Wigner's theorem

The Wigner’s theorem was formulated and demonstrated for the first time by Eugene Paul Wigner on Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum⁽¹⁾. It states that for each symmetry transformation in Hilbert’s space there exists a unitary or anti-unitary operator, uniquely determined less than a phase factor.
For symmetry transformation, we intend a space transformation that preserved the characteristics of a given physical system. Asymmetry transformation implies also a change of reference system.

Invariants

Invariants play a key role in physics, being the quantities that, in any reference system, are unchanged. With the advent of quantum physics, their importance increased, particularly in the formulation of a relativistic quantum field theory. One of the most important tools in the study of invariants is the Wigner’s theorem, an instrument of fundamental importance for all the development of quantum theory.
In particular, Wigner was interested in determining the properties of transformations that preserve the transition’s probability between two different quantum states. Given $\phi$ the wave function detected by the first observer, and $\bar {\phi}$ the wave function detected by the second observer, Wigner assumed that the equality \[|\langle \psi | \phi \rangle| = |\langle \bar \psi | \bar \phi \rangle|\] must be valid for all $\psi$ and $\phi$.
In the end, if we exclude time inversions, we find that the operator $\operatorname{O}_{R}$, such that $\bar{\phi} = \operatorname{O} _{R} \phi$, must be unitary and linear, but also anti-unitary and anti-linear. Consequence of this fact is that the two observers’ descriptions are equivalent. So the first observes $\phi$, the second $\bar{\phi}$, while the operator $\operatorname{H}$ for the first will be $\operatorname{O}_R \operatorname{H} \operatorname{O}_R^{-1}$ for the second.

The optimal version

There’s also an optimal version of the Wigner’s theorem. It states that a transformation preserve the transition probabilities if the trace of the product $PR$ is equal to the trace of the product $P’R’$, where $P’$, $R’$ are the transformed version of $P$ and $R$.
About this version I find a couple of interesting paper:

Gehér, G. P. (2017). Wigner’s theorem on Grassmann spaces. Journal of Functional Analysis. arXiv:1706.02329

Barvínek, J., & Hamhalter, J. (2017). Linear algebraic proof of Wigner Theorem and its consequences. Mathematica Slovaca, 67(2), 371-386. doi:10.1515/ms-2016-0273 (sci-hub)

In particular the second paper presents some applications to the quantum entropy of infinite quantum systems.

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1. Wigner, E.P. (1959), Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press Inc., New York ↩