In quantum field theory, one defines a particle as a unitary irreducible representations of the Poincaré group. The study of these representations allows to define the mass and the spin of the particle. However, the spin is not defined the same way for massive particles (where the eigenvalue of the Pauli-Lubanski vector squared are $-m^2 s(s+1)$ where $$s = -S, -S +1, \cdots, S,$$ and $S$ is the spin of the particle (and $m$ the mass)) and massless particles (where the helicity has eigenvalues $\pm \lambda$ and $S=\left|\lambda\right|$ – not to mention continuous spin representations).
With this definition, the "spin" $S$ that appears in both cases doesn't seem to be exactly the same thing. The eigenvalue that labels the irreducible representations are not from the same operator in the massive and the massless case … however, it's tempting (for me) to see the maximum value of these eigenvalues as the same physical quantity if both cases.
So I wonder if there is a more general definition that would embrace both massive and massless particles (even less practicable)?
Best Answer
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 (units are such that $c=1$):
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.