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?
  • 2
    $\begingroup$ Related: Why is the Yang-Mills gauge group assumed compact and semi-simple? $\endgroup$
    – Qmechanic
    Commented Sep 15, 2022 at 7:08
  • $\begingroup$ 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? $\endgroup$
    – Wihtedeka
    Commented Sep 15, 2022 at 7:40
  • $\begingroup$ @Wihtedeka yeah, I forgot this fact. $\endgroup$
    – M.N.Raia
    Commented Sep 15, 2022 at 8:09

1 Answer 1


A gauge transformation acts on the wave function and does not transform it in space and time but in an abstract internal space. The corresponding gauge covariant derivative has Lorentz indices that ensure local Lorentz invariance.


  • $\begingroup$ I see. I will think about your answer. Moreover, "internal spaces" are which bundle structure? Principal bundles or associated bundles? $\endgroup$
    – M.N.Raia
    Commented Sep 15, 2022 at 6:46
  • 1
    $\begingroup$ @M.N.Raia I don't know the terminology but each point in spacetime has an associated symmetry like U(1) for QED. So each spacetime location has a complex unit circle attached to it. I believe it is called a fiber. There is a very good book called Physics From Finance by Jakob Schwichtenberg on this. (and another one titled Physics From Symmetry). I'm not an true expert. I've only read a couple books on the subject. $\endgroup$
    – jelly ears
    Commented Sep 15, 2022 at 6:51
  • 2
    $\begingroup$ Thank you Jelly. Actually, I know Jakob's book and definetly I'll back to it. I'm studying a very nice book on differential geometry called Mathematical Gauge Theory by Mark J.D. Hamilton. This book puts all gauge theory into formal grounds. Sometimes, though, it lacks on interpretations. $\endgroup$
    – M.N.Raia
    Commented Sep 15, 2022 at 6:57

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