# Why standard model uses Lie groups like $SU(2)$ and not $SL(2,\mathbb{C})$?

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:

1. 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?
• Commented Sep 15, 2022 at 7:08
• This may be beyond my area of expertise, but might it be related to the fact that we want to gauge a global symmetry of the action? Commented Sep 15, 2022 at 7:40
• @Wihtedeka yeah, I forgot this fact. Commented Sep 15, 2022 at 8:09