Wigner's theorem is one of the most fundamental results in quantum theory, but I somehow didn't hear of it for the first time until the third year of my PhD. Even then, it took me another year or so to fully appreciate the theorem's importance. I suspect this experience is common — Wigner's theorem is thought of as being fairly technical or mathematical, and doesn't get covered in most quantum mechanics courses. But because it's so essential, I'd like to dedicate a post to explaining and proving it. The statement of the theorem is simple: every symmetry of a quantum system can be represented as a unitary or anti-unitary operator on Hilbert space . (Here we are implicitly thinking about quantum states as vectors in a Hilbert space — there are ways of thinking about Wigner's theorem from a more operator-algebraic point of view, but the Hilbert space picture is a good place to start.) The reason Wigner's theorem is so valuable is that if we believe a sy...