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. The Weyl–Brauer matrices provide an explicit construction of higher-dimensional gamma matrices for Weyl spinors. Gamma matrices also appear in generic settings in Riemannian geometry, particularly when a spin structure can be defined.
Consider a space-time of dimension with the flat Minkowski metric,
η=\|ηa\|=diag(+1,...,+1,-1,...,-1)~,
p
q
p+q=d
\{ea\}
Clp,q(C)\congClC(p,q)
\{\Gammaa\}
ea
By convention, the gamma group is realized as a collection of matrices, the gamma matrices, although the group definition does not require this. In particular, many important properties, including the C, P and T symmetries do not require a specific matrix representation, and one obtains a clearer definition of chirality in this way.[1] Several matrix representations are possible, some given below, and others in the article on the Weyl–Brauer matrices. In the matrix representation, the spinors are
N
Most of the properties of the gamma matrices can be captured by a group, the gamma group. This group can be defined without reference to the real numbers, the complex numbers, or even any direct appeal to the Clifford algebra.[1] The matrix representations of this group then provide a concrete realization that can be used to specify the action of the gamma matrices on spinors. For
(p,q)=(1,3)
(p,q)=(3,0)
(p,q)=(0,3).
The presentation of the gamma group
G=Gp,q
I
i
i4=I
i
\Gammaa
a=0,\ldots,p-1
2 | |
\Gamma | |
a |
=I~,
\Gammaa, a=p,\ldots,p+q-1
2 | |
\Gamma | |
a |
=i2~,
\Gammaa\Gammab=i2\Gammab\Gammaa
a\neb~.
These generators completely define the gamma group. It can be shown that, for all
x\inG
x4=I
x-1=x3~.
x\inG
x=in\Gammaa\Gammab … \Gammac
a<b< … <c
and
0\len\le3.
2p+q+2
The gamma group is a 2-group but not a regular p-group. The commutator subgroup (derived subgroup) is
[G,G]=\left\{I,i2\right\}~,
\alpha
Given elements
\Gammai
(\Gammaa\Gammab … \Gamma
sf{t}= | |
c) |
\Gammac … \Gammab\Gammaa
If there are
k
\Gammai
(\Gammaa\Gammab … \Gamma
sf{t}= | |
c) |
\left(i2\right)
| |||||
\Gammaa\Gammab … \Gammac
Another automorphism of the gamma group is given by conjugation, defined on the generators as
\dagger | |
\Gamma | |
a |
=\begin{cases} \Gammaa&for0\lea<p\\
2\Gamma | |
i | |
a |
&forp\lea<p+q\\ \end{cases}
supplemented with
i\dagger=i3
I\dagger=I.
(ab)\dagger=b\daggera\dagger~.
x\inG
xx\dagger=x\daggerx=I
xx\dagger=x\daggerx=i2~,
If one interprets the
p
q
\gamma0
\gammai
The main involution is the map that "flips" the generators:
\alpha(\Gammaa)=
2\Gamma | |
i | |
a |
i
\alpha(i)=i~.
Define the chiral element
\omega\equiv\Gammachir
\omega=\Gammachir=\Gamma0\Gamma1 … \Gammad-1
where
d=p+q
\Gammaa\omega=\left(i2\right)d-1\omega\Gammaa
It squares to
\omega2=\left(i2\right)
| |||||
For the Dirac matrices, the chiral element corresponds to
\gamma5~,
For the Pauli group, the chiral element is
\sigma1\sigma2\sigma3=i
G3,0
\Gamma1\Gamma2\Gamma3
i2~.
G0,3
ijk=i2~.
None of the above automorphisms (transpose, conjugation, main involution) are inner automorphisms; that is they cannot be represented in the form
CxC-1
C
C+\Gammaa
-1 | |
C | |
+ |
=
sf{t} | |
\Gamma | |
a |
and
C-\Gammaa
-1 | |
C | |
- |
=
sf{t} | |
i | |
a |
The above relations are not sufficient to define a group;
C2
The gamma group has a matrix representation given by complex
\{\Gammaa~,~\Gammab\}=\Gammaa\Gammab+\Gammab\Gammaa=2ηaIN~,
For the remainder of this article,it is assumed that
p=1
q=d-1
\begin{align}
\dagger | |
\Gamma | |
0 |
&=+\Gamma0~,&
\dagger | |
\Gamma | |
a |
&=-\Gammaa~(a=1,...,d-1)~. \end{align}
Transposition will be denoted with a minor change of notation, by mapping
sf{t}\mapsto | |
\Gamma | |
a |
sf{T} | |
\Gamma | |
a |
As before, the generators all generate the same group (the generated groups are all isomorphic; the operations are still involutions). However, since the are now matrices, it becomes plausible to ask whether there is a matrix that can act as a similarity transformation that embodies the automorphisms. In general, such a matrix can be found. By convention, there are two of interest; in the physics literature, both referred to as charge conjugation matrices. Explicitly, these are
\begin{align} C(+)\Gammaa
-1 | |
C | |
(+) |
&=+
sf{T}\\ | |
\Gamma | |
a |
C(-)\Gammaa
-1 | |
C | |
(-) |
&=-
sf{T}~. \end{align} | |
\Gamma | |
a |
They can be constructed as real matrices in various dimensions, as the following table shows. In even dimension both
C\pm
d |
=C(+) |
=C(-) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 |
=C(+)
=1 |
=-C(-)
=-1 | ||||||||||||||||||||||||
3 |
=-C(-)
=-1 | |||||||||||||||||||||||||
4 |
=-C(+)
=-1 |
=-C(-)
=-1 | ||||||||||||||||||||||||
5 |
=-C(+)
=-1 | |||||||||||||||||||||||||
6 |
=-C(+)
=-1 |
=C(-)
=1 | ||||||||||||||||||||||||
7 |
=C(-)
=1 | |||||||||||||||||||||||||
8 |
=C(+)
=1 |
=C(-)
=1 | ||||||||||||||||||||||||
9 |
=C(+)
=1 | |||||||||||||||||||||||||
10 |
=C(+)
=1 |
=-C(-)
=-1 | ||||||||||||||||||||||||
11 |
=-C(-)
=-1 |
* | |
C | |
(\pm) |
=C(\pm)
We denote a product of gamma matrices by
\Gammaabc=\Gammaa ⋅ \Gammab ⋅ \Gammac … {}
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
\Gammaa
\Gamma | |
a1...an |
=
1 | |
n! |
\sum | |
\pi\inSn |
\epsilon(\pi)
\Gamma | |
a\pi(1) |
…
\Gamma | |
a\pi(n) |
~,
Typically, provide the (bi)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
\Gammachir=
| |||||
i |
\Gamma0\Gamma1...m\Gammad-1~,
A matrix is called symmetric if
(C
\Gamma | |
a1...man |
)sf{T}=+(C
\Gamma | |
a1...man |
)~;
In the previous expression, can be either
C(+)
C(-)
C(+)
C(-)
d | C | Symmetric | Antisymmetric | ||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3 | C(-) | \gammaa | I2 | ||||||||||||||||||||||||||||
4 | C(-) | \gammaa~,~
| I4~,~\gammachir~,~\gammachir\gammaa | ||||||||||||||||||||||||||||
5 | C(+) |
| I4~,~\Gammaa | ||||||||||||||||||||||||||||
6 | C(-) | I8~,~\Gammachir
~,~
| \Gammaa~,~\Gammachir~,~\Gammachir\Gammaa~,~
| ||||||||||||||||||||||||||||
7 | C(-) | I8~,~
| \Gammaa~,~
| ||||||||||||||||||||||||||||
8 | C(+) | I16~,~\Gammaa~,~\Gammachir~,~\Gammachir\Gamma
~,~
| \Gammachir\Gammaa~,~
~,~\Gammachir
~,~
| ||||||||||||||||||||||||||||
9 | C(+) | I16~,~\Gammaa~,~
~,~
|
~,~
| ||||||||||||||||||||||||||||
10 | C(-) | \Gammaa~,~\Gammachir~,~\Gammachir\Gammaa~,~
~,~\Gammachir
~,~
| I32~,~\Gammachir
~,~
~,~
~,~\Gammachir
| ||||||||||||||||||||||||||||
11 | C(-) | \Gammaa~,~
~,~
| I32~,~
~,~
|
The proof of the trace identities for gamma matrices hold for all even dimension. One therefore only needs to remember the 4D case and then change the overall factor of 4 to
\operatorname{tr}(IN)
d
The matrices can be constructed recursively, first in all even dimensions, = 2, and thence in odd ones, 2 + 1.
Using the Pauli matrices, take
\begin{align} \gamma0&=\sigma1,&\gamma1&=-i\sigma2 \end{align}
and one may easily check that the charge conjugation matrices are
\begin{align} C(+)=\sigma1=
* | |
C | |
(+) |
=s(2,+)
sf{T}&= | |
C | |
(+) |
s(2,+)
-1 | |
C | |
(+) |
&s(2,+)&=+1\\ C(-)=i\sigma2=
* | |
C | |
(-) |
=s(2,-)
sf{T}&= | |
C | |
(-) |
s(2,-)
-1 | |
C | |
(-) |
&s(2,-)&=-1~. \end{align}
One may finally define the hermitian chiral chir to be
\gammachir=\gamma0\gamma1=\sigma3=
\dagger | |
\gamma | |
chir |
~.
One may now construct the, matrices and the charge conjugations (±) in + 2 dimensions, starting from the,, and (±) matrices in dimensions.
Explicitly,
\begin{align} \Gammaa'&=\gammaa' ⊗ \sigma3~~~\left(a'=0,...,d-1\right)~,& \Gammad&=I ⊗ (i\sigma1)~,& \Gammad+1&=I ⊗ (i\sigma2)~. \end{align}
One may then construct the charge conjugation matrices,
\begin{align} C(+)&=c(-) ⊗ \sigma1~,& C(-)&=c(+) ⊗ (i\sigma2)~, \end{align}
with the following properties,
\begin{align} C(+)=
* | |
C | |
(+) |
=s(d+2,+)
sf{T}&= | |
C | |
(+) |
s(d+2,+)
-1 | |
C | |
(+) |
&s(d+2,+)&=s(d,-)\\ C(-)=
* | |
C | |
(-) |
=s(d+2,-)
sf{T}&= | |
C | |
(-) |
s(d+2,-)
-1 | |
C | |
(-) |
&s(d+2,-)&=-s(d,+)~. \end{align}
Starting from the sign values for = 2, (2,+) = +1 and (2,−) = −1, one may fix all subsequent signs (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 +2 dimensions as
\begin{align} \Gammachir&=\alphad+2\Gamma0\Gamma1...m\Gammad+1=\gammachir ⊗ \sigma3~,& \alphad&=
| |||||
i |
~, \end{align}
which is diagonal by construction and transforms under charge conjugation as
\begin{align} C(\pm)\Gammachir
-1 | |
C | |
(\pm) |
&=\betad+2
sf{T}~,& | |
\Gamma | |
chir |
\betad&=
| |||||
(-) |
~. \end{align}
It is thus evident that = 0.
Consider the previous construction for − 1 (which is even) and simply take all matrices, to which append its . (The is required in order to yield an antihermitian matrix, and extend into the spacelike metric).
Finally, compute the charge conjugation matrix: choose between
C(+)
C(-)
C(s)\Gammachir
-1 | |
C | |
(s) |
=\betad
sf{T}= | |
\Gamma | |
chir |
s
sf{T}~. | |
\Gamma | |
chir |
As the dimension ranges, patterns typically repeat themselves with period 8. (cf. the Clifford algebra clock.)