| Wigner's theorem | |
|---|---|
| Name | Wigner's theorem |
| Field | Quantum mechanics |
| Statement | Symmetry transformations preserving transition probabilities are implemented by unitary or antiunitary operators on a Hilbert space. |
| Proved | 1931 |
| Proposer | Eugene Wigner |
Wigner's theorem
Wigner's theorem is a foundational result in Quantum mechanics establishing that any symmetry of a quantum system that preserves transition probabilities between pure states is represented on the underlying Hilbert space by either a linear unitary operator or a conjugate-linear antiunitary operator. The theorem underpins the connection between physical symmetries and mathematical operators used in quantum theory, and it justifies the ubiquitous use of unitary group representations in the formulation of quantum dynamics and conservation laws.
Wigner's theorem states that every bijective map on the set of rays (one-dimensional subspaces) of a complex Hilbert space that preserves the absolute value of the inner product (equivalently transition probabilities) is induced by either a unitary or an antiunitary operator on the Hilbert space. The result applies to pure states in quantum theory, where a ray corresponds to a physical pure state and the absolute value squared of the inner product gives the transition probability between states. The theorem was originally formulated in the context of nonrelativistic quantum theory by Eugene Wigner in 1931.
Wigner's theorem explains why symmetries in quantum systems are implemented by unitary or antiunitary operators, a principle that appears in the treatment of conserved quantities and selection rules. In practice this justifies the use of representations of symmetry groups such as the rotation group SO(3), the special unitary group SU(2), and the Poincaré group in relativistic quantum mechanics and quantum field theory. The dichotomy between unitary and antiunitary transformations distinguishes continuous symmetries (implemented by unitary one-parameter groups via Stone's theorem) from discrete symmetries such as time reversal (commonly represented by an antiunitary operator). The theorem is central to the formulation of Noether's theorem in the quantum context and to the theory of superselection sectors.
Let H be a complex separable Hilbert space and P(H) its projective space of rays. A map T: P(H) → P(H) that preserves transition probabilities satisfies |⟨ψ, φ⟩| = |⟨Tψ, Tφ⟩| for all representatives ψ, φ. Wigner's theorem asserts the existence of either a linear isometry U: H → H with U†U = I (unitary) or a conjugate-linear isometry K with K†K = I (antiunitary) inducing T on rays. Standard proofs proceed by choosing an orthonormal basis and defining images of basis vectors up to phase, then extending by linearity or conjugate-linearity while checking consistency using preserved inner-product magnitudes. Modern expositions use tools from projective geometry and functional analysis, invoking results about preservers of Hermitian forms and the structure of automorphisms of the projective space. Variants of the proof leverage the polar decomposition and properties of rank-one projections in the algebra of bounded operators B(H).
Projective Hilbert space P(H) is the set of equivalence classes of nonzero vectors under scalar multiplication; elements encode physical pure states. Transition probability between rays [ψ] and [φ] is given by |⟨ψ, φ⟩|^2. Symmetry transformations are maps on P(H) preserving these probabilities and hence the quantum mechanical statistics of measurement outcomes associated to projections and POVMs. By Wigner's theorem, such symmetries lift to either unitary or antiunitary operators on H, modulo an overall phase, which corresponds to the U(1) gauge symmetry of global phase in quantum states. This lifting is central in the classification of projective representations of groups, leading to the concept of central extensions and the role of ray representations in quantum mechanics.
Several generalizations relax assumptions or adapt Wigner's result to broader contexts. Extensions cover infinite-dimensional Hilbert spaces, non-bijective maps preserving transition probabilities, and preservers of orthogonality alone. Related theorems include Bargmann's theorem on projective unitary representations, which characterizes continuous projective representations and connects to group cohomology and central extensions. In quantum information theory, analogous results appear in studies of maps preserving fidelity or entanglement measures. Mathematical relatives occur in the theory of linear preservers and in results by Uhlhorn (Uhlhorn's theorem) which shows that maps preserving orthogonality of rays are unitary or antiunitary under mild hypotheses. Connections also exist to work by Boçk? and others on preservers of transition probabilities in operator algebras and to structural results in C*-algebra theory.
Concrete applications of Wigner's theorem include: - Time reversal in spin systems: time-reversal operator for half-integer spin is antiunitary (e.g., Kramers degeneracy in systems with time-reversal symmetry and half-integer spin). - Rotations and angular momentum: rotations are implemented by unitary representations of SU(2) on spinor Hilbert spaces leading to Clebsch–Gordan coefficients in addition of angular momenta. - Poincaré symmetry in relativistic quantum theory: representations of the Poincaré group on one-particle Hilbert spaces are unitary up to phase, leading to Wigner's classification of particles by mass and spin. - Quantum information: symmetry constraints on channels and operations often reduce admissible maps to combinations of unitary conjugations or antiunitary conjugations when fidelity or transition probabilities are preserved. These examples illustrate how Wigner's theorem constrains possible physical implementations of symmetries and shapes the mathematical structure used in spectral theory, scattering theory, and the representation theory of physical symmetry groups.
Category:Quantum mechanics Category:Mathematical physics Category:Eugene Wigner