Step 1: Induced representation via little group (Mackey's method). The Poincare group is a semidirect product $T \rtimes L$ where $T = \mathbb{R}^{3,1}$ (translations) and $L = \mathrm{SL}(2,\mathbb{C})$ (Lorentz). By Mackey's theorem on semidirect products, irreducible representations are classified by orbits of $L$ on the dual of $T$ (momentum space $\hat{T} \cong \mathbb{R}^{3,1}$) and irreducible representations of the stabilizer (little group) of a chosen point on each orbit.

Step 2: Orbits in momentum space. The Lorentz group acts on 4-momenta $p^\mu$ preserving $p^2 = p_\mu p^\mu$. The orbits of $L$ on $\mathbb{R}^{3,1}$ are:

• Massive: $p^2 = m^2 > 0$, $p^0 > 0$. Hyperboloid. Standard momentum: $p_0 = (m,0,0,0)$.
• Massless: $p^2 = 0$, $p^0 > 0$. Forward light cone. Standard: $p_0 = (E,0,0,E)$.
• Spacelike: $p^2 < 0$ (unphysical for single particles).
• Zero: $p^\mu = 0$ (vacuum representation).

Step 3: Little groups. The little group $W_{p_0} = \{L \in \mathrm{SL}(2,\mathbb{C}) : Lp_0 = p_0\}$:

• Massive ($p_0 = (m,\mathbf{0})$): $W = \mathrm{SU}(2)$ (spatial rotations). Irreducible representations are labeled by spin $s = 0, \frac{1}{2}, 1, \ldots$ with dimension $2s+1$. The Casimir $W^2 = -m^2 s(s+1)$.
• Massless ($p_0 = (E,0,0,E)$): $W = \mathrm{ISO}(2) = \mathbb{R}^2 \rtimes \mathrm{SO}(2)$ (the Euclidean group of the plane). The $\mathbb{R}^2$ translations must act trivially (otherwise continuous spin, not observed in nature). The remaining $\mathrm{SO}(2)$ contributes the helicity $\lambda \in \frac{1}{2}\mathbb{Z}$.

Step 4: Induced representation. Given orbit $\mathcal{O}$ with standard momentum $p_0$ and little group representation $\sigma$, the Hilbert space is $\mathcal{H} = L^2(\mathcal{O}, V_\sigma, d\mu)$ -- square-integrable sections of a vector bundle over the orbit. The Poincare group acts as: $[U(\Lambda,a)\psi](p) = e^{ip\cdot a}\,\sigma(W(\Lambda,p))\,\psi(\Lambda^{-1}p)$ where $W(\Lambda,p) = L_{\Lambda p}^{-1}\Lambda L_p$ is the Wigner rotation (an element of the little group) and $L_p$ is a standard boost taking $p_0$ to $p$. This construction yields all irreducible unitary representations. $\square$