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$SO(1,3)$ has an infinite number of representations, classified by the Casimir invariant $p^2$.

$SO(3)$ also has an infinite number of representations, classified by the Casimir invariant $\mathbf J^2$.

Since representations are diffeomorphic if and only if their Casimir invariants are the same, we are justified in this method of classification.

In the case of $SO(3)$, the physical interpretation is:

  • $\mathbf J$ generates rotations of the particle’s rest frame.

  • $\mathbf J^2$, the total spin of a particle, is the dimension of the vector space in which we have chosen to embed the particle.

Now I am baffled by the fact that we use $\mathbf J^2$, i.e. total spin, to classify $SO(1,3)$. That's the wrong Lie group! How is this not nonsense?

$p^2$ is the correct Casimir invariant - what happened to that?

  • Why isn't $p^2$ sufficient? - it's a Casimir invariant, and so it should give us all the classification information (i.e. tell us if reps are diffeomorphic)!

  • Now, suppose that we do things correctly (i.e. discard $\mathbf J^2$) and use $p^2$ to classify representations.

    • Are there “fermions” or “bosons” corresponding to $m$ taking on half or whole integer values in this case?

    • Finally, the representation $m^2=3$ is not isomorphic to $m^2=\pi$ (because $p^2$ is a Casimir invariant). Same with $m^2=2$ and $m^2=2.00000001$. However, in most field theory textbooks, $m>0$ is treated as one case. It's all a blob to them. What?!!!

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The Poincare group has two Casimir Invariants - namely $p^2$ and $W^2$ where $$ W_\mu = \frac{1}{2} \epsilon_{\mu\nu\rho\sigma} J^{\nu\rho} p^\sigma $$ is the Pauli-Lubanski pseudo-vector. Thus, representations of the Lorentz group are labelled by the eigenvalues of both $p^2$ and $W^2$.

  1. When $p^2 = -m^2$, we have the property $W^2 = -m^2 {\bf J}^2$. Thus, massive states are represented by their mass $m^2$ and their eigenvalue under ${\bf J}^2$ which by the representation theory of $SO(3)$ is $\hbar^2 s (s+1)$ for $s$ half-integer. Thus, all massive representations are labelled by $m^2$ and $s$. The spin $s$ representation is $2s+1$ dimensional.

  2. When $p^2 = 0$, there are generally two possibilities for $W_\mu$.

    • When $\vec{W} \not\propto \vec{p}$, then one obtains an infinite-dimensional representation which is not observed in nature (known as the continuous spin representations) and are therefore not considered in physics. However, it is precisely these representations that give rise to gauge invariance in a quantum theory.

    • When $\vec{W} \propto \vec{p}$, then consistency with the Poincare algebra implies that $\vec{W} = \vec{0}$ and the Casimiar invariant is simply $W^0 = \vec{J} \cdot \vec{P}$ (or rather $(W^0)^2$). Massless states are therefore labelled by their eigenvalue under $h = \frac{\vec{J} \cdot \vec{P}}{P^0} $ which is known as the helicity of the state.

In general, massless states are labelled by a single number $h$ and have one d.o.f. However, under parity $h \to - h$. Thus, in any theory with parity invariance, one must define a particle as a state with $h \oplus -h$ representation, thus giving two d.o.f. for each particle.

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