Restricting to theSU(2) subgroup ofSL(2,C), all these representations
are unitary, and equivalent. AsSL(2,C) representations, they are not unitary,
and while the representations are equivalent to their duals,SL andSRare
inequivalent (since as we have seen, one cannot complex conjugateSL(2,C)
matrices by a matrix conjugation).
For the case of the (^12 ,^12 ) representation, to see explicitly the isomorphism
betweenSL⊗SRand vectors, recall that we can identify Minkowski space with
2 by 2 self-adjoint matrices. Ω∈SL(2,C) acts by
(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3
)
→Ω
(
x 0 +x 3 x 1 −ix 2
x 1 +ix 2 x 0 −x 3)
Ω†
We can identify such matrices as linear maps fromSR∗toSL(and thus isomor-
phic to the tensor productSL⊗(SR∗)∗=SL⊗SR, see chapter 9).
41.2 Diracγmatrices and Cliff(3,1)
In our discussion of the fermionic version of the harmonic oscillator, we defined
the Clifford algebra Cliff(r,s) and found that elements quadratic in its gener-
ators gave a basis for the Lie algebra ofso(r,s) =spin(r,s). Exponentiating
these gave an explicit construction of the groupSpin(r,s). We can apply that
general theory to the case of Cliff(3,1) and this will give us the representations
(^12 ,0) and (0,^12 ).
If we complexify ourR^4 , then its Clifford algebra becomes the algebra of 4
by 4 complex matrices
Cliff(3,1)⊗C= Cliff(4,C) =M(4,C)We will represent elements of Cliff(3,1) as such 4 by 4 matrices, but should
keep in mind that we are working in the complexification of the Clifford algebra
that corresponds to the Lorentz group, so there is some sort of condition on the
matrices that needs to be kept track of to identify Cliff(3,1)⊂M(4,C). There
are several different choices of how to explicitly represent these matrices, and
for different purposes, different ones are most convenient. The one we will begin
with and mostly use is sometimes called the chiral or Weyl representation, and
is the most convenient for discussing massless charged particles. We will try
and follow the conventions used for this representation in [99]. Note that these
4 by 4 matrices act not on four dimensional space-time, but on spinors. It is
a special feature of 4 dimensions that these two different representations of the
Lorentz group have the same dimension.
Writing 4 by 4 matrices in 2 by 2 block form and using the Pauli matrices
σjwe assign the following matrices to Clifford algebra generators
γ 0 =−i
(
0 1
1 0
)
,γ 1 =−i(
0 σ 1
−σ 1 0)
,γ 2 =−i(
0 σ 2
−σ 2 0)
,γ 3 =−i(
0 σ 3
−σ 3 0