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,

$\eta = \parallel \eta_{a b} \parallel = \text{diag}(+1,-1, \dots, -1) ~,$

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×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),

$\{ \Gamma_a ~,~ \Gamma_b \} = \Gamma_a\Gamma_b + \Gamma_b\Gamma_a = 2 \eta_{a b} I_N ~,$

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 N² 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_chir²=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= 2k, and thence in odd ones, 2k+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

	$d=8 k$	$d=8 k+2$	$d=8 k+4$	$d=8 k+6$
$s_{(d,+)}$	+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 ≡ 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.)

See also

• Weyl–Brauer matrices
• Bispinor
• Clifford module

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