$2\times 2$ Lorentz group generators and the $\sigma$ matrices

Question

This will probably turn out to be a really simple issue, like bad notation in one of my sources confusing me, but here goes.

In Groups, Representations, and Physics (Jones) it's stated that the generators of rotations $X_i$, and boosts $Y_i$ (Jones' notation, I know they're usually written as $J_i$ and $K_i$) can be combined to give $\vec{X}^{(\pm)} = \frac{1}{2}(\vec{X}\pm i\vec{Y})$, and that these obey the commutation relations of spin, $[X^{(\pm)}_i,X^{(\pm)}_j]=i\varepsilon_{ijk}X^{(\pm)}_k$ and $[X^{(+)}_i,X^{(-)}_j]=0$. That's easy to confirm using the commutation relations of the $X$ and $Y$. So far, so good.

I've also seen it claimed (e.g. Physics of the Lorentz group by Baskal, Kim, and Noz) that the rotation and boost generators can be written in $2\times 2$ form as $X_i = \frac{1}{2}\sigma_i$ and $Y_i = \frac{i}{2}\sigma_i.$ However this seems to create a problem, to me, because we then get $$X^{(+)}_i = \frac{1}{2}(X_i + iY_i)=\frac{1}{4}(\sigma_i + (i^2)\sigma_i)=0.$$ So yes, it obeys the commutation relations of spin, but only trivially, as $[0,0]=0$. Conversely $$X^{(-)}_i = \frac{1}{2}(X_i - iY_i)=\frac{1}{4}(\sigma_i - (i^2)\sigma_i)=\frac{1}{2}\sigma_i = X_i.$$

I would have thought the $X^{(\pm)}$ would be non-trivial, and at least distinct from the rotation generators. What am I missing?

Answer 1

Seems like all your conclusions are correct. $X_i = \frac{1}{2}\sigma_i$ and $Y_i = \frac{i}{2}\sigma_i$ indeed provide one of the representations of the algebra of the Lorentz group, because they obey the commutation relations. The same expressions for generators are given in "Symmetry and the Standard Model" by Matthew Robinson, page 122, which I found in a question.

The definition of a Lie algebra representation (see here) does not prohibit the matrices $X_i^{(+)}$ or $Y_i^{(+)}$ to be zero or to be equal to rotation generators. Even more, for every Lie algebra (and for Lorentz algebra in particular), there exists a trivial representation in which all the matrices are zeros.