How did the group $SU(2)$ come into physics?

Question

It is natural for physicists to consider the group $SO(3)$. Presumably, $SU(2)$ came into physics because of quantum mechanics. How did people realize that when studying rotation of a physical system that sometimes, it is the group $SU(2)$ instead of the group $SO(3)$ that really matters?

Answer 1

It came to physics a bit earlier than quantum mechanics. The homomorphism $SU(2)\to SO(3)$ was discovered by Cayley (1843), Hamilton (1847), and Klein (1875) in their pure mathematical studies, and came to the attention of physicists through the theory of rigid body rotation (classical mechanics). It was Klein who brought it to the attention of physicists.

In old books on mechanics this is called the "Cayley-Klein's parameters". Klein explained this correspondence to physicists in his 1895/6 lectures which were later published as a book under the title "Theory of the top" (joint with his student A. Sommerfeld). (A concise exposition in English is in his 1897 Princeton lectures on the same subject. But the 4 volumes of Klein and Sommerfeld has been also translated into English).

In his Princeton lectures Klein says:

Instead of either of these commonly used systems of parameters, I propose to introduce another, which so far as I know has not yet been employed in dynamics.

Later he writes in the same lectures:

There is nothing essentially new in these considerations. I have merely attempted to throw a method already well known into the most convenient form for applications in mechanics.

Among the predecessors of Cayley, Hamilton, and Klein, Euler has to be mentioned: he discovered a rational parametrization of the rotation group which is equivalent to quaternion parametrization, in his work on arithmetic. It was apparently not noticed by physicists, but Klein credits the "quaternion parametrization" to Euler.

Answer 2

The group SU(2) is the universal cover of the real rotation group SO(3). The universal cover appears in quantum mechanics because of projective representations SO(3) on kets, in particular up to phase factors.

The laws of classical physics are invariant (equivariant) with respect to an action of the Galilei group in time dependent problems, and with respect to an action of the Euclidean group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$ in time independent problems. In particular, angular momentum as a conserved quantity corresponds to rotational invariance (of Lagrangians) by Noether's theorem. Vectors, tensors etc. are elements of a linear space (a real vector space), and the actions of SO(3) on such spaces gives linear representations on these linear space.

In quantum mechanics, however, there is an additional phase factor $e^{i \phi}$ by which every ket $|\psi>$ can be multiplied without changing the expectation values expresses by a bra-ket:

$$<\psi|\psi> = <\psi| e^{-i \phi} e^{i \phi} |\psi>$$

Therefore, it is not linear representations, which are important in quantum mechanics, but projective representations, i.e. representations "up to" scalar multiplication. In this sense, a ket corresponds rather to a ray (complex 1-dim subspace) and not just to a vector.

There's now a fundamental theorem in the representation theory of groups which says that every projective representation of a group G stems from a linear representation of the universal covering $\tilde{G}$ of G.

Therefore, instead of dealing with projective representations of SO(3), physicists prefer to deal with linear representations of its universal covering group SU(2). It is easier (for physicists) to perform linear algebra and linear tensor calculus instead of dealing with more complicated elements of projective geometry such as projective semi-invariants and similar quantities.

Typically, today's physicists are not trained in Projective Geometry or Projective Algebraic Geometry, but they are trained in Linear Algebra.