Chapter 41

Representations of the

Lorentz Group

Having seen the importance in quantum mechanics of understanding the repre-
sentations of the rotation groupSO(3) and its double coverSpin(3) =SU(2)
one would like to also understand the representations of the Lorentz group
SO(3,1) and its double coverSpin(3,1) =SL(2,C). One difference from the
SO(3) case is that all non-trivial finite dimensional irreducible representations
of the Lorentz group are non-unitary (there are infinite dimensional unitary ir-
reducible representations, of no known physical significance, which we will not
discuss). While these finite dimensional representations themselves only pro-
vide a unitary action of the subgroupSpin(3)⊂Spin(3,1), they will later be
used in the construction of quantum field theories whose state spaces will have
a unitary action of the Lorentz group.

41.1 Representations of the Lorentz group

In theSU(2) case we found irreducible unitary representations (πn,Vn) of di-
mensionn+1 forn= 0, 1 , 2 ,.... These could also be labeled bys=n 2 , called the
“spin” of the representation, and we will do that from now on. These represen-
tations can be realized explicitly as homogeneous polynomials of degreen= 2s
in two complex variablesz 1 ,z 2. For the case ofSpin(4) =SU(2)×SU(2), the
irreducible representations will be tensor products

Vs^1 ⊗Vs^2

ofSU(2) irreducibles, with the firstSU(2) acting on the first factor, the second
on the second factor. The cases 1 =s 2 = 0 is the trivial representation,s 1 =
1
2 ,s^2 = 0 is one of the half-spinor representations ofSpin(4) onC

(^2) ,s 1 = 0,s 2 =
1
2 is the other, ands^1 =s^2 =
1
2 is the representation on four dimensional
(complexified) vectors.