In general, quantum numbers are labels of irreducible representations of the relevant symmetry group, not primarily eigenvalues of an otherwise simply
defined operator.

But for every label that has a meaningful numerical value in every irreducible representation, one can define a Hermitian operator having it as an eigenvalue, simply by defining it as the sum of the projections to the irreducible subspaces multiplied by the label of this representation. It is not clear whether such an operator has any practical use.

This also holds for the spin. However, one can define the spin in a representation independent way, though not via eigenvalues.

The spin of an irreducible positive energy representation of the Poincare group is $s=(n-1)/2$, where $s$ is the smallest integer such that the representation occurs as part of the Foldy representation in $L^2(R^3,C^n)$
with inner product defined by

$~~~\langle \phi|\psi \rangle:= \displaystyle \int \frac{dp}{\sqrt{p^2+m^2}} \phi(p)^*\psi(p)$.

The Poincare algebra is generated by $p_0,p,J,K$ and acts on this space
as follows. $p$ is multiplication by $p$,

$~~~p_0 := \sqrt{m^2+p^2}$,

$~~~J := q \times p + S$,

$~~~K := \frac{1}{2}(p_0 q + q p_0) + \displaystyle\frac{p \times S}{m+p_0}$,

with the position operator $q := i \hbar \partial_p$ and the spin vector $S$ in a unitary irreducible representation of $so(3)$ on
the vector space $C^n$ of complex vectors of length $n$, with the same
commutation relations as the angular momentum vector.

The Poincare algebra is generated by $p_0,p,J,K$ and acts on this space irreducibly if $m>0$ (thus givning the spin $s$ representation), while it is reducible for $m=0$. Indeed, in the massless case, the helicity

$~~~\lambda := \displaystyle\frac{p\cdot S}{p_0}$,

is central in the universal envelope of the Lie algebra, and the possible eigenvalues of the helicity are $s,s-1,...,-s$, where $s=(n-1)/2$. Therefore, the eigenspaces of the helicity operator carry by restriction unitary
representations of the Poincare algebra (of spin $s,s-1,...,0$), which are easily seen to be irreducible.

The Foldy representation also exhibits the massless limit of the massive representations.

Edit: In the massless limit, the formerly irreducible representation becomes reducible. In a gauge theory, the form of the interaction (multiplication by a conserved current) ensures that only the irreducible representation with the highest helicity couples to the other degrees of freedom, so that the lower helicity parts have no influence on the dynamics, are therefore unobservable, and are therefore ignored.

This post imported from StackExchange Physics at 2016-02-15 17:16 (UTC), posted by SE-user Arnold Neumaier