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{{Short description|Gamma matrices for arbitrary Clifford algebras}}
{{redirect|gamma group|the surveillance technology company|Gamma Group}}
In [[mathematical physics]], '''higher-dimensional gamma matrices''' generalize to arbitrary dimension the four-dimensional [[Gamma matrices]] of [[Paul Dirac|Dirac]], which are a mainstay of relativistic quantum mechanics.  They are utilized in relativistically invariant wave equations for fermions (such as spinors)  in arbitrary space-time dimensions, notably in string theory and supergravity. The [[Weyl–Brauer matrices]] provide an explicit construction of higher-dimensional gamma matrices for [[Weyl spinor]]s. Gamma matrices also appear in generic settings in [[Riemannian geometry]], particularly when a [[spin structure]] can be defined.

== Introduction ==
Consider a space-time of dimension {{mvar|d}} with the flat [[Minkowski metric]],  
: <math> \eta = \|\eta_{a b}\| = \text{diag}(+1, \dots, +1, -1, \dots, -1) ~,</math>
with <math>p</math> positive entries, <math>q</math> negative entries, <math>p + q = d</math> and {{math|''a'', ''b'' {{=}} 0, 1, ..., ''d'' − 1}}.  Set {{math|''N'' {{=}} 2<sup>⌊{{sfrac|1|2}}''d''⌋</sup>}}.  The standard [[Dirac matrices]] correspond to taking {{math|''d'' {{=}} ''N'' {{=}} 4}} and {{math|''p'', ''q'' {{=}} 1, 3}} or {{math|3, 1}}. 

In higher (and lower) dimensions, one may define a [[group (mathematics)|group]], the '''gamma group''', behaving in the same fashion as the Dirac matrices.<ref name="petitjean">
{{Cite journal | last1=Petitjean | first1=Michel | title=Chirality of Dirac spinors revisited | journal=Symmetry | year=2020 | volume=12 | issue=4 | pages=616 | doi=10.3390/sym12040616 | doi-access=free}}
</ref> More precisely, if one selects a [[vector space basis|basis]] <math>\{e_a\}</math> for the (complexified) [[Clifford algebra#Physics|Clifford algebra]] <math>\mathrm{Cl}_{p,q}(\mathbb{C}) \cong \mathrm{Cl}^\mathbb{C}(p, q)</math>, then the gamma group [[group generator|generated by]] <math>\{\Gamma_a\}</math> is [[isomorphic]] to the ''multiplicative'' subgroup generated by the basis elements <math>e_a</math> (ignoring the additive aspect of the Clifford algebra).

By convention, the gamma group is realized as a collection of matrices, the gamma matrices, although the group definition does not require this. In particular, many important properties, including the [[C-symmetry|C]], [[P-symmetry|P]] and [[T-symmetry|T symmetries]] do not require a specific matrix representation, and one obtains a clearer definition of [[chirality]] in this way.<ref name="petitjean"/> Several matrix representations are possible, some given below, and others in the article on the [[Weyl–Brauer matrices]].  In the matrix representation, the spinors are <math>N</math>-dimensional, with the gamma matrices acting on the spinors. A detailed construction of spinors is given in the article on [[Clifford algebra]]. Jost provides a standard reference for spinors in the general setting of Riemmannian geometry.<ref>Jurgen Jost, (2002) "Riemannian Geometry and Geometric Analysis (3rd edition)", Springer. ''See Chapter 1, section 1.8.''</ref>

== Gamma group ==
Most of the properties of the gamma matrices can be captured by a [[group (mathematics)|group]], the '''gamma group'''. This group can be defined without reference to the real numbers, the complex numbers, or even any direct appeal to the [[Clifford algebra]].<ref name="petitjean"/> The matrix representations of this group then provide a concrete realization that can be used to specify the action of the [[gamma matrices]] on [[spinor]]s. For <math>(p,q)=(1,3)</math> dimensions, the matrix products behave just as the conventional [[Dirac matrices]]. The [[Pauli group]] is a [[group representation|representation]] of the gamma group for <math>(p,q)=(3,0)</math> although the Pauli group has more relationships (is [[free group|less free]]); see the note about the chiral element below for an example. The [[quaternion]]s provide a representation for <math>(p,q)=(0,3).</math>

The [[group presentation|presentation]] of the '''gamma group''' <math>G = G_{p,q}</math> is as follows.

* A [[neutral element]] is denoted as <math>I</math>.
* The element <math>i</math> with <math>i^4 = I</math> is a stand-in for the complex number <math>i</math>; it [[Center (group theory)|commutes with all other elements]],
* There is a collection of generators <math>\Gamma_a</math> indexed by <math>a = 0, \ldots, p - 1</math> with <math>\Gamma_a^2 = I~,</math>
* The remaining generators <math>\Gamma_a,\ a = p, \ldots, p + q - 1</math> obey <math>\Gamma_a^2 = i^2~,</math> 
* The anticommutator is defined as <math>\Gamma_a \Gamma_b = i^2 \Gamma_b \Gamma_a</math> for <math>a \ne b ~.</math>

These generators completely define the gamma group. It can be shown that, for all <math>x \in G</math> that <math>x^4 = I</math> and so <math>x^{-1} = x^3~.</math> Every element <math>x \in G</math> can be uniquely written as a product of a finite number of generators placed in canonical order as

:<math>x = i^n \Gamma_a\Gamma_b \cdots \Gamma_c</math>
with the indexes in ascending order

:<math>a < b < \cdots < c</math>

and <math>0 \le n \le 3.</math>  The gamma group is finite, and has at most <math>2^{p+q+2}</math> elements in it.  

The gamma group is a [[p-group|2-group]] but not a [[regular p-group]]. The [[commutator subgroup]] (derived subgroup) is <math>[G, G] = \left\{I, i^2\right\} ~,</math> therefore it is not a [[powerful p-group]]. In general, 2-groups have a large number of [[involution (mathematics)|involutions]]; the gamma group does likewise. Three particular ones are singled out below, as they have a specific interpretation in the context of [[Clifford algebra]]s, in the context of the representations of the gamma group (where transposition and Hermitian conjugation literally correspond to those actions on matrices), and in [[physics]], where the "main involution" <math>\alpha</math> corresponds to a combined [[P-symmetry]] and [[T-symmetry]].

===Transposition===
Given elements <math>\Gamma_i</math> of the generating set of the gamma group, the [[Transposition (mathematics)|transposition]] or ''reversal'' is given by
:<math>(\Gamma_a\Gamma_b\cdots\Gamma_c)^\textsf{t} = \Gamma_c\cdots\Gamma_b\Gamma_a</math>

If there are <math>k</math> elements <math>\Gamma_i</math> all distinct, then
:<math>(\Gamma_a\Gamma_b\cdots\Gamma_c)^\textsf{t} = \left(i^2\right)^{\frac{1}{2}k(k-1)}\Gamma_a\Gamma_b\cdots \Gamma_c</math>

===Hermitian conjugation===
Another [[automorphism]] of the gamma group is given by conjugation, defined on the generators as

:<math>\Gamma_a^\dagger = \begin{cases}
     \Gamma_a & \mbox{for } 0 \le a < p \\
  i^2\Gamma_a & \mbox{for } p \le a < p + q \\
\end{cases}</math>

supplemented with <math>i^\dagger = i^3</math> and <math>I^\dagger = I.</math> For general elements in the group, one takes the transpose: <math>(ab)^\dagger = b^\dagger a^\dagger~.</math>  From the properties of transposition, it follows that, for all elements <math>x\in G</math> that either <math>xx^\dagger = x^\dagger x = I</math> or that <math>xx^\dagger = x^\dagger x = i^2 ~,</math> that is, all elements are either Hermitian or unitary. 

If one interprets the <math>p</math> dimensions as being "time-like", and the <math>q</math> dimensions as being "space-like", then this corresponds to [[P-symmetry]] in physics. That this is the "correct" identification follows from the conventional Dirac matrices, where <math>\gamma_0</math> is associated with the time-like direction, and the <math>\gamma_i</math> the spatial directions, with the "conventional" (+−−−) metric.  Other metric and representational choices suggest other interpretations.

===Main involution===
The '''main involution''' is the map that "flips" the generators: <math>\alpha(\Gamma_a) = i^2\Gamma_a</math> but leaves <math>i</math> alone: <math>\alpha(i) = i ~.</math> This map corresponds to the combined [[P-symmetry]] and [[T-symmetry]] in physics; all directions are reversed.

===Chiral element===
Define the chiral element <math>\omega \equiv \Gamma_\mathrm{chir}</math> as
:<math>\omega = \Gamma_\mathrm{chir} = \Gamma_0\Gamma_1\cdots\Gamma_{d-1}</math>

where <math>d = p + q</math>. The chiral element commutes with the generators as

:<math>\Gamma_a\omega = \left(i^2\right)^{d-1}\omega \Gamma_a</math>

It squares to

:<math>\omega^2 = \left(i^2\right)^{q+\frac{1}{2}d(d-1)}</math>

For the Dirac matrices, the chiral element corresponds to <math>\gamma_5 ~,</math> thus its name, as it plays an important role in distinguishing the chirality of spinors.

For the [[Pauli group]], the chiral element is <math>\sigma_1\sigma_2\sigma_3 = i</math> whereas for the gamma group <math>G_{3,0}</math>, one cannot deduce any such relationship for <math>\Gamma_1\Gamma_2\Gamma_3</math> other than that it squares to <math>i^2~.</math> This is an example of where a representation may have more identities than the represented group.  For the [[quaternion]]s, which provide a representation of <math>G_{0,3}</math> the chiral element is <math>ijk = i^2~.</math>

===Charge conjugation===
None of the above automorphisms (transpose, conjugation, main involution) are [[inner automorphism]]s; that is they ''cannot'' be represented in the form <math>CxC^{-1}</math> for some existing element <math>C</math> in the gamma group, as presented above.  Charge conjugation requires extending the gamma group with two new elements; by convention, these are
:<math>C_{+}\Gamma_a C_{+}^{-1} = \Gamma_a^\textsf{t}</math>

and
:<math>C_{-}\Gamma_a C_{-}^{-1} = i^2\Gamma_a^\textsf{t}</math>

The above relations are not sufficient to define a group; <math>C^2</math> and other products are undetermined.

===Matrix representation===
The gamma group has a matrix representation given by complex <math>N \times N</math> matrices with <math>N = 2^{\left\lfloor \frac{1}{2}d\right\rfloor}</math> and <math>d = p + q</math> and <math>\lfloor x \rfloor</math> the [[floor function]], the largest integer less than <math>x.</math> The group presentation for the matrices can be written compactly in terms of the [[Anticommutator#Ring theory|anticommutator]] relation from the [[Clifford algebra#Physics|Clifford algebra]] {{math|''Cℓ''<sub>''p'',''q''</sub>('''R''')}}

: <math>\{ \Gamma_a ~,~ \Gamma_b \} = \Gamma_a\Gamma_b + \Gamma_b\Gamma_a = 2 \eta_{a b} I_N ~,</math>
where the matrix {{math|''I<sub>N</sub>''}} is the [[identity matrix]] in {{mvar|N}}  dimensions. Transposition and Hermitian conjugation correspond to their usual meaning on matrices.

== Charge conjugation ==
For the remainder of this article,it is assumed that <math>p = 1</math> and so <math>q = d - 1</math>.  That is, the [[Clifford algebra#Physics|Clifford algebra]] {{math|''Cℓ''<sub>''1'',''d''−1</sub>('''R''')}} is assumed.{{efn|It is possible and even likely that many or most of the formulas and tables in this and later sections hold in the general case; however, this has not been verified. This and later sections were originally written with the assumption of a (1,d−1) metric.}} In this case, the gamma matrices have the following property under [[Hermitian conjugation]],
: <math>\begin{align}
  \Gamma_0^\dagger &= +\Gamma_0 ~, &
  \Gamma_a^\dagger &= -\Gamma_a ~(a = 1, \dots, d - 1) ~.
\end{align}</math>

Transposition will be denoted with a minor change of notation, by mapping <math>\Gamma_a^\textsf{t} \mapsto \Gamma_a^\textsf{T}</math> where the element on the left is the abstract group element, and the one on the right is the literal [[matrix transpose]].

As before, the generators {{math|Γ<sub>''a''</sub>, −Γ<sub>''a''</sub><sup>{{font|T}}</sup>, Γ<sub>''a''</sub><sup>{{font|T}}</sup>}} all generate the same group (the generated groups are all [[isomorphic]]; the operations are still [[involution (mathematics)|involutions]]). However, since the {{math|Γ<sub>''a''</sub>}} are now matrices, it becomes plausible to ask whether there is a matrix that can act as a [[matrix similarity|similarity transformation]] that embodies the automorphisms. In general, such a matrix can be found. By convention, there are two of interest; in the physics literature, both referred to as [[charge conjugation]] matrices. Explicitly, these are

: <math>\begin{align}
  C_{(+)} \Gamma_a C_{(+)}^{-1} &= + \Gamma_a^\textsf{T} \\
  C_{(-)} \Gamma_a C_{(-)}^{-1} &= - \Gamma_a^\textsf{T} ~.
\end{align}</math>

They can be constructed as real matrices in various dimensions, as the following table shows.  In even dimension both <math>C_\pm</math> exist, in odd dimension just one.
{| class="wikitable"
|-
! ''d''
! <math> C^*_{(+)} = C_{(+)} </math>
! <math> C^*_{(-)} = C_{(-)} </math>
|-
| <math> 2 </math>
| <math> C^\textsf{T}_{(+)} = C_{(+)};~~~C^2_{(+)} = 1 </math>
| <math> C^\textsf{T}_{(-)} = -C_{(-)};~~~C^2_{(-)} = -1 </math>
|-
| <math> 3 </math>
|
| <math> C^\textsf{T}_{(-)} = -C_{(-)};~~~C^2_{(-)} = -1 </math>
|-
| <math> 4 </math>
| <math> C^\textsf{T}_{(+)} = -C_{(+)};~~~C^2_{(+)} = -1 </math>
| <math> C^\textsf{T}_{(-)} = -C_{(-)};~~~C^2_{(-)} = -1 </math>
|-
| <math> 5 </math>
| <math> C^\textsf{T}_{(+)} = -C_{(+)};~~~C^2_{(+)} = -1 </math>
|
|-
| <math> 6 </math>
| <math> C^\textsf{T}_{(+)} = -C_{(+)};~~~C^2_{(+)} = -1 </math>
| <math> C^\textsf{T}_{(-)} = C_{(-)};~~~C^2_{(-)} = 1 </math>
|-
| <math> 7 </math>
|
| <math> C^\textsf{T}_{(-)} = C_{(-)};~~~C^2_{(-)} = 1 </math>
|-
| <math> 8 </math>
| <math> C^\textsf{T}_{(+)} = C_{(+)};~~~C^2_{(+)} = 1 </math>
| <math> C^\textsf{T}_{(-)} = C_{(-)};~~~C^2_{(-)} = 1 </math>
|-
| <math> 9 </math>
| <math> C^\textsf{T}_{(+)} = C_{(+)};~~~C^2_{(+)} = 1 </math>
|
|-
| <math> 10 </math>
| <math> C^\textsf{T}_{(+)} = C_{(+)};~~~C^2_{(+)} = 1 </math>
| <math> C^\textsf{T}_{(-)} = -C_{(-)};~~~C^2_{(-)} = -1 </math>
|-
| <math> 11 </math>
|
| <math> C^\textsf{T}_{(-)} = -C_{(-)};~~~C^2_{(-)} = -1 </math>
|}
Note that <math> C^*_{(\pm)} = C_{(\pm)} </math> is a basis choice.

== Symmetry properties ==
We denote a product of gamma matrices by

:<math>\Gamma_{abc \dotsm} = \Gamma_a \cdot \Gamma_b \cdot \Gamma_c \cdots {}</math>

and note that the anti-commutation property allows us to simplify any such sequence to one in which the indices are distinct and increasing.  Since distinct <math>\Gamma_a</math> anti-commute this motivates the introduction of an anti-symmetric "average".  We introduce the anti-symmetrised products of distinct {{mvar|n}}-tuples from 0, ..., {{mvar|d}} − 1:
:<math>\Gamma_{a_1 \dots a_n} = \frac{1}{n!} \sum_{\pi \in S_n} \epsilon(\pi) \Gamma_{a_{\pi(1)}} \cdots \Gamma_{a_{\pi(n)}} ~,</math>
where {{mvar|π}} runs over all the [[permutations]] of {{mvar|n}} symbols, and {{mvar|ϵ}} is the [[alternating character]].  There are 2<sup>''d''</sup> such products, but only {{mvar|N}}<sup>2</sup> are independent, spanning the space of {{mvar|N}}×{{mvar|N}} matrices.

Typically, {{math|Γ<sub>''ab''</sub>}} provide the (bi)spinor representation of the {{math|{{sfrac|1|2}}''d''(''d'' − 1)}}  generators of the [[higher-dimensional]] [[Lorentz group]], {{math|SO<sup>+</sup>(1, ''d'' − 1)}}, generalizing the [[Representation theory of the Lorentz group#Reducible representations|6 matrices ''σ<sup>μν</sup>'']] of the [[spin representation]] of the Lorentz group in four dimensions.

For even {{mvar|d}},  one may further define the hermitian ''chiral matrix'' 
: <math> \Gamma_\text{chir} = i^{\frac{d}{2}-1} \Gamma_0 \Gamma_1 \dotsm \Gamma_{d-1} ~,</math>
such that {{math|{Γ<sub>chir</sub>, Γ<sub>''a''</sub>} {{=}} 0}} and {{math|Γ<sub>chir</sub><sup>2</sup> {{=}} 1}}.   (In odd dimensions, such a matrix would commute with all {{mvar|Γ}}<sub>''a''</sub>s and would thus be proportional to the identity, so it is not considered.)

A {{mvar|Γ}} matrix is called symmetric if
: <math>( C \Gamma_{a_1 \dotsm a_n} )^\textsf{T} = +( C \Gamma_{a_1 \dotsm a_n} ) ~;</math>
otherwise, for a − sign,  it is called antisymmetric.

In the previous expression, {{mvar|C}} can be either <math> C_{(+)} </math> or <math> C_{(-)} </math>. In odd dimension, there is no ambiguity, but in even dimension it is better to choose whichever one of <math> C_{(+)} </math> or <math> C_{(-)} </math> allows for Majorana spinors. In {{mvar|d}} = 6, there is no such criterion and therefore we consider both.

{| class="wikitable"
|-
! ''d''
! ''C''
! Symmetric
! Antisymmetric
|-
| <math> 3 </math>
| <math> C_{(-)} </math>
| <math> \gamma_{a} </math>
| <math> I_2 </math>
|-
| <math> 4 </math>
| <math> C_{(-)} </math>
| <math> \gamma_{a} ~,~ \gamma_{a_1 a_2} </math>
| <math> I_4 ~,~  \gamma_\text{chir} ~,~ \gamma_\text{chir} \gamma_a </math>
|-
| <math> 5 </math>
| <math> C_{(+)} </math>
| <math> \Gamma_{a_1 a_2} </math>
| <math> I_4 ~,~  \Gamma_a </math>
|-
| <math> 6 </math>
| <math> C_{(-)} </math>
| <math> I_8 ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3} </math>
| <math> \Gamma_a ~,~ \Gamma_\text{chir}~,~ \Gamma_\text{chir} \Gamma_a ~,~ \Gamma_{a_1 a_2}</math>
|-
| <math> 7 </math>
| <math> C_{(-)} </math>
| <math> I_8 ~,~ \Gamma_{a_1 a_2 a_3} </math>
| <math> \Gamma_a ~,~ \Gamma_{a_1 a_2}</math>
|-
| <math> 8 </math>
| <math> C_{(+)} </math>
| <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>
| <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>
|-
| <math> 9 </math>
| <math> C_{(+)} </math>
| <math> I_{16} ~,~ \Gamma_{a} ~,~ \Gamma_{a_1 \dotsm a_4} ~,~ \Gamma_{a_1 \dotsm a_5} </math>
| <math> \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3}</math>
|-
| <math> 10 </math>
| <math> C_{(-)} </math>
| <math> \Gamma_{a} ~,~ \Gamma_\text{chir} ~,~ \Gamma_\text{chir} \Gamma_a ~,~ \Gamma_{a_1 a_2}
~,~ \Gamma_\text{chir} \Gamma_{a_1 \dotsm a_4}  ~,~ \Gamma_{a_1 \dotsm a_5}</math>
| <math> I_{32} ~,~ \Gamma_\text{chir} \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 a_2 a_3}
~,~ \Gamma_{a_1 \dotsm a_4}  ~,~  \Gamma_\text{chir} \Gamma_{a_1 a_2 a_3} </math>
|-
| <math> 11 </math>
| <math> C_{(-)} </math>
| <math> \Gamma_a ~,~ \Gamma_{a_1 a_2} ~,~ \Gamma_{a_1 \dotsm a_5} </math>
| <math> I_{32} ~,~ \Gamma_{a_1 a_2 a_3} ~,~ \Gamma_{a_1 \dotsm a_4}</math>
|}

==Identities==
The proof of the trace identities for gamma matrices hold for all even dimension. One therefore only needs to remember the [[Gamma matrices#Trace identities|4D case]] and then change the overall factor of 4 to <math>\operatorname{tr}(I_N)</math>. For [[Gamma matrices#Miscellaneous identities|other identities]] (the ones that involve a contraction), explicit functions of <math>d</math> will appear.
{{ordered list
| <math>\gamma^\mu\gamma_\mu = d I_N</math>
| <math>\gamma^\mu\gamma^\nu\gamma_\mu = (2 - d) \gamma^\nu</math>
| <math>\gamma^\mu\gamma^\nu\gamma^\rho\gamma_\mu = 2 \gamma^\rho\gamma^\nu + (d - 2) \gamma^\nu\gamma^\rho</math>
| <math>\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma_\mu = 2 \gamma^\rho\gamma^\sigma\gamma^\nu - 2 \gamma^\nu\gamma^\sigma\gamma^\rho - (d - 2) \gamma^\nu\gamma^\rho\gamma^\sigma </math>
}}
Even when the number of physical dimensions is four, these more general identities are ubiquitous in loop calculations due to [[dimensional regularization]].

== Example of an explicit construction in the [[Gamma matrices#Weyl (chiral) basis|chiral basis]] ==
The {{mvar|Γ}}  matrices can be constructed recursively, first in all even dimensions, {{mvar|d}} = 2{{mvar|k}}, and thence in odd ones, 2{{mvar|k}} + 1.

=== ''d'' = 2 ===
Using the [[Pauli matrices]], take
: <math>\begin{align}
  \gamma_0 &= \sigma_1, & \gamma_1 &= -i \sigma_2
\end{align}</math>

and one may easily check that the charge conjugation matrices are
: <math>\begin{align}
  C_{(+)} =   \sigma_1 = C_{(+)}^* = s_{(2,+)} C_{(+)}^\textsf{T} &= s_{(2,+)} C_{(+)}^{-1} & s_{(2,+)} &= +1 \\
  C_{(-)} = i \sigma_2 = C_{(-)}^* = s_{(2,-)} C_{(-)}^\textsf{T} &= s_{(2,-)} C_{(-)}^{-1} & s_{(2,-)} &= -1~.
\end{align}</math>

One may finally define the hermitian chiral {{mvar|γ}}<sub>chir</sub> to be
: <math>\gamma_\text{chir} = \gamma_0 \gamma_1 = \sigma_3 = \gamma_\text{chir}^\dagger ~.</math>

=== Generic even ''d'' = 2''k''  ===
One may now construct the  {{math|Γ<sub>''a''</sub>,  (''a'' {{=}} 0, ... , ''d'' + 1)}}, matrices and the charge conjugations {{mvar|C}}<sub>(±)</sub> in {{mvar|d}} + 2 dimensions, starting from the {{math|''γ<sub>a'</sub>''}} , ({{math| ''a' '' {{=}} 0, ... , ''d'' − 1}}), and {{mvar|c}}<sub>(±)</sub> matrices in {{mvar|d}} dimensions.

Explicitly, 
: <math>\begin{align}
   \Gamma_{a'} &= \gamma_{a'} \otimes \sigma_3 ~~~ \left(a' = 0, \dots, d - 1\right)~, &
    \Gamma_{d} &= I \otimes (i \sigma_1) ~, &
  \Gamma_{d+1} &= I \otimes (i \sigma_2) ~.
\end{align}</math>

One may then construct the charge conjugation matrices,
: <math>\begin{align}
  C_{(+)} &= c_{(-)} \otimes \sigma_1 ~, &
  C_{(-)} &= c_{(+)} \otimes (i \sigma_2) ~,
\end{align}</math>

with the following properties,
: <math>\begin{align}
  C_{(+)} = C_{(+)}^* = s_{(d+2,+)} C_{(+)}^\textsf{T} &= s_{(d+2,+)} C_{(+)}^{-1} & s_{(d+2,+)} &=  s_{(d,-)} \\
  C_{(-)} = C_{(-)}^* = s_{(d+2,-)} C_{(-)}^\textsf{T} &= s_{(d+2,-)} C_{(-)}^{-1} & s_{(d+2,-)} &= -s_{(d,+)} ~.
\end{align}</math>

Starting from the sign values for {{mvar|d}} = 2, {{mvar|s}}<sub>(2,+)</sub> = +1 and {{mvar|s}}<sub>(2,−)</sub> = −1, one may fix all subsequent signs {{math|s}}<sub>(''d'',±) </sub> which have periodicity 8; explicitly, one finds
{| class="wikitable"
|-
!
! <math> d = 8 k </math>
! <math> d = 8 k + 2 </math>
! <math> d = 8 k + 4 </math>
! <math> d = 8 k + 6 </math>
|-
| <math> s_{(d,+)} </math>
| +1
| +1
| −1
| −1
|-
| <math> s_{(d,-)} </math>
| +1
| −1
| −1
| +1
|}

Again, one may define the hermitian chiral matrix in {{mvar|d}}+2  dimensions as
: <math>\begin{align}
  \Gamma_\text{chir} &= \alpha_{d+2} \Gamma_0 \Gamma_1 \dotsm \Gamma_{d+1} = \gamma_\text{chir} \otimes \sigma_3~, &
            \alpha_d &= i^{\frac{1}{2}d-1} ~,
\end{align}</math>

which is diagonal by construction and transforms under charge conjugation as
: <math>\begin{align}
  C_{(\pm)} \Gamma_\text{chir} C_{(\pm)}^{-1} &= \beta_{d+2} \Gamma_\text{chir}^\textsf{T}~, &
  \beta_d &= (-)^{\frac{1}{2}d(d-1)} ~.
\end{align}</math>

It is thus evident that {{math|{Γ<sub>chir</sub> , Γ<sub>''a''</sub>} }} = 0.

=== Generic odd ''d'' = 2''k'' + 1 ===
Consider the previous construction for {{mvar|d}} − 1 (which is even) and simply take all {{math|Γ<sub>''a''</sub>  (''a'' {{=}} 0, ..., ''d'' − 2)}} matrices, to which append its {{math|''i''Γ<sub>chir</sub> ≡ Γ<sub>''d''−1</sub>}}.  (The {{mvar|i}} is required in order to yield an antihermitian matrix, and extend into the spacelike metric).

Finally, compute the charge conjugation matrix:  choose between <math> C_{(+)} </math> and <math> C_{(-)} </math>, in such a way that {{math|Γ<sub>''d''−1</sub>}} transforms as all the other  {{math|Γ}} matrices. Explicitly, require
: <math> C_{(s)} \Gamma_\text{chir} C_{(s)}^{-1} = \beta_{d} \Gamma_\text{chir}^\textsf{T} = s \Gamma_\text{chir}^\textsf{T} ~.</math>

As the dimension {{mvar|d}} ranges, patterns typically repeat themselves with period 8. (cf. the [[Bott periodicity theorem#Geometric model of loop spaces|Clifford algebra clock]].)

==See also==
* [[Weyl–Brauer matrices]]
* [[Bispinor]]
* [[Clifford module]]

==Notes==
{{notelist}}

==References==
{{reflist}}

==General reading==
* {{cite journal | last1=Brauer | first1=Richard | author1-link=Richard Brauer | last2=Weyl | first2=Hermann | author2-link=Hermann Weyl | title=Spinors in ''n'' dimensions | journal=Am. J. Math. | volume=57 | pages=425–449 | year=1935 | zbl=0011.24401 | jfm=61.1025.06 | doi=10.2307/2371218 | jstor=2371218 }} 
* {{Cite journal | doi = 10.1063/1.1703856| title = On Spinors in n Dimensions| journal = Journal of Mathematical Physics| volume = 3| issue = 6| pages = 1135| year = 1962| last1 = Pais | first1 = Abraham | bibcode = 1962JMP.....3.1135P| doi-access = free}}
* {{Cite journal | doi = 10.1016/0550-3213(77)90206-1| title = Supersymmetry, supergravity theories and the dual spinor model| journal = Nuclear Physics B| volume = 122| issue = 2| pages = 253| year = 1977| last1 = Gliozzi | first1 = F.| last2 = Scherk | first2 = Joel| last3 = Olive | first3 = D.| bibcode = 1977NuPhB.122..253G| url = https://cds.cern.ch/record/203097/files/CM-P00061869.pdf}}
* {{Cite journal | doi = 10.1063/1.525069| title = Clifford algebras in 2ω dimensions| journal = Journal of Mathematical Physics| volume = 22| issue = 7| pages = 1330| year = 1981| last1 = Kennedy | first1 = A. D.| bibcode = 1981JMP....22.1330K}}
* de Wit, Bryce and  Smith, J. (1986).  ''Field Theory in Particle Physics'' (North-Holland Personal Library), Volume 1, Paperback, [http://www.staff.science.uu.nl/~wit00103/ftip/AppendixE.pdf Appendix E] ([https://web.archive.org/web/20180107085820/http://www.staff.science.uu.nl/~wit00103/ftip/AppendixE.pdf Archived from original]), {{ISBN|978-0444869999}}
* Murayama, H. (2007).  [http://hitoshi.berkeley.edu/230A/clifford.pdf   "Notes on Clifford Algebra and Spin(N) Representations"]
* Pietro Giuseppe Frè (2012). "Gravity, a Geometrical Course: Volume 1: Development of the Theory and Basic Physical Applications." Springer-Verlag. {{ISBN|9400753608}}. ''See pp 315ff.''

[[Category:Spinors]]
[[Category:Matrices]]
[[Category:Clifford algebras]]