# Transformation of matter field in different representations in Yang Mills theory

I've read this post and also this one but I couldn't find my answer.

My question is a stupid one. I know that matter fields in Yang mills theory can be transformed in any representation of gauge group, for example in fundamental representation of in adjoint representation.

I know also that in fundamental representation scalar fields $$\phi$$ transform according to this rule:

$$\phi_i \xrightarrow{} \phi_i + i \theta^a (T^a_{fund})_{ij} \phi_j.$$

But since dimension of adjoint representation is $$N^2-1$$ and we have $$N$$ scalar fields in above expression, how we could transform fields in adjoint representation? Could anyone give me the rule for transformation of matter field in adjoint representation as above?

• " I know that matter fields in Yang mills theory can be transformed in any representation of gauge group" where did you get this from? Usually the matter field has a specific representation. Often it is the fundamental representation. Commented Feb 7, 2022 at 9:46

A fundamental field $$\phi_i$$ transforms as $$\phi_i \to \phi_i - i \theta^a (T^a_{fund})_{ij} \phi_j + O(\theta^2).$$ An adjoint field $$\phi^a$$ transforms as $$\phi^a \to \phi^a - i \theta^a(T^a_{adj})^{bc} \phi^c + O(\theta^2) = \phi^a + f^{abc} \theta^b \phi^c + O(\theta^2) .$$ etc.
As a side note, it is often said that a gauge field transforms in the adjoint, but its transformation is not the same as above. This is because the gauge field is a connection. It transforms as $$A_\mu^a \to A_\mu^a + f^{abc} \theta^b \phi^c + \partial_\mu \theta^a + O(\theta^2)$$ Notice that this transformation is basically the same as that of $$\phi^a$$ but it has an extra term.
• so a field that transforms in adjoint representation must contain $N^2-1$ components? Commented Feb 7, 2022 at 10:18
• yes! A field transforming in representation $R$ has $\text{dim}R$ number of components. Commented Feb 7, 2022 at 10:18