result of the complexification, the finite-dimensional representations we will obtain this way may
not be unitary.
The finite-dimensional representations of the algebrasAandBare perfectly well-known. Each
is labeled by a non-negative integer or half-integer, whichwe shall denote byafor the algebraA,
andbfor the algebraB. The corresponding magnetic quantum numbers will be denotedmaand
mb, and have the customary ranges,
−a≤ma≤+a a−ma integer
−b≤mb≤+b b−mb integer (21.80)The states may be labeled byaandma, as follows,|a,ma〉, and the quantum numbers are defined
by
A~^2 |a,ma〉 = a(a+ 1)|a,ma〉
B~^2 |b,mb〉 = b(b+ 1)|b,mb〉
A 3 |a,ma〉 = ma|a,ma〉
B 3 |b,mb〉 = mb|b,mb〉 (21.81)Thus, we may label the finite-dimensional representations of the Lorentz algebra by a pair of non-
negative integers or half-integers,
(a,b) a,b≥ 0 , , 2 a, 2 b∈Z (21.82)SinceA~andB~ are neither Hermitian conjugates of one another, and not necessarily Hermitian
themselves, the representation (a,b) will in general be complex. However, the complex conjugate
of the representation (a,b) is effectively obtained by interchangingA~andB~, and thus transforms
under the representation (b,a). One may thus identify these two representations,
(a,b)∗= (b,a) (21.83)In particular, this means that the following representations,
(a,a) (a,b) + (b,a) (21.84)are always real, for anya,b. The dimension of a representation (a,b) is given by
dim(a,b) = (2a+ 1)(2b+ 1) (21.85)The representations (a,b) are genuine single-valued ifa+bis integer, and they are double valued
whena+bis a half-integer, so that
bosons (a,b) a+binteger +1
2
fermions (a,b) a+binteger (21.86)