Wigner's theorem

Wigner's theorem states that a transformation of pure quantum states preserving transition probabilities is induced by a unitary or antiunitary operator on the Hilbert space.

Quantum state: A physical state in quantum theory, represented by a ray in Hilbert space or by a density operator. The theorem acts on pure states, which are rays rather than individual state vectors.
Hilbert space: A complete inner-product vector space that provides the mathematical setting for quantum state vectors. The theorem's unitary and antiunitary operators act on this state space.
Time reversal: A transformation that reverses the direction of time in physical laws and processes. In quantum theory, time reversal is represented antiunitarily under standard assumptions.
Projective representation: A group representation whose multiplication law holds up to scalar phase factors. Operator representatives of ray symmetries need only compose up to an unobservable phase.
Eugene Wigner: A Hungarian-American theoretical physicist who established foundational results on symmetries in quantum mechanics. He proved the theorem and applied symmetry principles to quantum physics.
Transition probability: The probability that a quantum system prepared in one pure state is found in another, given by the squared inner product magnitude. Preserving this quantity is the theorem's defining condition.
Inner product: A sesquilinear form that assigns lengths and angles to vectors in a complex vector space. Its magnitude determines the transition probabilities that the theorem requires symmetries to preserve.
Parity (physics): The spatial inversion transformation that sends position coordinates to their negatives. Its quantum implementation is a unitary symmetry operator.
Central extension: A group built by adjoining a central subgroup so that a projective representation can become an ordinary representation. It provides a group-theoretic way to handle the phases arising from Wigner symmetries.
Group theory in physics: The use of abstract groups and their representations to describe physical symmetries and conservation laws. Wigner's result formalizes how symmetry groups act on quantum states.

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Kramers' theoremRelated: It explains why time reversal is represented antiunitarily in quantum mechanics.