Wigner's theorem

Wigner's theorem states that any symmetry transformation in quantum mechanics—changing the state of a system without affecting measurable outcomes—must be represented mathematically by either a unitary or an anti-unitary operator. In simpler terms, whenever a symmetry preserves the probabilities of measurement outcomes, it corresponds to a specific type of mathematical operation that maintains the structure of the quantum system’s states. This theorem underpins how symmetries are incorporated into quantum theory, ensuring that physical laws remain consistent under transformations like rotations or reflections, and clarifies the mathematical form these symmetries must take.