Is representation theory in QM a real physical thing or just a mathematical tool? I was studying group theory and representation theory in Quantum Mechanics and I really don't understand yet if it is just a mathematical tool of seen the operators as a representation of a group action, or if it is really conected to the physical reallity. I guess Representation Theory is not the only thing that botters me in this sense, because every time I learn some deeper math in a physical theory I have the feeling that I am only using an abstract tool that happens to help me in my problem.
 A: All of physics is not real. The universe is real. Physics is a mathematical model of the universe.
We see patterns of behavior in the universe. The purpose of physics is to describe those patterns. So the mathematical tools of physics are chosen to match the behavior. We call these the laws of physics.
Physicists use the laws to predict the behavior of the universe. Then they do experiments to confirm the predictions. Physicists quickly get used to the correspondence between the laws and the behavior. They begin to think of the laws and the behavior as the same thing. That is, when they think of the behavior described by Newton's $2^{nd}$ law, they thing of $F = ma$. This is a mathematical statement with two vectors and a number. But physicists think of this as the same thing. Force is a vector. Mass is a number.
So you are right. Representation theory is an abstract tool that helps describe the universe. But it is also connected to the physical reality.
This way of thinking dates back to Galileo, who first connected systematic mathematics to the behavior of the universe. Later Newton greatly expanded this ideas and transformed physics from philosophy to science.
Before this, it was not possible to think precisely about the behavior. Before this, people thought like Aristotle. He said rocks come to rest because it is the nature of rocks (things with lots of earth in them) to be at rest. The rock might move as it rolls down a hill, but soon the nature of the rock asserts itself. On the other hand, it is the nature of things like stars and planets (things with lots of fire in them) to move. You can see the advantage math brings.
