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## The spin-statistics theorem and the Berry-Robbins problem

In ​1​ , 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 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.

It turns out that the Berry-Robbins problem does indeed have a solution. Even more, Atiyah and Bielawski in ​2​ show the existence of a smooth 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.

In ​3​ and ​4​ , 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.

I am planning to introduce this problem in a subsequent post.

1. 1.
Berry MV, Robbins JM. Indistinguishability for quantum particles: spin, statistics and the geometric phase. Proc R Soc Lond A. Published online August 8, 1997:1771-1790. doi:10.1098/rspa.1997.0096
2. 2.
Atiyah M, Bielawski R. Nahm’s equations, configuration spaces and flag manifolds. Bull Braz Math Soc. Published online July 2002:157-176. doi:10.1007/s005740200007
3. 3.
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. Published online July 15, 2001:1375-1387. doi:10.1098/rsta.2001.0840
4. 4.
Atiyah M, Sutcliffe P. The geometry of point particles. Proc R Soc Lond A. Published online May 8, 2002:1089-1115. doi:10.1098/rspa.2001.0913