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  "body": "\n## Statement\n\nLet $\\tilde P = \\R^{1,3} \\rtimes \\mathrm{SL}_2(\\C)$ be the universal cover of the identity component of the [[Poincaré group|8ZQ9TA4Q2Q]].\nHere $\\R^{1,3}$ is Minkowski space with the form $p \\cdot p = p_0^2 - p_1^2 - p_2^2 - p_3^2$, and [[SL(2,C)|J58ED1GV6N]] acts on it through the covering map onto the restricted Lorentz group.\nThe group $\\mathrm{SL}_2(\\C)$ also acts on the momentum space $\\widehat{\\R^{1,3}} \\cong \\R^{1,3}$ of characters $x \\mapsto e^{i p \\cdot x}$ of the translations.\nFor a momentum $p$, let $H_p \\subseteq \\mathrm{SL}_2(\\C)$ be its stabilizer, the little group.\n\n1. The irreducible [[unitary representations|74BPYJNZTM]] of $\\tilde P$, up to equivalence, are in bijection with certain pairs $(\\mathcal{O}, \\sigma)$. Here $\\mathcal{O}$ is an [[orbit|0YQPZB5Q1G]] of $\\mathrm{SL}_2(\\C)$ in momentum space. And $\\sigma$ is an [[irreducible|FX3NEFYFFN]] unitary representation of $H_p$ for a chosen $p \\in \\mathcal{O}$, up to equivalence.\n2. The representation attached to $(\\mathcal{O}, \\sigma)$ is the [[induced representation|6Q7AHAZ6Q4]] $\\mathrm{Ind}_{\\R^{1,3} \\rtimes H_p}^{\\tilde P} (e^{i p \\cdot x} \\sigma)$.\n3. For the orbit $p \\cdot p = m^2$ with $m \u003e 0$ and $p_0 \u003e 0$, the little group is isomorphic to [[SU(2)|E5P8AG6CK7]], and $\\sigma$ is labelled by a spin $s \\in \\set{0, \\tfrac12, 1, \\tfrac32, \\dots}$.\n4. For the orbit $p \\cdot p = 0$ with $p_0 \u003e 0$, the little group is isomorphic to a double cover of the Euclidean group of the plane. The representations $\\sigma$ that are trivial on its translation part are characters labelled by a helicity $h \\in \\tfrac12 \\Z$; the others form the continuous spin representations.\n\n## Remarks\n\nThe translations form an abelian normal subgroup, so the [[Mackey machine|12CEWPMSG4]] applies.\nThe orbits of $\\mathrm{SL}_2(\\C)$ on momentum space are the hyperboloids $p \\cdot p = m^2$, the two halves of the light cone, the hyperboloids $p \\cdot p \u003c 0$ of one sheet, and the point $0$.\nThese orbits are locally closed, which is the condition under which Mackey's theory gives a complete classification by orbits and little groups.\nThe remaining orbits ($p \\cdot p \u003c 0$, $p_0 \u003c 0$, and $p = 0$) give representations that are not used for physical particles.\nFor example, an electron is described by the orbit with mass $m \u003e 0$ and spin $s = \\tfrac12$, and a photon by the light cone orbit with helicities $\\pm 1$.\nThe [[semidirect product|0H3WZXCZNF]] structure is essential: without the normal abelian subgroup of translations there is no momentum space to start from.\n\n## Explanations\n\n### L1 human\n\nSymmetry of spacetime under shifts, rotations and boosts forms the Poincaré group.\nWigner found all the elementary ways in which this group can act on the states of a quantum system.\nEach such way is labelled by a mass and by a spin or a helicity.\nThis is why physicists say that a particle \"is\" an irreducible representation of the Poincaré group.\n\n### L2 human\n\nThe method is the general one for semidirect products $A \\rtimes H$ with $A$ abelian.\nFirst choose a character of $A$, here a momentum $p$, then a representation of its stabilizer in $H$, and induce.\nFor massive particles the stabilizer of $(m, 0, 0, 0)$ is $\\mathrm{SU}(2)$, which explains why spin takes half-integer values.\nThe same method classifies the representations of the Euclidean motion groups and of the Heisenberg-type semidirect products.\n",
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