QuantumPhysics.dvi

coordinates of space-time, but usually, we may ultimately restrict to its real subalgebra when
needed. The relevant linear complexified combinations ofJkandKkare given as follows,

Ak≡

1

2

(Jk−iKk) Jk=Ak+Bk

Bk≡

1

2

(Jk+iKk) Kk=i(Ak−Bk) (21.76)

Their commutation relations are as follows,

[Ai,Bj] = 0

[Ai,Aj] = iεijkAk

[Bi,Bj] = iεijkBk (21.77)

The relation on the top line signifies that the sets of generatorsA = {A 1 ,A 2 ,A 3 } andB =
{B 1 ,B 2 ,B 3 } mutually commute, so that the full complexifiedLorentz algebra is a direct sum
of twoSU(2) algebras, considered with complex coefficients.

Finally, the representations of the Lorentz algebra have certain reality properties. If the finite-
dimensional representation, given by the generators Lμν is real, so that the matricesLμν are
real, andD(Λ) is real for all Λ, then, the rotation and boost generators,Jk,Kk must be purely
imaginary. As a result of the standard commutation relations of angular momentum, the generators
Jkmust be (equivalent to) Hermitian matrices. For boosts, however, the structure relations allow
for either anti-HermitianKkor HermitianKk. HermitianKkcorresponds to the rotation algebra
in 4 dimensions, not the Lorentz algebra. The Lorentz algebra corresponds to the other alternative
: HermitianKk. As a result, we then have the following complex conjugationrelation,

A†k=Bk (21.78)

Thus, the operation of complex conjugation interchanges the subalgebrasAandB.

21.10Representations of the Lorentz algebra

We have already encountered several representations of theLorentz algebra,

scalar dimension 1 φ

vector dimension 4 xμ, ∂μ, Aμ

anti−symmetric rank 2 tensor dimension 6 Fμν, Lμν (21.79)

Here, we wish to construct all finite-dimensional representations of the Lorentz algebra in a sys-
tematic way. This problem is greatly simplified by the observation that the complexified Lorentz
algebra is a direct sum of two complexifiedSU(2) algebras. Thus, our procedure will consist in
first constructing allcomplexfinite-dimensional representations of the complexified Lorentz alge-
bra, and then restrict those complex representations to real ones, when possible. Note that, as a