Motivation for splitting the Lorentz Algebra using $J_{\pm i}$

Question

On page 116, Zee (in QFT in a nutshell) introduces the combinations: $$J_{\pm i}\equiv \frac{1}{2} \left( J_i\pm iK_i \right) \tag{1}$$ Where the $J_i $'s are the 3 generators of the rotation group and the $K_i $'s are the generators for the 3 Lorentz Boosts in each direction. They are used to split the algebra into "two separate $SU(2)$ algebras"

It is clear why the $J_{\pm i} $'s are convenient, but what's the motivation behind such a combination? For instance, is there a way to methodically come up with such a combination that's able to separate the algebra?

Answer 1

This is one of those kinds of stuff that people cannot explain how they arrive at it, only that they can work it out and discover that it is better for us to start from these things than otherwise.

For example, either you can try Qmechanic's complexification idea, or you can study the Dirac equation for chirality, especially in the massless case, and you will arrive at the spin-half Weyl equations. The right chiral and left chiral Weyl equations both behave the same way under rotations, i.e. the $J_i$ operators, but have the opposite boost $K_i$ behaviours. When you combine them as a way to deal with Lorentz transformations in general, the resulting spinor representations of the Lorentz group lead to both types of Weyl equations, and their combination gets you back the Dirac equation.

And then between Weyl and Wigner, their two different methods of using spin-half representations of Poincaré group to figure out all possible relativistic wave equations then led to the most natural way to classify and identify them all.

This is the logical flow of the argumentation that would be most pedagogically sensible. If you try to intuit it some other way, it is likely to be much more confused. We can only ever know how to teach a subject from the hindsight of having sorted out the landscape and then thought through how best to explain things.

Answer 2

Before considering a real Lie algebra $\mathfrak{g}$ [in this case the Lorentz Lie algebra $so(3,1;\mathbb{R})\cong sl(2;\mathbb{C})$] it is a good idea to consider its complexification $\mathfrak{g}_{\mathbb{C}}$ [in this case $so(3,1;\mathbb{C})\cong sl(2;\mathbb{C})\oplus sl(2;\mathbb{C})$], as the classification is easier. This will naturally take you to the complex linear combinations (1).