Wigner's theorem on Grassmann spaces

作者：Geher Gyorgy Pal^*

来源：Journal of Functional Analysis, 2017, 273(9): 2994-3001.

DOI：10.1016/j.jfa.2017.06.011

• Transition probability
• Isometries Grassmann space
• Rank-n projections

Wigner's celebrated theorem, which is particularly important in the mathematical foundations of quantum mechanics, states that every bijective transformation on the set of all rank-one projections of a complex Hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator. This vital theorem has been generalised in various ways by several scientists. In 2001, Molnar provided a natural generalisation, namely, he provided a characterisation of (not necessarily bijective) maps which act on the Grassmann space of all rank-n projections and leave the system of Jordan principal angles invariant (see [17] and [20]). In this paper we give a very natural joint generalisation of Wigner's and Molnar's theorems, namely, we prove a characterisation of all (not necessarily bijective) transformations on the Grassmann space which fix the quantity Tr PQ (i.e. the sum of the squares of cosines of principal angles) for every pair of rank-n projections P and Q.

• 出版日期2017-11-1