Wigner’s Theorem

Full treatment. Read Wigner’s Theorem for the finite- and infinite-dimensional proof architecture, uniqueness, projective group lifts, and counterexamples.

Statement

Let $\mathcal H$ be a complex Hilbert space with $\dim\mathcal H\geq2$. A bijection $S$ of its ray space that preserves every transition probability,

$$\mathcal P(S[\psi],S[\phi]) = \mathcal P([\psi],[\phi]),$$

is implemented by either a unitary or an antiunitary operator. The implementing operator is unique up to one global phase.

Check before use

• The map acts on rays and is bijective.
• It preserves all transition probabilities, not only orthogonality.
• No continuity assumption is needed for this isolated full-preserver form.
• Without surjectivity, isometric embeddings such as the unilateral shift are possible and need not be unitary.
• Uhlhorn’s orthogonality-only theorem normally uses dimension at least three; that restriction is not the standard full-preserver Wigner hypothesis.

For a symmetry group, individual lifts can compose only up to a phase. Wigner’s theorem alone gives a projective representation, not automatically an ordinary unitary representation or a Hamiltonian symmetry.

References

• V. Bargmann, “Note on Wigner’s theorem on symmetry operations,” Journal of Mathematical Physics 5, 862–868, 1964, doi:10.1063/1.1704188.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.