I've just started studying quantum mechanics, and I've come across this correlation between Pauli matrices ($\sigma_i$) and quaternions which I can't grasp: namely, that $i\sigma_1$, $i\sigma_2$ and $i\sigma_3$, along with the 2x2 identity matrix $I$, correspond identically to the four 2x2 matrix representation of unit quaternions.
My first guess was that this should have something to do with quaternions being useful for representing orientations and rotations of objects in three dimensions and Pauli matrices being related to the three spatial components of spin, but I didn't really know how to put together those two ideas. Google wasn't much help either: the relation is mentioned, for instance, in this Wikipedia article, but no further explanation is given.
Although I suspect there is no direct answer to this question, I would appreciate if someone could enlighten me on the subject. In particular, what is the role of the $i$ factor?
Answer
At the level of formulas, the three quaternionic units $i_a$, $a\in~\{1,2,3\}$, in $\mathbb{H}\cong \mathbb{R}^4$ satisfy $$i_a i_b ~=~ -\delta_{ab} + \sum_{c=1}^3\varepsilon_{abc} i_c, \qquad\qquad a,b~\in~\{1,2,3\}, \tag{1}$$ while the three Pauli matrices $\sigma_a \in {\rm Mat}_{2\times 2}(\mathbb{C})$, $a\in~\{1,2,3\}$, $\mathbb{C}=\mathbb{R}+\mathrm{i}\mathbb{R}$, satisfy $$\sigma_a \sigma_b ~=~ \delta_{ab} {\bf 1}_{2\times 2} + \mathrm{i}\sum_{c=1}^3\varepsilon_{abc} \sigma_c\quad\Leftrightarrow \quad \sigma_{4-a} \sigma_{4-b} ~=~ \delta_{ab} {\bf 1}_{2\times 2} - \mathrm{i}\sum_{c=1}^3\varepsilon_{abc} \sigma_{4-c}, $$ $$ \qquad\qquad a,b~\in~\{1,2,3\},\tag{2}$$ with complex unit $\mathrm{i}\in\mathbb{C}.$ In other words, we evidently have an $\mathbb{R}$-algebra monomorphism $$\Phi:~~\mathbb{H}~~\longrightarrow ~~{\rm Mat}_{2\times 2}(\mathbb{C}).\tag{3}$$ by extending the definition $$\Phi(1)~=~{\bf 1}_{2\times 2},\qquad \Phi(i_a)~=~\mathrm{i}\sigma_{4-a}, \qquad\qquad a~\in~\{1,2,3\},\tag{4}$$ via $\mathbb{R}$-linearity. This observation essentially answers OP title question (v2).
However OP's question touches upon many beautiful and useful mathematical facts about Lie groups and Lie algebras, some of which we would like to mention. The image of the $\mathbb{R}$-algebra monomorphism (3) is $$\Phi(\mathbb{H}) ~=~ \left\{\left. \begin{pmatrix} \alpha & \beta \cr -\bar{\beta} & \bar{\alpha} \end{pmatrix}\in {\rm Mat}_{2\times 2}(\mathbb{C}) \right| \alpha,\beta \in\mathbb{C}\right\}$$ $$~=~ \left\{ M\in {\rm Mat}_{2\times 2}(\mathbb{C}) \left| \overline{M} \sigma_2=\sigma_2 M\right. \right\}.\tag{5}$$ Let us for the rest of this answer identify $\mathrm{i}=i_1$. Then the $\mathbb{R}$-algebra monomorphism (3) becomes $$ \mathbb{C}+\mathbb{C}i_2~=~\mathbb{H}~\ni~x=x^0+\sum_{a=1}^3 i_a x^a ~=~\alpha+\beta i_2$$ $$~~\stackrel{\Phi}{\mapsto}~~ \begin{pmatrix} \alpha & \beta \cr -\bar{\beta} & \bar{\alpha} \end{pmatrix} ~=~ x^0{\bf 1}_{2\times 2}+\mathrm{i}\sum_{a=1}^3 x^a \sigma_{4-a}~\in~ {\rm Mat}_{2\times 2}(\mathbb{C}),$$ $$ \alpha~=~x^0+\mathrm{i}x^1~\in~\mathbb{C},\qquad \beta~=~x^2+\mathrm{i}x^3~\in~\mathbb{C},\qquad x^0, x^1, x^2, x^3~\in~\mathbb{R}.\tag{6}$$
One may show that $\Phi$ is a star algebra monomorphism, i.e. the Hermitian conjugated matrix satisfies $$ \Phi(x)^{\dagger}~=~\Phi(\bar{x}), \qquad x~\in~\mathbb{H}. \tag{7}$$ Moreover, the determinant becomes the quaternionic norm square $$\det \Phi(x)~=~ |\alpha|^2+|\beta|^2~=~\sum_{\mu=0}^3 (x^{\mu})^2 ~=~|x|^2, \qquad x~\in~\mathbb{H}.\tag{8}$$ Let us for completeness mention that the transposed matrix satisfies $$\Phi(x)^t~=~\Phi(x|_{x^2\to-x^2})~=~ \Phi(-j\bar{x}j), \qquad x~\in~\mathbb{H}. \tag{9} $$
Consider the Lie group of quaternionic units, which is also the Lie group $$U(1,\mathbb{H})~:=~\{x\in\mathbb{H}\mid |x|=1 \} \tag{10}$$ of unitary $1\times 1$ matrices with quaternionic entries. Eqs. (7) and (8) imply that the restriction $$\Phi_|:~U(1,\mathbb{H})~~\stackrel{\cong}{\longrightarrow}~~ SU(2)~:=~\{g\in {\rm Mat}_{2\times 2}(\mathbb{C})\mid g^{\dagger}g={\bf 1}_{2\times 2},~\det g = 1 \} $$ $$~=~\left\{\left. \begin{pmatrix} \alpha & \beta \cr -\bar{\beta} & \bar{\alpha} \end{pmatrix} \in {\rm Mat}_{2\times 2}(\mathbb{C}) \right| \alpha, \beta\in\mathbb{C}, |\alpha|^2+|\beta|^2=1\right\}\tag{11}$$ of the monomorphism (3) is a Lie group isomorphism. In other words, we have shown that
$$ U(1,\mathbb{H})~\cong~SU(2).\tag{12}$$
Consider the corresponding Lie algebra of imaginary quaternionic number $$ {\rm Im}\mathbb{H}~:=~\{x\in\mathbb{H}\mid x^0=0 \}~\cong~\mathbb{R}^3 \tag{13}$$ endowed with the commutator Lie bracket. [This is (twice) the usual 3D vector cross product in disguise.] The corresponding Lie algebra isomorphism is $$\Phi_|:~{\rm Im}\mathbb{H}~~\stackrel{\cong}{\longrightarrow}~~ su(2)~:=~\{m\in {\rm Mat}_{2\times 2}(\mathbb{C})\mid m^{\dagger}=-m \} ~=~\mathrm{i}~{\rm span}_{\mathbb{R}}(\sigma_1,\sigma_2,\sigma_3),\tag{14}$$ which brings us back to the Pauli matrices. In other words, we have shown that
$$ {\rm Im}\mathbb{H}~\cong~su(2).\tag{15}$$
It is now also easy to make contact to the left and right Weyl spinor representations in 4D spacetime $\mathbb{H}\cong \mathbb{R}^4$ endowed with the quaternionic norm $|\cdot|$, which has positive definite Euclidean (as opposed to Minkowski) signature, although we shall only be sketchy here. See also e.g. this Phys.SE post.
Firstly, $U(1,\mathbb{H})\times U(1,\mathbb{H})$ is (the double cover of) the special orthogonal group $SO(4,\mathbb{R})$.
The group representation $$\rho: U(1,\mathbb{H}) \times U(1,\mathbb{H}) \quad\to\quad SO(\mathbb{H},\mathbb{R})~\cong~ SO(4,\mathbb{R}) \tag{16}$$ is given by $$\rho(q_L,q_R)x~=~q_Lx\bar{q}_R, \qquad q_L,q_R~\in~U(1,\mathbb{H}), \qquad x~\in~\mathbb{H}. \tag{17}$$ The crucial point is that the group action (17) preserves the norm, and hence represents orthogonal transformations. See also this math.SE question.
Secondly, $U(1,\mathbb{H})\cong SU(2)$ is (the double cover of) the special orthogonal group $SO({\rm Im}\mathbb{H},\mathbb{R})\cong SO(3,\mathbb{R})$.
This follows via a diagonal restriction $q_L=q_R$ in eq. (17).
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