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I am studying the $SU(2)$ symmetric Lagrangian in particle physics.

$${\mathcal{L}} = (\partial_\mu \Phi)^\dagger (\partial^\mu \Phi) - (\mu^2 \Phi^ \dagger \Phi + \lambda (\Phi^ \dagger \Phi)^2).$$

In this the $SU(2)$ triplet taken consists of 3 real fields.

$$\Phi = \begin{bmatrix} \phi_1 \\ \phi_2 \\ \phi_3 \end{bmatrix}.$$

I am bit confused about why this triplet needs to be real? Can't it be also complex? Instead of a triplet can we take a doublet? If so, will it be real or complex?

In general if we are given a $SU(N)$ symmetric Lagrangian, is there any convention to choose real multiplet and which order multiplet should we choose so that it is a real multiplet?

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Here is an incomplete list to start with:

  • The fundamental/defining/spinor representation ${\bf 2}$ of $SU(2)$ is a quaternionic/pseudoreal representation.

  • More generally the ${\bf 2j\!+\!1}$ irreducible representation of $SU(2)$ is (pseudo)real if the spin $j$ is a (half)integer, respectively.

  • The fundamental/defining representation ${\bf N}$ of $SU(N)$ for $N>2$ is a complex representation.

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