Phrase 'the field is in [certain] representation of a [certain] group'

[Stan324]

I have a question about quantum field theory.

What does the phrase 'the field is in [certain, e. g. fundamental] representation of a [certain, e. g. SU(2)] group' mean?

I know mathematical definitions of groups and their representations, but what does this specific phrase mean?

 

[The_Duck]

Let's take the Lorentz group. When we say that "the field $\phi_a(x)$ is in the representation $R$ of the Lorentz group" we mean that if we perform a Lorentz transformation $\Lambda$ of our system, the new version of the field is given by $\phi_a'(x) = {D(\Lambda)_a}^b \phi_b(\Lambda^{-1} x)$. The $D(\Lambda)$ is a matrix (there is one for each possible Lorentz transformation $\Lambda$) and the set of $D(\Lambda)$ matrices are a representation $R$ of the Lorentz group. For example, if $D(\Lambda) = 1$ for all $\Lambda$ then we have the trivial representation. If the $D(\Lambda)$ are the familiar 4x4 Lorentz matrices, we call this the "vector" representation. And so on.

 

[Stan324]

Thank you very much!