A non-uniqueness result for the canonical structure in quantum theory shows that the classical part of the Copenhagen interpretation contains physically important information not contained in its quantum part. As a consequence, we cannot compute the symmetry group of a quantum theory considering only the quantum part. The unavoidable vagueness of the classical part therefore leads to a similar vagueness in the definition of the symmetry group. This makes it at least problematic, if not impossible, to establish the true symmetry group of a quantum theory in the Copenhagen interpretation. Different from the old measurement problem, the symmetry group is to important physically to be rejected as a metaphysical pseudoproblem.
What is the common thing between some integers and symmetries of an equilateral triangle? How studying rotations on a circle helps us understand quantum mechanics? Groups and their representations provide an amazing framework for many physics and chemistry theories. Understanding how certain groups act on physical systems, both classical and quantum, helps us to build models and make predictions. In this talk, we discover how abstract mathematical notions such as groups and representation theory and counterintuitive nature of quantum mechanics come together to guide us in our journey to explore the world we live in. This talk is designed to be accessible to any person with some understanding of calculus and a little bit of linear algebra.