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I know that the SU(2) group describes internal symmetries such as isospin and the weak interaction. But isospin and weak interactions are quite different, so more precise what is the role of SU(2) in the two cases?

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Saying that $\textrm{SU}(2)$ describes internal symmetries as the isospin is somewhat incorrect. Rather, it is the gauge group describing the weak interaction.

In quantum field theory the equations of motion for the fields and the particles involved therein are described by means of a Lagrangian which is supposed to be invariant under some gauge group of transformations. According to how you choose the gauge group you have different couplings among field-field interactions, field-particle, particle-particle and so on and so forth. For an initial description you can have a look at the related wikipedia article.

Electromagnetism is fully described by taking a Lagrangian that is invariant under the action of $G=\textrm{U}(1)$; likewise the weak interaction is described along the same line by taking $G=\textrm{SU}(2)$ and the dimensionality of the gauge group gives back the number of different force carriers in each case: for instance one (the photon) for the electromagnetic field and three (the weak $W_{\pm},Z$ bosons) for the weak one. When exploiting the calculations you will find that, similarly to the electric charge, there is an associated conserved quantity under the action of $G=\textrm{SU}(2)$ which we refer to as isospin, but that is not an inner consequence of the gauge group, it is an additional consequence of the Noether theorem and is apart from the description of the weak interactions as gauge theories themselves.

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