Definition(Spinor representation). The spinor representation ofSpin(3) =
SU(2)is the representation onC^2 given by
g∈SU(2)→πspinor(g) =gElements of the representation spaceC^2 are called “spinors”.
The spin representation ofSU(2) is not a representation ofSO(3). The
double cover map Φ :SU(2)→SO(3) is a homomorphism, so given a rep-
resentation (π,V) ofSO(3) one gets a representation (π◦Φ,V) ofSU(2) by
composition. One cannot go in the other direction: there is no homomorphism
SO(3)→SU(2) that would allow one to make the spin representation ofSU(2)
onC^2 into anSO(3) representation.
One could try and define a representation ofSO(3) by
g∈SO(3)→π(g) =πspinor( ̃g)∈SU(2)where ̃gis some choice of one of the two elements ̃g∈SU(2) satisfying Φ( ̃g) =g.
The problem with this is that it won’t quite give a homomorphism. Changing
the choice of ̃gwill introduce a minus sign, soπwill only be a homomorphism
up to sign
π(g 1 )π(g 2 ) =±π(g 1 g 2 )
The nontrivial nature of the double covering map Φ implies that there is no
way to completely eliminate all minus signs, no matter how one chooses ̃g(since
a continuous choice of ̃gis not possible for allgin a non-contractible loop of
elements ofSO(3)). Examples like this, which satisfy the representation prop-
erty only one up to a sign ambiguity, are known as “projective representations”.
So, the spinor representation ofSU(2) =Spin(3) can be used to construct a
projective representation ofSO(3), but not a true representation ofSO(3).
Quantum mechanics texts sometimes deal with this phenomenon by noting
that there is an ambiguity in how one specifies physical states inH, since mul-
tiplying a vector inHby a scalar doesn’t change the eigenvalues of operators
or the relative probabilities of observing these eigenvalues. As a result, the
sign ambiguity noted above has no physical effect since arguably one should be
working with states modulo the scalar ambiguity. It seems more straightfor-
ward though to not try and work with projective representations, but just use
the larger groupSpin(3), accepting that this is the correct group reflecting the
action of rotations on three dimensional quantum systems.
The spin representation is more fundamental than the vector representa-
tion, in the sense that the spin representation cannot be found only knowing
the vector representation, but the vector representation ofSO(3) can be con-
structed knowing the spin representation ofSU(2). We have seen this using the
identification ofR^3 with 2 by 2 complex matrices, with equation 6.5 showing
that rotations ofR^3 correspond to conjugation by spin representation matrices.
Another way of seeing this uses the tensor product, and is explained in section
9.4.3. Note that taking spinors as fundamental entails abandoning the descrip-
tion of three dimensional geometry purely in terms of real numbers. While the