Higher-dimensional gamma matrices
Encyclopedia
In mathematical physics
, higher-dimensional gamma matrices are the matrices which satisfy the Clifford algebra
with the metric given by
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, higher-dimensional gamma matrices are the matrices which satisfy the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
with the metric given by
-
where
and 
the identity matrix in 
dimensions.
They have the following property under hermitian conjugation
-
Charge conjugation
Since the groups generated by
,
,
are the same we deduce from Schur's lemma
that there must exist a similarity transformation which connect them.
This transformation is generated by the 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
D 
Symmetry properties
A
matrix is called symmetric if
otherwise it is called antisymmetric.
In the previous expression
can be either 
or
. In odd dimension there is not ambiguity but
in even dimension it is better to choose whichever one of
or
which allows
for Majorana spinors. In
there is not such
criterion and therefore we consider both.
D C Symmetric Antisymmetric 
Example of an explicit construction in chiral base
We construct the
matrices in a recursive way, first in all even dimensions and then in odd ones.
d = 2
We take
and we can easily check that the charge conjugation matrices are
We can also define the hermitian chiral
to be
generic even d = 2k
We now construct the
( 
) matrices and the charge conjugations 
in 
dimensions starting from the 
(
) and 
matrices in 
dimensions.
Explicitly we have
Then we can construct the charge conjugation matrices
with the following properties
Starting from the values for
, 
we can compute all the signs 
which have a periodicity of 8, explicitly we find
+1 +1 −1 −1 
+1 −1 −1 +1
Again we can define the hermitian chiral matrix in
dimensions as
-
which is diagonal by construction and transforms under charge conjugation as
-
generic odd d = 2k + 1
We consider the previous construction for
(which is even) and then we simply take all 
(
) matrices to which we add 
( the 
is there in order to have an antihermitian matrix).
Finally we can compute the charge conjugation matrix: we have to choose between
and 
in such a way that 
transforms as all the others 
matrices. Explicitly we require
-
-
-
-
