alternative standard notation to the two-component van der Waerden notation
is to use the four components ofC^4 with the action of theγmatrices. The
relation between the two notations is given by
ΨA↔
(
ψB
φ
B ̇)
where the indexAon the left takes values 1, 2 , 3 ,4 and the indicesB,B ̇on the
right each take values 1,2.
Note that identifying Minkowski space with elements of the Clifford algebra
by
(x 0 ,x 1 ,x 2 ,x 3 )→/x=x 0 γ 0 +x 1 γ 1 +x 2 γ 2 +x 3 γ 3
identifies Minkowski space with certain 4 by 4 matrices. This again gives the
identification used earlier of Minkowski space with linear maps fromS∗RtoSL,
since the upper right two by two block of the matrix will be given by
−i(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 −x 0 −x 3)
and takesS∗RtoSL.
An important element of the Clifford algebra is constructed by multiplying
all of the basis elements together. Physicists traditionally multiply this byito
make it self-adjoint and define
γ 5 =iγ 0 γ 1 γ 2 γ 3 =(
−1 0
0 1
)
This can be used to produce projection operators from the Dirac spinors onto
the left and right-handed Weyl spinors
1
2(1−γ 5 )Ψ =ψL,1
2
(1 +γ 5 )Ψ =ψR∗There are two other commonly used representations of the Clifford algebra
relations, related to the one above by a change of basis. The Dirac representation
is useful to describe massive charged particles, especially in the non-relativistic
limit. Generators are given by
γ 0 D=−i(
1 0
0 − 1
)
,γD 1 =−i(
0 σ 1
−σ 1 0)
γ 2 D=−i(
0 σ 2
−σ 2 0)
,γD 3 =−i(
0 σ 3
−σ 3 0)
and the projection operators for Weyl spinors are no longer diagonal, since
γ 5 D=