{"id":38,"date":"2021-04-14T22:19:10","date_gmt":"2021-04-14T22:19:10","guid":{"rendered":"https:\/\/josephmalkoun.com\/blog\/?p=38"},"modified":"2021-04-14T22:19:10","modified_gmt":"2021-04-14T22:19:10","slug":"the-spin-statistics-theorem-and-the-berry-robbins-problem","status":"publish","type":"post","link":"https:\/\/josephmalkoun.com\/blog\/the-spin-statistics-theorem-and-the-berry-robbins-problem\/","title":{"rendered":"The spin-statistics theorem and the Berry-Robbins problem"},"content":{"rendered":"\n

In \u200b1\u200b<\/sup><\/span> , Berry and Robbins propose an interesting way to obtain the spin-statistics theorem, which is close to the famous belt trick, though expressed more mathematically. They completely explain their construction for $2$ particles, but while attempting to extend their construction to $n$ particles, they faced a technical obstruction. This led to the Berry-Robbins problem.<\/p>\n\n\n\n

Let $C_n(\\mathbb{R}^3)$ denote the configuration space of $n$ distinct particles. Then the symmetric group $S_n$ acts on $C_n(\\mathbb{R}^3)$ by permuting the components of any configuration. Moreover, let $U(n)$ denote the group of unitary $n \\times n$ matrices and let $T^n$ denote the subgroup of diagonal unitary matrices. Then $S_n$ acts on $U(n)\/T^n$ by permuting the columns of a representive of any left coset $gT^n$ (where $g \\in U(n)$). The Berry-Robbins asks if there is a continuous mapping from $C_n(\\mathbb{R}^3)$ into $U(n)$, which is $S_n$ equivariant.<\/p>\n\n\n\n

It turns out that the Berry-Robbins problem does indeed have a solution. Even more, Atiyah and Bielawski in \u200b2\u200b<\/sup><\/span> show the existence of a smooth<\/em> solution of the Berry-Robbins. This then shows that the Berry and Robbins’ argument does carry through for $n$ particles. While Atiyah and Bielawski’s article is quite a nice article, involving Nahm’s equations, Lie algebras and even proposing a conjectural link with the work of Kazhdan and Lusztig, yet there is another attempt at solving the Berry-Robbins problem, which I personally find quite interesting. It is what is known as the Atiyah problem on configurations of points, or sometimes the Atiyah-Sutcliffe conjectures.<\/p>\n\n\n\n

In \u200b3\u200b<\/sup><\/span> and \u200b4\u200b<\/sup><\/span> , Atiyah, and then Atiyah and Sutcliffe, propose an alternative approach to the Berry-Robbins problem, which is very explicit and elementary. Indeed, to state the problem only involves the Hopf map. But for the approach to succeed in providing a solution to the Berry-Robbins problem, a linear independence conjecture has to hold. This is the so-called Atiyah-Sutcliffe conjecture 1.<\/p>\n\n\n\n

I am planning to introduce this problem in a subsequent post.<\/p>\n\n\n\n

  1. \n
    1. <\/div>
    Berry MV, Robbins JM. Indistinguishability for quantum particles: spin, statistics and the geometric phase. Proc R Soc Lond A<\/i>. Published online August 8, 1997:1771-1790. doi:10.1098\/rspa.1997.0096<\/a><\/div>\n <\/div>\n<\/li>
  2. \n
    2. <\/div>
    Atiyah M, Bielawski R. Nahm\u2019s equations, configuration spaces and flag manifolds. Bull Braz Math Soc<\/i>. Published online July 2002:157-176. doi:10.1007\/s005740200007<\/a><\/div>\n <\/div>\n<\/li>
  3. \n
    3. <\/div>
    Atiyah M. Configurations of points. Arnold VI, Bruce JW, Moffatt HK, Pelz RB, eds. Philosophical Transactions of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences<\/i>. Published online July 15, 2001:1375-1387. doi:10.1098\/rsta.2001.0840<\/a><\/div>\n <\/div>\n<\/li>
  4. \n
    4. <\/div>
    Atiyah M, Sutcliffe P. The geometry of point particles. Proc R Soc Lond A<\/i>. Published online May 8, 2002:1089-1115. doi:10.1098\/rspa.2001.0913<\/a><\/div>\n <\/div>\n<\/li><\/ol><\/section>\n\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    In \u200b1\u200b , Berry and Robbins propose an interesting way to obtain the spin-statistics theorem, which is close to the famous belt trick, though expressed more mathematically. They completely explain their construction for $2$ particles, but while attempting to extend their construction to $n$ particles, they faced a technical obstruction. This led to the Berry-Robbins problem. Let $C_n(\\mathbb{R}^3)$ denote the configuration space of $n$ distinct particles. Then the symmetric group $S_n$ acts on $C_n(\\mathbb{R}^3)$ by permuting the components of any…<\/p>\n

    Read More Read More<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,4,2],"tags":[14,13,12],"class_list":["post-38","post","type-post","status-publish","format-standard","hentry","category-geometry","category-mathematical-physics","category-mathematics","tag-atiyah-problem-on-configurations","tag-berry-robbins-problem","tag-spin-statistics-theorem"],"_links":{"self":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts\/38","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/comments?post=38"}],"version-history":[{"count":13,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts\/38\/revisions"}],"predecessor-version":[{"id":54,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/posts\/38\/revisions\/54"}],"wp:attachment":[{"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/media?parent=38"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/categories?post=38"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josephmalkoun.com\/blog\/wp-json\/wp\/v2\/tags?post=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}