关于Wigner定理的注记

Some remarks on Wigner′s theorem

研究了Wigner定理的几种不同表述形式之间的关系,给出了该定理在物理、几何等不同方面的描述.应用算子论与算子代数的方法,证明了这些不同形式命题之间的等价性.结果表明,若满射T:R1(H)→R1(K)保持单位射线的内积,满射S:R(H)→R(K)保持射线的内积,满射Φ:P1(H)→P1(K)保持1-秩投影乘积的迹,满射W:H→K保持向量的内积,则存在相应的酉算子或反酉算子U:H→K,使得Uy∈Tx,r(Uy)=Sx,Φ(Px)=UPxU*及W(x)=φ(x)U(x),其中φ:H→C满足|φ(x)|=1.

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 algebra, the equivalence between these different forms of propositions is proved. The results show that if the surjection T： R1 （H） R1 （K） preserves the inner product between unit rays, the surjection S：R（H）→R（K） preserves the inner prod- uct between rays, the surjection Ф ： P1 （H） →P1 （K） preserves the trace of the product of rank-one projections, and the surjection W：H→K preserves the inner product between vectors, then there exists unit or anti-unitary operators U：H→K, such that Uy∈Tx,r（Uy）=Sx,Φ（Px）=UPxU＊andW（x）=φ（x）U（x） ,in which φ：H→C satisfies ｜φ（x）｜ =1.