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Some remarks on Wigner’s theorem. (Chinese. English summary) Zbl 1212.47123

Summary: The relations between several different forms of Wigner’s theorem are investigated, the descriptions of the theorem in terms of physics and geometry are given. By using operator theory and operator algebras, the equivalence between these different forms of propositions is proved. The results show that, if the surjection \(T : R_1(H)\rightarrow R_1(K)\) preserves the inner product between unit rays, the surjection \(S : R(H)\rightarrow R(K)\) preserves the inner product between rays, the surjection \(\varPhi : P_1(H)\rightarrow P_1(K)\) preserves the trace of the product of rank-one projections, and the surjection \(W : H\rightarrow K\) preserves the inner product between vectors, then there exist unitary or anti-unitary operators \(U : H\rightarrow K\) such that \(Uy \in Tx\), \(r(Uy)=Sx\), \(\varPhi(P_x)=UP_xU^*\) and \(W(x)=\varphi(x)U(x)\), where \(\varphi : H\rightarrow C\) satisfies \(|\varphi(x)|=1\).

MSC:

47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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