Since the groups generated by 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.
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 -tuples from 0,...,−1: where runs over all the permutations of symbols, and is the alternating character. There are 2d such products, but only 2 are independent, spanning the space of × matrices. Typically, provide the spinor representation of the generators of the higher-dimensional Lorentz group,, generalizing the 6 matrices σμν of the spin representation of the Lorentz group in four dimensions. For even, one may further define the hermitian chiral matrix such that = 0 and =1. A matrix is called symmetric if otherwise, for a − sign, it is called antisymmetric. In the previous expression, 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 =6, there is no such criterion and therefore we consider both.
d
C
Symmetric
Antisymmetric
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. For other identities, explicit functions of will appear. 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">Gamma matrices#Weyl basis">chiral basis
The matrices can be constructed recursively, first in all even dimensions, = 2, and thence in odd ones, 2+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 chir to be
Generic even ''d'' = 2''k''
One may now construct the , matrices and the charge conjugations in +2 dimensions, starting from the ,, and matrices in dimensions. Explicitly, One may then construct the charge conjugation matrices, with the following properties, Starting from the sign values for =2, =+1 and =−1, one may fix all subsequent signs which have periodicity 8; explicitly, one finds
+1
+1
−1
−1
+1
−1
−1
+1
Again, one may define the hermitian chiral matrix in +2 dimensions as which is diagonal by construction and transforms under charge conjugation as It is thus evident that = 0.
Generic odd ''d'' = 2''k'' + 1
Consider the previous construction for −1 and simply take all matrices, to which append its . . Finally, compute the charge conjugation matrix: choose between and, in such a way that transforms as all the other matrices. Explicitly, require As the dimension ranges, patterns typically repeat themselves with period 8.
