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  "body": "\n## Example\n\nLet $G = \\mathrm{SO}^+(1, 3)$ be the restricted [[Lorentz group|QWK3BYVJ9W]]: the linear maps of $\\R^4$ that preserve $t^2 - x^2 - y^2 - z^2$, have determinant $1$ and preserve the direction of time.\nIts universal cover is the group [[SL(2, C)|J58ED1GV6N]] $\\mathrm{SL}_2(\\C)$, a double cover with kernel $\\set{\\pm I}$.\n\nFor half-integers $m, n \\in \\set{0, \\frac12, 1, \\frac32, \\dots}$, let $V_{2m}$ be the space of homogeneous polynomials of degree $2m$ in two variables, with $\\mathrm{SL}_2(\\C)$ acting by $(\\pi_{2m}(g) P)(z) = P(g^{-1} z)$.\nRestricted to [[SU(2)|E5P8AG6CK7]], this is the irreducible representation of dimension $2m + 1$.\nPut\n$$\\pi_{m,n}(g) = \\pi_{2m}(g) \\otimes \\pi_{2n}(\\bar g) \\quad \\text{on} \\quad V_{2m} \\otimes V_{2n},$$\nwhere $\\bar g$ is the matrix with complex conjugate entries.\nThis is a continuous homomorphism, so it is a [[representation|7JA7WH92K0]] of $\\mathrm{SL}_2(\\C)$ of dimension $(2m + 1)(2n + 1)$.\n\nThe classification: the $\\pi_{m,n}$ are [[irreducible|FX3NEFYFFN]] and pairwise non-isomorphic.\nView $\\mathrm{SL}_2(\\C)$ as a real Lie group.\nThen each irreducible finite-dimensional complex representation of it is isomorphic to exactly one $\\pi_{m,n}$.\nThe element $-I$ acts by $(-1)^{2m + 2n}$, so $\\pi_{m,n}$ is a representation of $G$ exactly when $m + n$ is an integer.\n\nCheck on the Lie algebra: the complexification of $\\mathfrak{so}(1, 3)$ is $\\mathfrak{sl}_2(\\C) \\oplus \\mathfrak{sl}_2(\\C)$.\nOn it, $\\pi_{m,n}$ becomes the tensor product of the irreducible representations of highest weights $2m$ and $2n$ of the two factors.\n\n## Remarks\n\nThe named cases are used throughout physics.\nThe representation $(0, 0)$ is the scalar one, $(\\frac12, 0)$ and $(0, \\frac12)$ are the left- and right-handed Weyl spinors, and $(\\frac12, \\frac12)$ is the defining action on four-vectors, after complexification.\nThe sum $(1, 0) \\oplus (0, 1)$ is the action on antisymmetric tensors such as the electromagnetic field tensor, and $(\\frac12, 0) \\oplus (0, \\frac12)$ is the Dirac spinor.\n\nNone of the $\\pi_{m,n}$ other than $(0, 0)$ is unitary, because $G$ is a noncompact simple Lie group.\nThe irreducible unitary representations of $\\mathrm{SL}_2(\\C)$ are infinite-dimensional, apart from the trivial one, and they were classified by Gelfand and Naimark.\n\n## Explanations\n\n### L1 human\n\nThe Lorentz group is the group of changes of reference frame in special relativity.\nPhysical quantities change in definite ways under it: numbers stay put, vectors rotate and boost, spinors change by a \"square root\" of this rule.\nThe finite-dimensional ways in which the group can act are labelled by two half-integers $(m, n)$.\nEach label gives one kind of field or particle, from scalars to spinors to the electromagnetic field.\n",
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