{"id":21908,"date":"2025-04-18T16:09:35","date_gmt":"2025-04-18T14:09:35","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=21908"},"modified":"2025-05-25T15:08:20","modified_gmt":"2025-05-25T13:08:20","slug":"s3-archimedes-su2-and-the-semicircle","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2025\/04\/18\/s3-archimedes-su2-and-the-semicircle\/","title":{"rendered":"S3, Archimedes, SU2, and the semicircle"},"content":{"rendered":"
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Archimedes death, by Edouard Vimont (1846 - 1930)<\/figcaption><\/figure>\n

This post is about the semicircle distribution, and the way it appears in special unitary $2\\times 2$ matrices. The link passes through the sphere $S^3$ and the Archimedes theorem.<\/p>\n

Semicircle.<\/strong> The semicircle distribution on the interval $[-2\\sigma,2\\sigma]$, $\\sigma > 0$, is \\[ \\mathrm{d}\\mu^{\\mathrm{SC}}_\\sigma(x) =\\frac{\\sqrt{4\\sigma^2-x^2}}{2\\pi\\sigma^2}\\mathbf{1}_{x\\in[-2\\sigma,2\\sigma]}(x)\\mathrm{d}x. \\] It has mean $0$ and variance $\\sigma^2$. It is a symmetric $\\mathrm{Beta}_{[-2\\sigma,2\\sigma]}(\\frac{3}{2},\\frac{3}{2})$ distribution since \\[ \\sqrt{4\\sigma^2-x^2}=(2\\sigma-x)^{\\frac{3}{2}-1}(2\\sigma+x)^{\\frac{3}{2}-1}. \\] It is the projection on $\\mathbb{R}$ of the uniform distribution on the disc of radius $2\\sigma$.<\/p>\n

Archimedes and projection of $S^3$.<\/strong> The famous extended Archimedes theorem on the sphere and the cylinder states that if $(U_1,\\ldots,U_{n+2})$ is uniformly distributed on the unit sphere $S^{n+1}=\\{x\\in\\mathbb{R}^{n+2}:x_1^2+\\cdots+x_{n+2}^2=1\\}$ of $\\mathbb{R}^{n+2}$, then its projection $(U_1,\\ldots,U_n)$ on $\\mathbb{R}^n$ is uniformly distributed on the unit ball $\\{x\\in\\mathbb{R}^n:x_1^2+\\cdots+x_n^2\\leq1\\}$. In particular, the projection on $\\mathbb{R}^2$ of the uniform distribution on $S^3$ is the uniform distribution on the unit disc. Also its projection on $\\mathbb{R}$ is the semicircle distribution $\\mu^{\\mathrm{SC}}_{\\frac{1}{2}}$.<\/p>\n

Let us retain that if a random vector is uniformly distributed on $S^3$ then each of its $4$ coordinates are distributed according to the semicircle distribution $\\mu^{\\mathrm{SC}}_{\\frac{1}{2}}$.<\/p>\n

Note that $S^3$ can also be understood as the quaternionic unit sphere via \\[ |x|=\\sqrt{x_1^2+x_2^2+x_3^2+x_4^2} \\quad\\text{where}\\quad x=x_1+x_2\\mathrm{i}+x_3\\mathrm{j}+x_4\\mathrm{k}\\in\\mathbb{H}\\equiv\\mathbb{R}^4, \\] just like $S^1$ and $S^2$ are the unit spheres of $\\mathbb{R}$ and $\\mathbb{C}$ respectively.<\/p>\n

$S^3$ is diffeomorphic to $\\mathrm{SU}(2)$.<\/strong> The special unitary group $\\mathrm{SU}(2)$ is the set of $2\\times 2$ unitary matrices with determinant equal to $1$. Let us write $U\\in\\mathrm{SU}(2)$ as \\[ U=\\begin{pmatrix} a & c\\\\ b & d\\end{pmatrix},\\quad a,b,c,d\\in\\mathbb{C}. \\] The condition $U\\in\\mathrm{U}(2)$ gives $|a|^2+|b|^2=1$, $|c|^2+|d|^2=1$, and $(c,d)\\perp(a,b)$, hence $(c,d)=\\mathrm{e}^{\\mathrm{i}\\theta}(-\\overline{b},\\overline{a})$, for some phase $\\theta\\in[0,2\\pi)$. Next, the unit determinant condition reads $1=ad-bc=(|a|^2+|b|^2)\\mathrm{e}^{\\mathrm{i}\\theta}$, hence $\\theta=0$. We have obtained the parametrization \\[ U=\\begin{pmatrix} a & -\\overline{b}\\\\ b & \\overline{a}\\end{pmatrix},\\quad a,b\\in\\mathbb{C},\\quad |a|^2+|b|^2=1. \\] In other words $\\mathrm{SU}(2)$ is diffeomorphic to $S^3$, which is also the quaternionic unit sphere. The uniform probability measure on $S^3$ gives the uniform probability measure on $\\mathrm{SU}(2)$, which is the normalized Haar measure on this group. Also, combining with what we have already seen on $S^3$, if $U$ is a random matrix distributed according to the normalized Haar measure on $\\mathrm{SU}(2)$, then $\\Re U_{11}$ follows the semicircle distribution $\\mu^{\\mathrm{SC}}_{\\frac{1}{2}}$.<\/p>\n

Eigenvalues of $\\mathrm{SU}(2)$.<\/strong> If $U\\in\\mathrm{SU}(2)$, then its eigenvalues $\\lambda_1$ and $\\lambda_2$ belong to the unit circle of the complex plane, while $1=\\det(U)=\\lambda_1\\lambda_2$, hence $\\lambda_1=\\mathrm{e}^{\\mathrm{i}\\theta}$ and $\\lambda_2=\\mathrm{e}^{-\\mathrm{i}\\theta}$, for some $\\theta\\in[0,\\pi)$. Thus, using the fact that $U_{22}=\\overline{U_{11}}$, we get \\[ 2\\Re U_{11}=\\mathrm{Tr}(U)=2\\cos(\\theta). \\] It follows that if $U$ is a random matrix distributed according to the normalized Haar measure on $\\mathrm{SU}(2)$, then $\\mathrm{Tr}(U)=2\\cos(\\theta)$ follows the semicircle distribution $\\mu^{\\mathrm{SC}}_1$.<\/p>\n

Conjugacy classes of $\\mathrm{SU}(2)$.<\/strong> Two elements $U,V\\in\\mathrm{SU}(2)$ have same conjugacy class iff $U=WVW^*$ for some $W\\in\\mathrm{SU}(2)$. This is equivalent to say that they have the same eigenvalues $\\{\\mathrm{e}^{\\pm\\mathrm{i}\\theta}\\}$, in other words, the same trace, as seen previously. If we parametrize the set of conjugacy classes by the trace, then it follows the semicircle distribution $\\mu^{\\mathrm{SC}}_1$.<\/p>\n

Elliptic curves and Sato-Tate conjecture.<\/strong> Formulated independently by Mikio Sato (1928 - 2023) and John Torrence Tate Jr (1925 - 2019) around 1960, it states that for any fixed elliptic curve over the rationals, without complex multiplication, denoting $N$ its conductor or complexity, $N_p$ the number of its points mod $p$, $\\mathcal{P}_n$ the set of prime numbers $\\leq n$ not dividing $N$, and $|\\mathcal{P}_n|$ its cardinal, we have the asymptotic equipartition \\[ \\frac{1}{|\\mathcal{P}_n|}\\sum_{p\\in\\mathcal{P}_n}\\delta_{a_p} \\xrightarrow[n\\to\\infty]{\\mathrm{weak}} \\mu^{\\mathrm{SC}}_1 \\quad\\text{where}\\quad a_p=\\frac{(p+1)-N_p}{\\sqrt{p}}. \\] The weak (or narrow) convergence is with respect to continuous and bounded test functions, namely the convergence in law of probability theory. The link with $\\mathrm{SU}(2)$ is via the interpretation of $a_p$ as a normalized trace of a Frobenius endomorphism. This conjecture was essentially proved around 2008 by Laurent Clozel (1953 - ), Michael Harris (1954 - ), Nicholas Shepherd-Barron (1955 - ), and Richard Taylor (1962 - ).<\/p>\n

Elliptic curves are fascinating objects from algebraic number theory, connected to modular form by the Shimura-Taniyama correspondence, after Gor\u014d Shimura (1930 - 2019) and Yutaka Taniyama (1927 - 1958), at the heart of the proof by Andrew Wiles (1953 - ) and Richard Taylor (1962 - ) of the last theorem of Pierre de Fermat (196? - 1665). They are also used in cryptography - in particular for cryptocurrencies such as Bitcoin - an application that motivated a conjecture formulated in 1976 by Serge Lang (1927 - 2005) and Hale Trotter (1931 - 2022).<\/p>\n

Further reading.<\/strong><\/p>\n

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