Who discovered the covering homomorphism between SU(2) and SO(3)?

Who discovered this? It is quite nontrivial and very important in quantum mechanics.

• In old books on classical mechanics parametrization of $SO(3)$ by $SU(2)$ is called the Klein parametrization. – Alexandre Eremenko Oct 9 at 23:58

Hamilton and Klein, Klein was more explicit about it. Hamilton in Lectures on Quaternions (1853) realized that his representation of rotations of rigid bodies by the unit quaternions was not $$1$$-$$1$$, but $$2$$-$$1$$. Klein in Lectures on the Ikosahedron and the Solution of Equations of the Fifth Degree (1888) replaced the unit quaternions by $$2 × 2$$ unitary matrices with the determinant $$1$$, now denoted $$SU(2)$$. He then more or less spelled out that the unit quaternions and $$SU(2)$$ are isomorphic groups, which are $$2$$-$$1$$ epimorphic onto the group of 3D rotations $$SO(3)$$.

Pauli proposed the "two-valuedness not describable classically", which was later identified with the electron spin, in 1924, and formalized it in the matrix form in 1927. In 1932 Heisenberg and Ivanenko guessed that the same effect regulates protons/neutrons as the states of a single particle, later dubbed nucleon, and incorporated it into their proton–neutron model of the nucleus.

Steiner cites this homomorphism as a prime example of "unreasonable effectiveness" of mathematics. Both times the mathematical machinery developed was not aimed, even indirectly, at the application it ended up being useful for. In the case of nucleus, any visible connection to rotations and 3D space is missing altogether.

• But, in case of the isospin symmetry, it is not actually an exact symmetry of nature either. – Vladimir F Oct 10 at 19:32

Before Hamilton (1847) one should cite Euler (1771), Gauss (1819), Rodrigues (1840), and Cayley (1845). Detailed references in e.g.

Pujol, J., Hamilton, Rodrigues, Gauss, quaternions, and rotations: a historical reassessment, Commun. Math. Anal. 13, No. 2, 1-14 (2012). ZBL1268.01010.

Specifically, to four numbers $$p,q,r,s$$ with $$pp+qq+rr+ss=u$$, Euler attached

which is precisely (the transpose of) the rotation attached to $$(a,b,c,d)=\dfrac{(p,q,r,s)}{\sqrt u}$$ in Wikipedia: $$R = \begin{pmatrix} a^2+b^2-c^2-d^2&2bc-2ad &2bd+2ac \\ 2bc+2ad &a^2-b^2+c^2-d^2&2cd-2ab \\ 2bd-2ac &2cd+2ab &a^2-b^2-c^2+d^2\\ \end{pmatrix}. \tag1$$ So he had the map — though not AFAIK the group law on 4-tuples that makes it a homomorphism; that law is commonly attributed to Gauss or Rodrigues.1 If you want it in terms of literally $$SU(2) =\left\{\begin{pmatrix}a+bi &-c+di\\c+di&\phantom{-}a-bi\end{pmatrix}: (a,b,c,d)\in S^3\right\}, \tag2$$ the question becomes who first represented quaternions this way: the possibility is mentioned in Cayley (1858) and Laguerre (1867)’s first memoirs on matrices, and maybe first made explicit in Peirce (1882) and four papers of Sylvester (1882-83). Alternatively, one can ask who first identified rotations of the Riemann sphere as homographies $$\smash{z\mapsto\frac{Az+B}{Cz+D}}$$ with $$\smash{\left(\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right)}$$ in $$(2)$$: Klein credits Cayley (1879), anticipated again by Gauss (c.1819).

1. Although puzzingly, Euler two pages on displays a 4 x 4 array which is very nearly a quaternion product: it is $$\left(\begin{array}{rr|rr} p&-q&-r&-s\\ q&p&s&-r\\ \hline r&-s&p&q\\ s&r&-q&p \end{array}\right) \begin{pmatrix} 1&\\ &-1\\ &&-1\\ &&&-1 \end{pmatrix} \left(\begin{array}{rr|rr} a&-b&-c&-d\\ b&a&d&-c\\ \hline c&-d&a&b\\ d&c&-b&a \end{array}\right). \tag3$$ (The first column of this array is already in his famous letter to Goldbach, 4 May 1748.) Euler’s paper and map are cited by Cayley (1855), Hankel (1867), Jacobi (1884), Darboux (1887), Koenigs (1897), Schoenflies (1902), Cartan-Study (1908), Bourbaki. With alternative notation $$(a,b,c,d)=\left(\sqrt{\tfrac{M\vphantom Q}4},\sqrt{\tfrac{N\vphantom Q}4},\sqrt{\tfrac{P\vphantom Q}4},\sqrt{\tfrac{Q}4}\,\right) \tag4$$ his rotation $$(1)$$ is also in Monge (1786), Monge (1787), Lacroix (1797), Encke (1830), Rodrigues (1840). Rodrigues is cited by Cayley (1845) who is cited by Hamilton (1847).