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

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?

• Not all Lie algebras separate like that, themselves as real algebras or their complexifications (a case being the $\mathfrak{so}(1,3)$). So no "methodically coming up with such a separation" is generally possible. Commented May 29, 2023 at 8:49

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.
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).