Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states.
for all has the form for all , where has modulus one, and is either unitary or anti unitary, depending on the symmetry considered.
Symmetry in quantum mechanics
In quantum mechanics and quantum field theory, the quantum state that characterizes one or more particles or fields is a vector (ket) in a complex Hilbert space. Any symmetry operation, for example "translate all particles and fields forward in time by five seconds", or "Lorentz transform all particles and fields by a 5 m/s boost in the x direction", corresponds to an operator T on that Hilbert space. This operator, T, must be bijective because every quantum state must have a unique corresponding transformed state and vice-versa. Also, the probability of finding a system in state when it is initially in state is given by . Since T is a symmetry operation, the probability of finding the system in state Tx when it is initially in state Ty must be the same; therefore . It follows that T satisfies the hypotheses of Wigner's theorem.
Thus, according to Wigner's theorem, T is either unitary or anti-unitary. In the two examples above (time translations and Lorentz boosts), T corresponds to a unitary symmetry operator. The time-reversal symmetry operator is the archetypal example of an anti-unitary symmetry operator.
- E. P. Wigner, Gruppentheorie (Friedrich Vieweg und Sohn, Braunschweig, Germany, 1931), pp. 251–254; Group Theory (Academic Press Inc., New York, 1959), pp. 233–236.
- Bargmann, V. "Note on Wigner's Theorem on Symmetry Operations". Journal of Mathematical Physics Vol 5, no. 7, Jul 1964.
- Mouchet, Amaury. "An alternative proof of Wigner theorem on quantum transformations based on elementary complex analysis". Physics Letters A 377 (2013) 2709-2711. hal.archives-ouvertes.fr:hal-00807644
- Molnar, Lajos. "An Algebraic Approach to Wigner's Unitary-Antiunitary Theorem". arXiv:math/9808033
- Simon, R., Mukunda, N., Chaturvedi, S., Srinivasan, V., 2008. "Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics". Phys. Lett. A 372, 6847–6852.
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