I have been studying a course on Lie algebras in particle physics and I could never understand how complexifying helps us understand the original Lie algebra.

For example, consider $\mathfrak{su}(2)$: I complexify this to give me $\mathfrak{su}(2)_\mathbb{C}$ which allows me to form a Cartan-Weyl basis of ladder operators and a Cartan subalgebra, so I can generate highest weight representations. This is fine, but these are representations of $\mathfrak{su}(2)_\mathbb{C}$ not $\mathfrak{su}(2)$ because we cannot form a Cartan-Weyl basis unless we complexify. So how does this help us construct representations of $\mathfrak{su}(2)$?

All across particle physics we talk about particles living in representations of particular Lie algebras, but in fact should they actually be living in the complexifications of these? i.e. we talk about spinors of the Lorentz group, but the way to come across these is by complexifying the Lie algebra $\mathfrak{so}(3,1)$ so that it decomposes as

$$\mathfrak{so}(3,1)_\mathbb{C}=\mathfrak{su}(2)_\mathbb{C} \oplus \mathfrak{su}(2)_\mathbb{C}.$$

At which point I can label representations by $(A,B)$ where $A,B$ label the highest weights of the two subalgebras, and I would say that left-handed spinors live in $(1/2,0)$ and right handed in $(0,1/2)$. But again, these are complexified Lie algebras. How does this tell me that spinors exist with respect to the real Lorentz group, the one which the universe uses.

In summary, my questions:

  1. How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$ when the highest weight method only works with complexified Lie algebras?

  2. How do I know that the things which I discover after complexifying, like spinors and particle multiplets, are valid with respect to the orginal Lie algebra? i.e. We live in a universe where Lorentz transformations are real NOT complex, so how can we discover spinors without complexifying our Lorentz algebra?


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