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

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

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

Brauer, Richard; Weyl, Hermann (1935). "Spinors in n dimensions". Am. J. Math. 57: 425–449. doi:10.2307/2371218. JFM 61.1025.06. JSTOR 2371218. Zbl 0011.24401.
Pais, A. (1962). "On Spinors in n Dimensions". Journal of Mathematical Physics. 3 (6): 1135. doi:10.1063/1.1703856.
Gliozzi, F.; Scherk, J.; Olive, D. (1977). "Supersymmetry, supergravity theories and the dual spinor model". Nuclear Physics B. 122 (2): 253. doi:10.1016/0550-3213(77)90206-1.
Kennedy, A. D. (1981). "Clifford algebras in 2ω dimensions". Journal of Mathematical Physics. 22 (7): 1330. 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, ISBN 978-0444869999

This article is issued from Wikipedia - version of the 2/23/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.