is the change of basis matrix relating the representation and its complex
conjugate.
This is no longer true forSL(2,C). Conjugation by a fixed matrix will not
change the eigenvalues of the matrix, and these can be complex (unlike
SU(2) matrices, which have real eigenvalues). So such a (matrix) conju-
gation cannot change allSL(2,C) matrices to their complex conjugates,
since in general (complex) conjugation will change their eigenvalues.The classification of irreducible finite dimensionalSU(2) representation was
done in chapter 8 by considering its Lie algebrasu(2), complexified to give
us raising and lowering operators, and this complexification issl(2,C). If one
examines that argument, one finds that it mostly also applies to irreducible finite
dimensionalsl(2,C) representations. There is a difference though: now flipping
positive to negative weights (which corresponds to change of sign of the Lie
algebra representation matrices, or conjugation of the Lie group representation
matrices) no longer takes one to an equivalent representation. It turns out that
to get all irreducibles, one must take both the representations we already know
about and their complex conjugates. One can show (we won’t prove this here)
that the tensor product of one of each type of irreducible is still an irreducible,
and that the complete list of finite dimensional irreducible representations of
sl(2,C) is given by:
Theorem(Classification of finite dimensionalsl(2,C) representations). The
irreducible representations ofsl(2,C)are labeled by(s 1 ,s 2 )forsj= 0,^12 , 1 ,....
These representations are given by the tensor product representations
(πs 1 ⊗πs 2 ,Vs^1 ⊗Vs^2 )where(πs,Vs)is the irreducible representation of dimension 2 s+ 1and(πs,Vs)
its complex conjugate. Such representations have dimension(2s 1 + 1)(2s 2 + 1).
All these representations are also representations of the groupSL(2,C) and
one has the same classification theorem for the group, although we will not try
and prove this. We will also not try and study these representations in general,
but will restrict attention to the cases of most physical interest, which are
- (0,0): The trivial representation onC, also called the “spin 0” or scalar
representation. - (^12 ,0): These are called left-handed (for reasons we will see later on) “Weyl
spinors”. We will often denote the representation spaceC^2 in this case as
SL, and write an element of it asψL. - (0,^12 ): These are called right-handed Weyl spinors. We will often denote
the representation spaceC^2 in this case asSR, and write an element of it
asψR. - (^12 ,^12 ): This is called the “vector” representation since it is the complexifi-
cation of the action ofSL(2,C) asSO(3,1) transformations of space-time