Why representations instead of just groups?

This question is essentially asking for a clarification on what has already been said in this one. What I don't understand is why it is the representations that are important in Quantum Field Theory rather than just the group itself. Do the representations add extra structure that the group didn't possess? If so how do we know that we want that structure in our theory? (if there is a reason other than empirical observation)

Or to state the question differently: Is there a way to write Quantum Field Theories with certain symmetries in a representation free way? (e.g. the Standard Model).

• What use would a group without a representation be? Any time a group acts in a "nice" way on a vector, you get a representation. No representation means no group actions, and then of what relevance is the group? I don't understand the question. – ACuriousMind Apr 1 '16 at 20:26
• I thought the same thing at one point, thinking that a representation was just like a choice of basis, and that everything could be reformulated in a basis or representation-independent way. But this is not true, even in quantum mechanics: the different representations of $SU(2)$ give you particles with different spins! The common group structure just tells you that you're dealing with rotation; you need the representation to tell you the rest. – knzhou Apr 1 '16 at 20:27
• @knzhou I think this almost answers my question already. May I ask what exactly it is that the representations add in structural sense? As in how do the different spins come from the group? (if that makes any sense) – Wolpertinger Apr 1 '16 at 20:32
• Well, you have no idea what elements of your group $G$ do without taking a representation. For example, you might think some element $g$ represents, say, a 45 degree rotation about the $z$ axis. But to say that, you've implicitly chosen the vector representation, i.e. this is how $g$ acts on $v$ if $v \in \mathbb{R}^3$ and is in the vector representation. Without doing this, $g$ has no physical meaning at all. – knzhou Apr 1 '16 at 20:38
• Part of the confusion comes from the fact that Lie groups are often presented as matrix groups, i.e. $SU(n)$ is written in terms of $n \times n$ unitary matrices, and so on. This is not "the group" in the most abstract sense; it is a particular representation of the group, called the fundamental representation. – knzhou Apr 1 '16 at 20:42