# Why use $SU(3)$ and not $SL(3, \mathbb{R})$ for color charge? [duplicate]

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Why do we use the group $$SU(3)$$ and not $$SL(3, \mathbb{R})$$ for color charge?

As far as I can tell, the $$SL(3, \mathbb{R})$$ is volume and orientation preserving, by the fact that it has unit determinant, and if I'm not mistaken the only thing that $$SU(3)$$ adds is that it is inner product conserving?

And now to play devils advocate, does that still matter? Ever since the Klein-Gordon equation we have stopped viewing fields as wave-functions, and the expectation values come from calculating things like

$$< F > = \frac{\int \mathcal{D}\phi F[\phi] e^{i S[\phi]}}{\int \mathcal{D}\phi e^{i S[\phi]}},$$

so wouldn't such differences be divided out?

I've also calculated the structural constants and noticed that the anti-symmetric ones are all identical apart from a sign change in $$f_{257}$$, although for the symmetric ones there are actually a lot of sign changes.

Also, I kind of struggle to see why $$SL(3, \mathbb{R})$$ would not be norm conserving, is it possible to show this fairly easily?