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


2 Answers 2


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.

  • $\begingroup$ But, in case of the isospin symmetry, it is not actually an exact symmetry of nature either. $\endgroup$ Commented Oct 10, 2019 at 19:32
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    $\begingroup$ Now that I have the points to comment, here is what I’d have been content to simply say: No! To answer "Hamilton and Klein" is to ignore swaths of well-documented history — and talk of Pauli, Heisenberg, etc., is besides the OP’s focused question. Less anecdotally, the fact that this answer stood not only unquestioned, but heavily upvoted, praised as "fantastic" (a since-deleted comment) and likely soon "accepted", epitomizes (IMO) a systemic dysfunction of the site. $\endgroup$ Commented Nov 22, 2019 at 23:50
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    $\begingroup$ Exactly. And that's the dysfunction right there. $\endgroup$ Commented Nov 23, 2019 at 12:54
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    $\begingroup$ Thank you for taking criticism with grace, but why would I? As far as I can tell, these users (capable of debating and moderating questions like this, and rooting out agenda-driven ones, a lot better) are at MO. Had professional historians flocked here, it could have been a place upholding the standards of both groups. But this never happened, and as it is, it upholds the standards of neither. $\endgroup$ Commented Nov 24, 2019 at 14:03
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    $\begingroup$ Note that I wrote 3 sentences. As to perennity, I wouldn't be so sure; unlike MO, HSM is for-profit (and routinely deletes comment threads; 3, 2, 1,...) $\endgroup$ Commented Nov 25, 2019 at 23:41

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

enter image description here

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).

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.

  • $\begingroup$ Very very interesting. $\endgroup$ Commented Feb 24, 2020 at 16:19
  • $\begingroup$ I just read your exchanges with @Conifold. I understand both positions. I participate mainly to Math Stack Exchange. Yes, a small percentage of answers smell "amateurism" [consider that some of them come from High school "gifted students"]. IMHO, such sites participate to the evolution of education : they give the opportunity to non-yet-confirmed scientists to participate, to improve themselves by confronting their ideas/writing to other ones. Very scarcely, erroneous facts remain. This said, it is very important that experts [as you evidently are] participate to such sites. $\endgroup$ Commented Feb 24, 2020 at 17:54
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    $\begingroup$ @JeanMarieBecker Thanks for the kind words. It may or may not be important that they participate, but the fact is that (unlike at MO) they overwhelmingly don't; and it's getting worse, not better. My next to last comment in that thread tried to better explain why I think this is, but it's been deleted (not by me). $\endgroup$ Commented Feb 24, 2020 at 21:59

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