What is the motivation for complexifying a Lie algebra?
In quantum mechanical angular momentum the commutation relations
$$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$
become, on complexifying (arbitrarily defining $J_{\pm} = J_x \pm iJ_y$)
$$[J_+,J_-] = 2J_z,\quad [J_z,J_\pm] = \pm 2J_z.$$
and then everything magically works in quantum mechanics. This complexification is done for the Lorentz group also, as well as in the conformal algebra.
There should be a unified reason for doing this in all cases explaining why it works, & further some way to predict the answers once you do this (without even doing it), though I was told by a famous physicist there is no motivation 🙁
Best Answer
Short answer: complexifications facilitate representation theory.
In physics, we typically want to find representations of a Lie algebra $\mathfrak g$, and often times determining the representations of its complexification $\mathfrak g_\mathbb C$ is easier. Moreover, we have the following theorem (see ref 1. Proposition 4.6) which tells us that determining the representations of the complexification allows us to determine the representations of the original algebra.
Theorem. Let $\mathfrak g$ be a real Lie algebra, and let $g_\mathbb C$ be its complexification. Every finite-dimensional complex representation $\pi$ of $\mathfrak g$ has a unique extension to a complex-linear representation $\pi_\mathbb C$ of $\mathfrak g_\mathbb C$ \begin{align} \pi_\mathbb C(X+iY) = \pi(X) + i\pi(Y) \end{align} for all $X,Y\in\mathfrak g$. Furthermore, $\pi_\mathbb C$ is irreducible as a representation of $\mathfrak g_\mathbb C$ if and only if $\pi$ it is irreducible as a representation of $\mathfrak g$.
Example. Angular momentum in QM
In the case of the angular momentum in quantum mechanics, what physics books are doing mathematically is attempting to find representations of $\mathfrak {su}(2)$ acting on the Hilbert space of a given physical system. The complexification of $\mathfrak{su}(2)$ is $\mathfrak{sl}(2,\mathbb C)$, and $\mathfrak{sl}(2,\mathbb C)$ has a nice basis $J_\pm, J_z$ which has no counterpart in $\mathfrak{su}(2)$ and which makes determining representations much easier. The structure relations in the $J_\pm, J_z$ basis allow one to use "raising" and "lowering" operators.
Example. Lorentz algebra
In relativistic quantum field theory, we look for representations of $\mathfrak{so}(1,3)$. It turns out, quite happily, that when we complexify this algebra, it splits into a direct sum of complexified angular momentum algebras: \begin{align} \mathfrak{so}(1,3)_\mathbb C \cong \mathfrak{sl}(2,\mathbb C)\oplus \mathfrak{sl}(2,\mathbb C), \end{align} and since we already know the representation theory of the complexified angular momentum algebra so well, this makes studying the representations of the Lorentz algebra easy.
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