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,

where a,b = 0,1, ..., d−1. Set N= 2d/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 which satisfy the anticommutator relation from the Clifford algebra Cℓ1,d−1(R) (generating a representation for it),

where IN 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,

Charge conjugation

Since the groups generated by Γa, −ΓaT, ΓaT 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

They can be constructed as real matrices in various dimensions, as the following table shows. In even dimension both exist, in odd dimension just one.

d

Symmetry properties

We denote a product of gamma matrices by

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 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:

where π runs over all the permutations of n symbols, and ϵ is the alternating character. There are 2d such products, but only N2 are independent, spanning the space of N×N matrices.

Typically, Γab 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 σμν of the spin representation of the Lorentz group in four dimensions.

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

such that {Γchir , Γa} = 0 and Γchir2=1. (In odd dimensions, such a matrix would commute with all Γas and would thus be proportional to the identity, so it is not considered.)

A Γ matrix is called symmetric if

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

d C Symmetric Antisymmetric

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

and one may easily check that the charge conjugation matrices are

One may finally define the hermitian chiral γchir to be

Generic even d = 2k

One may now construct the Γa , (a=0, ... , d+1), matrices and the charge conjugations C(±) in d+2 dimensions, starting from the γa' , ( a' =0, ... , d−1), and c(±) matrices in d dimensions.

Explicitly,

One may then construct the charge conjugation matrices,

with the following properties,

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

+1 +1 1 1
+1 1 1 +1

Again, one may define the hermitian chiral matrix in d+2 dimensions as

which is diagonal by construction and transforms under charge conjugation as

It is thus evident that {Γchir , Γa} = 0.

Generic odd d = 2k + 1

Consider the previous construction for d−1 (which is even) and simply take all Γa (a=0, ..., d−2) matrices, to which append its chirΓd−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 and , in such a way that Γd−1 transforms as all the other Γ matrices. Explicitly, require

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

See also

References

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