[Physics] An identity of Pauli matrices

angular momentumlie-algebraquantum mechanicsquantum-spinspinors

I am studying spin recently, and textbook gives some identities of Pauli matrices, one said that

for any two unit vectors $\bf m$ and $\bf n$, $[\bf m \cdot \bf{\sigma},\bf {n \cdot \sigma}]= 2i(m\times n)\cdot\sigma$

I know how to derive it, but is there any physical meaning of this identity?

Best Answer

Yes. This commutation relation is that of the Lie algebra $\mathfrak{so}(3)$ corresponding to the rotation group in three dimensions. Thus the commutation relation states that the Pauli matrices generate rotations.

To understand why this is the commutation relation of $\mathfrak{so}(3)$, one can draw a diagram showing that the commutator of two infinitesimal rotations is an infinitesimal rotation around the perpendicular axis. It is of course also possible to reason without the picture. To work that out, you need that the effect of an infinitesimal rotation around $\mathbf n$ on $\mathbf v$ is $$\mathbf v \mapsto \mathbf v + \varepsilon \mathbf n \times \mathbf v.$$