Symmetry in quantum mechanics

Summary

A symmetry of a quantum system is a transformation of its states that preserves all transition probabilities. A symmetry of the dynamics also commutes with the time evolution. By Wigner's theorem, every such transformation is given by a unitary or an antiunitary operator on the Hilbert space. A group of symmetries therefore acts by a projective unitary representation. Representation theory then organises the states: degenerate energy levels carry representations of the symmetry group. Continuous symmetries give conserved quantities, such as momentum from translations and angular momentum from rotations.