values for <math>d=2</math>, <math> s_{(2,+)}=+1,~~~ s_{(2,-)}=-1</math> we can compute all the signs <math>s_{(d,\pm)} </math> which have a periodicity of 8, explicitly we find
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| {| class="wikitable"
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| |-
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| !
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| ! <math> d=8 k </math>
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| ! <math> d=8 k+2 </math>
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| ! <math> d=8 k+4 </math>
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| ! <math> d=8 k+6 </math>
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| | <math> s_{(d,+)} </math>
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| | +1
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| | +1
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| | −1
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| | −1
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| |-
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| | <math> s_{(d,-)} </math>
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| | +1
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| | −1
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| | −1
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| | +1
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| |}
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| Again we can define the hermitian chiral matrix in <math>d+2</math> dimensions as
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| : <math> \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}</math>
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| which is diagonal by construction and transforms under charge conjugation as
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| : <math> C_{(\pm)} \Gamma_\text{chir} C_{(\pm)}^{-1} = \beta_{d+2} \Gamma_\text{chir}^T
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| ~~~~ \beta_d= (-)^{d(d-1)/2} </math>
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| === generic odd ''d'' = 2''k'' + 1 ===
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| We consider the previous construction for <math> d-1 </math> (which is even) and then we simply take all <math> \Gamma_{a} </math> (<math> a=0, \dots, d-2 </math>) matrices to which we add <math> \Gamma_{d-1}= i \Gamma_\text{chir} </math> ( the <math> i </math> is there in order to have an antihermitian matrix).
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| Finally we can compute the charge conjugation matrix: we have to choose between <math> C_{(+)} </math> and <math> C_{(-)} </math> in such a way that <math> \Gamma_{d-1} </math> transforms as all the others <math> \Gamma </math> matrices. Explicitly we require
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| : <math> C_{(s)} \Gamma_\text{chir} C_{(s)}^{-1} = \beta_{d} \Gamma_\text{chir}^T = s \Gamma_\text{chir}^T </math>
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| [[Category:Quantum field theory]]
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| [[Category:Spinors]]
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| [[Category:Matrices]]
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| [[Category:Clifford algebras]]
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