Wigner's theorem

The title of this article is ambiguous. The Wigner theorem must not be confused with the Wigner-Eckart theorem .

The Wigner theorem , proven by Eugene Paul Wigner in 1931, is a milestone in the mathematical foundations of quantum physics . The theorem describes how symmetries operate in the Hilbert space of quantum mechanical states . Examples of such symmetries are rotations, displacements in spatial space, Lorentz boosts , point symmetries or the CPT symmetry . According to the theorem, every symmetry can be represented as a unitary operator or an antiunitary operator of the Hilbert space.

Exactly expressed, it says that every surjective (but not necessarily linear) mapping on a complex Hilbert space of the condition ${\displaystyle T\colon H\to H}$ ${\displaystyle H}$

${\displaystyle |\langle Tx,Ty\rangle |=|\langle x,y\rangle |}$

for all is enough shape for all has. Here has the amount one and is a unitary or anti-unit operator. ${\displaystyle x,y\in H}$ ${\displaystyle Tx=\varphi Ux}$ ${\displaystyle x\in H}$ ${\displaystyle \varphi \in \mathbb {C} }$ ${\displaystyle U\colon H\rightarrow H}$

Evidence sketch

The proof is based on the fact that vectors of the Hilbert space, which differ only by a complex prefactor of the magnitude one, describe physically identical states due to the probability interpretation . By making a suitable choice for these pre-factors, each mapping can then be made unitary or anti-unitary. ${\displaystyle T}$

Continuous symmetries in quantum mechanics

With Wigner's theorem one can assume that the group elements of a continuous symmetry are unitary. Based on the single element of the group, these are then displayed in the vicinity of the single element with the generators of the continuous group parameterized ( Lie theory ): ${\displaystyle U(\varphi _{i})}$ ${\displaystyle I}$ ${\displaystyle J_{i}}$

${\displaystyle U(\varphi _{i})=\exp(i\varphi _{i}J_{i})}$

If these group elements are unitary, then they are necessarily Hermitian operators of the Hilbert space and thus observables. In general, these are not interchangeable, but operators can be constructed that commute with all of them ( Casimir operators ). That is, thanks to the theorem, a set of obtained quantities can be directly assigned to each continuous symmetry. Conversely, an underlying physical symmetry of the problem can also be guessed from the existence of conserved quantities, which has been successfully used to construct physical theories (example: Eightfold Way ). ${\displaystyle J_{i}}$ ${\displaystyle J_{i}}$

literature

• 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 Unit-Anti-Unit 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.

Individual evidence

1. ^ EP Wigner, Group Theory (Friedrich Vieweg and Son, Braunschweig, Germany, 1931), pp. 251-254; Group Theory (Academic Press Inc., New York, 1959), pp. 233-236.
2. ↑ Hendrik van Hees: The Wigner Theorem, Archived Copy ( Memento of the original from March 4, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.