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

Question

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

Answer 1

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.

Answer 2

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 (1771, §33) 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 maybe not the group law on 4-tuples that makes it a homomorphism:¹ that (or less anachronistically, a "formula for the parameters of a composite rotation") is commonly attributed to Rodrigues (1840, p. 408), who put everything in the notation $$ (a,b,c,d)=\left(\cos\tfrac\theta2,\ \sin\tfrac\theta2\cos g,\ \sin\tfrac\theta2\cos h,\ \sin\tfrac\theta2\cos l\right). \tag2 $$ Then Cayley (1845, pp. 123-124) identified Rodrigues' multiplication of 4-tuples $(2)$ as quaternion multiplication, and the map $(a,b,c,d)\mapsto R$ as what we would call the adjoint representation of $\mathit{Sp}(1)$; and Hamilton (1847, pp. 13-14) concurred — as did also Boole (1848) and Donkin (1851).

One may wonder why Euler wasn't cited at the time. As far as I can tell, it's because Monge (1786) had (ostensibly independently) published $(1)$ in the alternative notation $$ (a,b,c,d)=\left(\sqrt{\tfrac{M\vphantom Q}4},\sqrt{\tfrac{N\vphantom Q}4},\sqrt{\tfrac{P\vphantom Q}4},\sqrt{\tfrac{Q}4}\,\right), \tag3 $$ and for many years everyone,² up to and including Rodrigues (p. 405), cited that instead. Only once Euler's paper was reprinted in a book (1849, p. 440) did everyone³ switch to citing him.

So far everything has been in terms of the sphere $S^3\subset\mathbb R^4$, or unit quaternions. If you want the homomorphism in terms of literally $$ SU(2) =\left\{\begin{pmatrix}a+bi &-c+di\\c+di&\phantom{-}a-bi\end{pmatrix}: (a,b,c,d)\in S^3\right\}, \tag4 $$ the question becomes who first represented quaternions this way. The first memoirs on matrices by Cayley (1858), Laguerre (1867), and Frobenius (1877) all mention the possibility, but apparently left it to be done explicitly by Peirce (1882) and four papers by Sylvester (1882-83). On the other hand, one might argue that Hermite (1850, footnote) had it "before matrices", or 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 $(4)$: for this, Klein clearly credits Cayley (1879).

Finally, as is often the case, it later emerged that unpublished papers of Gauss (c.1819) already had both quaternion multiplication (p. 359) and rotations as homographies (p. 355).

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^{1. Opinions differ: e.g. Cartan-Study (link below) say that Euler had the composition formula. Maybe they consider he'd have found it obvious, or thought of the bijection $\smash{\mathbb{RP}^3}\to SO(3)$ rather than the covering $\smash{S^3}\to SO(3)$, making the question moot; or they understand, better than I do, why his next § displayed a 4 x 4 array which is very nearly a quaternion product: it is $$ \left(\begin{array}{rr|rr} p&-q&-r&-s\\ q&p&s&-r\\ \hline r&-s&p&q\\ s&r&-q&p \end{array}\right) \begin{pmatrix} 1&\\ &-1\\ &&-1\\ &&&-1 \end{pmatrix} \left(\begin{array}{rr|rr} a&-b&-c&-d\\ b&a&d&-c\\ \hline c&-d&a&b\\ d&c&-b&a \end{array}\right). \tag5 $$ (The first column of this array is right out of his famous letter to Goldbach (1748), and is also in his papers E242 (1760) and E445 (1773), as well as in Lagrange (1772), Legendre (1797) and the English translation of his Algebra (1810).)}

^{2. E.g. Monge (1787), Lacroix (1797), Hachette (1813), Encke (1830), Grunert (1832), Grunert (1833), Cayley (1862).}

^{3. E.g. Hamilton (1853), Cayley (1855), Lebesgue (1856), Salmon (1866), Hankel (1867), Hoüel (1874), Jacobi (1884), Darboux (1887), Study (1890), Beez (1896), Koenigs (1897), Schoenflies (1902), Cartan-Study (1908), Müller (1910), Muir (1911), Whittaker (1917), Bourbaki.}