First of all, the question is written in section $2)$. Also, I known that the $SU(2)$ group do not appears "alone" in standard model, rather, inside the Glashow-Salam-Weinberg model.
1) Introduction
The heuristic picture of the mathematical structure of standard model (SM) lies on Lie group theory. Moreover, SM is a big gauge theory and therefore uses the technology of fiber bundles.
1.1) Bundles and Gauge theory
- Given a manifold $\mathcal{M}$ (a spacetime) and a Lie group $G$ (a gauge group), we can readly construct another manifold using lie group a $G$: the principal bundles $P_{G}$.
- Once you constructed the $P_{G}$, you can stablish Ehresmann connections and therefore the connection $1$-form $A$: the gauge field (in fact the one can put the gauge field in spacetime using the local connection $1$-form $A_{s} = s^{*}A$. The $s$ is precisely a choose of the local gauge).
- Given the $P_{G}$, the implementation of matter fields $\Phi$ in spacetime (spinors, scalars, vectors and tensors) lies on another bundle called Associated bundle: $A_{P_{G}}$. Its definition is given by the quotient: $$A_{P_{G}} := P_{G} \times_{\rho}V = \frac{P_{G} \times V}{G}, \tag{1}$$ where $\rho$ is the representation map between groups: $\rho: G \to GL(V)$ and $V$ is a vector space. Also, $\Phi$ are sections of $A_{P_{G}}$;
- Inside $A_{P_{G}}$ the one can define, in a formal way, our beloved gauge covariant derivatives that acts (locally) on a matter field $\Phi$ as: $$D_{\mu}\Phi = \partial_{\mu}\Phi + \rho_{*}(A_{s}(X))\Phi \tag{2}.$$
Where $X$ is a vector field and the map $\rho_{*}$ is the representation map acting on Lie algebra elements that follows the diagram:
With the exponential map, $\mathrm{exp}$, you can "translate" the technology of standard Lie group theory, into lie algebra representations as:
$\require{AMScd}$ \begin{CD} \mathfrak{g}@>{\rho_{*}}>> \hspace{0.4cm}\mathrm{End}(V)\\ @V{\mathrm{exp}}VV @VV{\mathrm{exp}}V\\ G @>{\rho}>> GL(V) \end{CD}
2) My Question
The section $1.1)$ shows mathematical structures that are highly dependent on Lie groups, Lie algebras and its representations. Also, complex representations of $SU(2)$ represent non-relativistic spinors and representations of $SL(2,\mathbb{C})$ represent relativistic spinors $(*)$.
Therefore, my question is: why do we use groups like $SU(2)$ to represent gauge symmetry, instead of groups like $SL(2, \mathbb{C})$?
Another possible way to ask the question:
- Knowing the steatment $(*)$, we realize that $SU(2)$ represent non-relativistic fields and $SL(2,\mathbb{C})$ represent relativistic fields. Since the standard model is a relativistic theory shouldn't be better to deal with "things" that represent relativistic behaviour?
