# Student Colloquium: Observables and Representation Theory, v/Frederik Rosendal Rytter

*Supervisor: Georg Bruun*

Representation theory is the study of how we can represent abstract algebraic objects by linear transformations on vector spaces. The hope being that we can transform complicated algebraic problems, into simpler linear algebra problems. This field is widely used in Physics, but we will focus on its use in studying ob-servables in non-relativistic quantum mechanics.

We will find that representations are natural objects to study, when working with symmetries of quantum systems. We will see that if a representation commutes with of our Hamiltonian on our Hilbert space, then we get a well-structured way of studying the degeneracy of the energy levels. In fact we will see, that the mere presence of a representation on our Hilbert space, provides us with observables to study, such as xˆ and pˆ. We know that some observables has a discrete spectrum, while others has a continuous one. Representation theory can explain this using the simplest Lie group of them all, the circle (U(1)).