# How to go from a Higgs which transforms in the adjoint representation to a 2x2 matrix? [closed]

I have a triplet transforming in the adjoint map of the lie albegra of su(2) but I don´t know how to include it in to a Lagrangian where I have two lepton doublets. It should be a 2x2 matrix but I don´t understand the meaning of the transformation nor how it is done (I think it is done by means of the Pauli matrices). Could someone explain it through the concepts of group theory?

• physics.stackexchange.com/q/29192 – AMS Jun 17 '16 at 16:34
• Let $a$ be the adjoint index and $i$ the fundamental index. Then, a $2\times2$ adjoint matrix can be written as $\phi_{ij} = \sum_{a=1}^3 T^a_{ij} \phi^a$ where $T^a_{ij}$ are the generators in the fundamental representation. – Prahar Jun 17 '16 at 16:34
• Yes, you can construct such a matrix but why is it still the Higgs? I mean you had a vector with three components and now you get a matrix through the generators of SU(2). Why is it still something used as if it were the Higgs? (My question is why do you introduce sigma.Higgs in the Lagrangian) – florpi Jun 17 '16 at 16:39
• I don't understand the problem here. The Lie algebra $\mathfrak{su}(2)$ consists of 2x2 matrices, what is the issue? – ACuriousMind Jun 17 '16 at 22:02
• The triplet Higgs transforms as a triplet under the action of the group SU(2). That means it transforms in the adjoint representation of SU(2), but it is not an element of the group. Why should then the Higgs be an element of SU(2)? – florpi Jun 18 '16 at 10:50