Higher-dimensional gamma matrices

In mathematical physics, higher-dimensional gamma matrices generalize to arbitrary dimension the four-dimensional Gamma matrices of 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.

Consider a space-time of dimension $d$ with the flat Minkowski metric,

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

where $a,b = 0,1, ..., d -1$ . Set $N = 2 ⌊ d /2⌋$ . The standard Dirac matrices correspond to taking $d = N = 4$ .

The higher gamma matrices are a $d$ -long sequence of complex $N \times N$ matrices $\Gamma _{i},\ i=0,\ldots ,d-1$ which satisfy the anticommutator relation from the Clifford algebra $Cℓ 1,d-1 (R)$ (generating a representation for it),

\{\Gamma _{a}~,~\Gamma _{b}\}=\Gamma _{a}\Gamma _{b}+\Gamma _{b}\Gamma _{a}=2\eta _{ab}I_{N}~,

where $I N$ is the identity matrix in $N$ dimensions. (The spinors acted on by these matrices have $N$ components in $d$ dimensions.) Such a sequence exists for all values of $d$ and can be constructed explicitly, as provided below.

The gamma matrices have the following property under hermitian conjugation,

\Gamma _{0}^{\dagger }=+\Gamma _{0}~,~\Gamma _{i}^{\dagger }=-\Gamma _{i}~(i=1,\dots ,d-1)~.

Charge conjugation

Since the groups generated by $Γ a, -Γ a T, Γ a T$ are the same, we can look for a similarity transformation which connects them all. This transformation is generated by a respective 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. In even dimension both $C_{\pm }$ exist, in odd dimension just one.

d	$C_{(+)}^{*}=C_{(+)}$	$C_{(-)}^{*}=C_{(-)}$
$2$	$C_{(+)}^{T}=C_{(+)};~~~C_{(+)}^{2}=1$	$C_{(-)}^{T}=-C_{(-)};~~~C_{(-)}^{2}=-1$
$3$		$C_{(-)}^{T}=-C_{(-)};~~~C_{(-)}^{2}=-1$
$4$	$C_{(+)}^{T}=-C_{(+)};~~~C_{(+)}^{2}=-1$	$C_{(-)}^{T}=-C_{(-)};~~~C_{(-)}^{2}=-1$
$5$	$C_{(+)}^{T}=-C_{(+)};~~~C_{(+)}^{2}=-1$
$6$	$C_{(+)}^{T}=-C_{(+)};~~~C_{(+)}^{2}=-1$	$C_{(-)}^{T}=C_{(-)};~~~C_{(-)}^{2}=1$
$7$		$C_{(-)}^{T}=C_{(-)};~~~C_{(-)}^{2}=1$
$8$	$C_{(+)}^{T}=C_{(+)};~~~C_{(+)}^{2}=1$	$C_{(-)}^{T}=C_{(-)};~~~C_{(-)}^{2}=1$
$9$	$C_{(+)}^{T}=C_{(+)};~~~C_{(+)}^{2}=1$
$10$	$C_{(+)}^{T}=C_{(+)};~~~C_{(+)}^{2}=1$	$C_{(-)}^{T}=-C_{(-)};~~~C_{(-)}^{2}=-1$
$11$		$C_{(-)}^{T}=-C_{(-)};~~~C_{(-)}^{2}=-1$

Note that $C_{(\pm )}^{*}=C_{(\pm )}$ is a basis choice.

Symmetry properties

We denote a product of gamma matrices by

\Gamma _{abc\ldots }=\Gamma _{a}\cdot \Gamma _{b}\cdot \Gamma _{c}\cdots \

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 $\Gamma _{a}$ anti-commute this motivates the introduction of an anti-symmetric "average". We introduce the anti-symmetrised products of distinct $n$ -tuples from 0,..., $d$ −1:

\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)}}~,

where $π$ runs over all the permutations of $n$ symbols, and $ϵ$ is the alternating character. There are 2^d such products, but only $N$ ² are independent, spanning the space of $N$ × $N$ matrices.

Typically, $Γ ab$ provide the (bi)spinor representation of the $d(d-1) /2$ generators of the higher-dimensional Lorentz group, $SO + (1, d -1)$ , generalizing the 6 matrices σ^μν of the spin representation of the Lorentz group in four dimensions.

For even $d$ , one may further define the hermitian chiral matrix

\Gamma _{\text{chir}}=i^{d/2-1}\Gamma _{0}\Gamma _{1}\dots \Gamma _{d-1}~,

such that ${Γ chir, Γ a}$ = 0 and $Γ chir 2$ =1. (In odd dimensions, such a matrix would commute with all $Γ$ _as and would thus be proportional to the identity, so it is not considered.)

A $Γ$ matrix is called symmetric if

(C\Gamma _{a_{1}\dots a_{n}})^{T}=+(C\Gamma _{a_{1}\dots a_{n}})~;

otherwise, for a − sign, it is called antisymmetric. In the previous expression, $C$ can be either $C_{(+)}$ or $C_{(-)}$ . In odd dimension, there is no ambiguity, but in even dimension it is better to choose whichever one of $C_{(+)}$ or $C_{(-)}$ allows for Majorana spinors. In $d$ =6, there is no such criterion and therefore we consider both.

d	C	Symmetric	Antisymmetric
$3$	$C_{(-)}$	$\gamma _{a}$	$I_{2}$
$4$	$C_{(-)}$	$\gamma _{a}~,~\gamma _{a_{1}a_{2}}$	$I_{4}~,~\gamma _{\text{chir}}~,~\gamma _{\text{chir}}\gamma _{a}$
$5$	$C_{(+)}$	$\Gamma _{a_{1}a_{2}}$	$I_{4}~,~\Gamma _{a}$
$6$	$C_{(-)}$	$I_{8}~,~\Gamma _{\text{chir}}\Gamma _{a_{1}a_{2}}~,~\Gamma _{a_{1}a_{2}a_{3}}$	$\Gamma _{a}~,~\Gamma _{\text{chir}}~,~\Gamma _{\text{chir}}\Gamma _{a}~,~\Gamma _{a_{1}a_{2}}$
$7$	$C_{(-)}$	$I_{8}~,~\Gamma _{a_{1}a_{2}a_{3}}$	$\Gamma _{a}~,~\Gamma _{a_{1}a_{2}}$
$8$	$C_{(+)}$	$I_{16}~,~\Gamma _{a}~,~\Gamma _{\text{chir}}~,~\Gamma _{\text{chir}}\Gamma _{a_{1}a_{2}a_{3}}~,~\Gamma _{a_{1}\dots a_{4}}$	$\Gamma _{\text{chir}}\Gamma _{a}~,~\Gamma _{a_{1}a_{2}}~,~\Gamma _{\text{chir}}\Gamma _{a_{1}a_{2}}~,~\Gamma _{a_{1}a_{2}a_{3}}$
$9$	$C_{(+)}$	$I_{16}~,~\Gamma _{a}~,~\Gamma _{a_{1}\dots a_{4}}~,~\Gamma _{a_{1}\dots a_{5}}$	$\Gamma _{a_{1}a_{2}}~,~\Gamma _{a_{1}a_{2}a_{3}}$
$10$	$C_{(-)}$	$\Gamma _{a}~,~\Gamma _{\text{chir}}~,~\Gamma _{\text{chir}}\Gamma _{a}~,~\Gamma _{a_{1}a_{2}}~,~\Gamma _{\text{chir}}\Gamma _{a_{1}\dots a_{4}}~,~\Gamma _{a_{1}\dots a_{5}}$	$I_{32}~,~\Gamma _{\text{chir}}\Gamma _{a_{1}a_{2}}~,~\Gamma _{a_{1}a_{2}a_{3}}~,~\Gamma _{a_{1}\dots a_{4}}~,~\Gamma _{\text{chir}}\Gamma _{a_{1}a_{2}a_{3}}$
$11$	$C_{(-)}$	$\Gamma _{a}~,~\Gamma _{a_{1}a_{2}}~,~\Gamma _{a_{1}\dots a_{5}}$	$I_{32}~,~\Gamma _{a_{1}a_{2}a_{3}}~,~\Gamma _{a_{1}\dots a_{4}}$

Identities

The proof of the trace identities for gamma matrices is independent of dimension. One therefore only needs to remember the 4D case and then change the overall factor of 4 to $\operatorname {tr} (I_{N})$ . For other identities (the ones that involve a contraction), explicit functions of $d$ will appear.

$\gamma ^{\mu }\gamma _{\mu }=dI_{N}$
$\gamma ^{\mu }\gamma ^{\nu }\gamma _{\mu }=(2-d)\gamma ^{\nu }$
$\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma _{\mu }=2\gamma ^{\rho }\gamma ^{\nu }+(d-2)\gamma ^{\nu }\gamma ^{\rho }$
$\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\sigma }\gamma _{\mu }=2\gamma ^{\rho }\gamma ^{\sigma }\gamma ^{\nu }-2\gamma ^{\nu }\gamma ^{\sigma }\gamma ^{\rho }-(d-2)\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\sigma }$

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 chiral basis

The $Γ$ matrices can be constructed recursively, first in all even dimensions, $d$ = 2 $k$ , and thence in odd ones, 2 $k$ +1.

d = 2

Using the Pauli matrices, take

\gamma _{0}=\sigma _{1}~,~\gamma _{1}=-i\sigma _{2}

and one may easily check that the charge conjugation matrices are

C_{(+)}=\sigma _{1}=C_{(+)}^{*}=s_{(2,+)}C_{(+)}^{T}=s_{(2,+)}C_{(+)}^{-1}\qquad s_{(2,+)}=+1

C_{(-)}=i\sigma _{2}=C_{(-)}^{*}=s_{(2,-)}C_{(-)}^{T}=s_{(2,-)}C_{(-)}^{-1}\qquad s_{(2,-)}=-1~.

One may finally define the hermitian chiral $γ$ _chir to be

\gamma _{\text{chir}}=\gamma _{0}\gamma _{1}=\sigma _{3}=\gamma _{\text{chir}}^{\dagger }~.

Generic even d = 2k

One may now construct the $Γ a, (a =0, ... , d +1)$ , matrices and the charge conjugations $C$ _(±) in $d$ +2 dimensions, starting from the $γ a'$ , ( $a' =0, ... , d -1$ ), and $c$ _(±) matrices in $d$ dimensions.

Explicitly,

\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})~.

One may then construct the charge conjugation matrices,

C_{(+)}=c_{(-)}\otimes \sigma _{1}~,\qquad C_{(-)}=c_{(+)}\otimes (i\sigma _{2})~,

with the following properties,

C_{(+)}=C_{(+)}^{*}=s_{(d+2,+)}C_{(+)}^{T}=s_{(d+2,+)}C_{(+)}^{-1}\qquad s_{(d+2,+)}=s_{(d,-)}

C_{(-)}=C_{(-)}^{*}=s_{(d+2,-)}C_{(-)}^{T}=s_{(d+2,-)}C_{(-)}^{-1}\qquad s_{(d+2,-)}=-s_{(d,+)}~.

Starting from the sign values for $d$ =2, $s$ _(2,+)=+1 and $s$ _(2,−)=−1, one may fix all subsequent signs $s$ _(d,±) which have periodicity 8; explicitly, one finds

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

Again, one may 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}~.

It is thus evident that ${Γ chir, Γ a}$ = 0.

Generic odd d = 2k + 1

Consider the previous construction for $d$ −1 (which is even) and simply take all $Γ a (a =0, ..., d -2)$ matrices, to which append its $iΓ chir \equiv Γ d-1$ . (The $i$ is required in order to yield an antihermitian matrix, and extend into the spacelike metric).

Finally, compute the charge conjugation matrix: choose between $C_{(+)}$ and $C_{(-)}$ , in such a way that $Γ d-1$ transforms as all the other $Γ$ matrices. Explicitly, require

C_{(s)}\Gamma _{\text{chir}}C_{(s)}^{-1}=\beta _{d}\Gamma _{\text{chir}}^{T}=s\Gamma _{\text{chir}}^{T}~.

As the dimension $d$ ranges, patterns typically repeat themselves with period 8. (cf. the Clifford algebra clock.)

References

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Gliozzi, F.; Scherk, J.; Olive, D. (1977). "Supersymmetry, supergravity theories and the dual spinor model" (PDF). Nuclear Physics B. 122 (2): 253. Bibcode:1977NuPhB.122..253G. doi:10.1016/0550-3213(77)90206-1.
Kennedy, A. D. (1981). "Clifford algebras in 2ω dimensions". Journal of Mathematical Physics. 22 (7): 1330. Bibcode:1981JMP....22.1330K. doi:10.1063/1.525069.
de Wit, B. and Smith, J. (1986). Field Theory in Particle Physics (North-Holland Personal Library), Volume 1, Paperback, Appendix E (Archived from original), ISBN 978-0444869999
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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.

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