# How come the Pauli Matrices are the generators of SU(2)

The special unitary group's elements fulfil the unitary condition, i.e. $aa^\dagger=1$, and have determinant 1 (special). I have read, that the Pauli matrices generate SU(2). Since all Pauli matrices have determinant -1, how can they generate the group, when they are not contained.

I'm referring to the common representation $\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ $\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$ $\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$

• Do you know about Lie groups and Lie algebras? Apr 1, 2017 at 11:00
• The Pauli matrices are not themselves part of $SU(2)$ (as a Lie group), but rather they are part of its generating Lie algebra, in the sense that they form a basis from which you can write down (using a map from the algebra to the group) any element of $SU(2)$. Apr 1, 2017 at 11:19

Pauli matrices are not generators of the Lie group $$SU(2)$$, but after multiplication by$$~\mathbf i$$ they are so-called infinitesimal generators. These are elements of the Lie algebra $$\mathfrak{su}(2)$$ of $$SU(2)$$, and for such elements $$X$$ the exponential mapping $$t\mapsto\exp(tX)$$ gives a one-parameter subgroup of the Lie group. In view of this relation, the determinant of the Pauli matrices is not important; what is relevant is that they are Hermitian (so after multiplication by$$~\mathbf i$$ they become anti-Hermitian, and the exponential mapping gives unitary matrices) and have trace zero (so that exponential mapping gives matrices of determinant$$~1$$).
No countable set of group elements could generate the uncountable groups $$SU(2)$$, but the one-parameter subgroups for the Pauli matrices do generate $$SU(2)$$. Maybe more pertinent is the fact that the Pauli matrices generate the real vector space of traceless Hermitian matrices. Thus every element of $$SU(2)$$ is the image under the exponential mapping of a real linear combination of the multiples by $$~\mathbf i$$ of the Pauli matrices (although the fact that $$SU(2)$$ is connected and compact is instrumental for this fact).