Wigner's theorem on Grassmann spaces
Wigner's theorem on Grassmann spaces
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DOI：
10.1016/j.jfa.2017.06.011
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发表时间：
2017-06
期刊：
影响因子：
--
通讯作者：
G. Geh'er
中科院分区：
文献类型：
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作者：
G. Geh'er
维格纳著名的定理在量子力学的数学基础中特别重要，他指出，复希尔伯特空间中保持转移概率的所有秩一投影的集合上的每个双射变换都是由酉算子或反酉算子诱导的。这一重要定理已被几位科学家以不同的方式推广。2001年，Molnár给出了一个自然的推广，即他刻画了作用于所有秩n投影的Grassmann空间并保持Jordan主角系统不变的映射(不一定是双射映射)(见[17]和[20])。本文给出了Wigner和Molnár定理的一个非常自然的联合推广，即我们证明了Grassmann空间上的所有(不一定是双射的)变换的一个特征，这些变换确定了每一对秩n投影P和Q的数量TrPQ(即主角余弦的平方和)。
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, Molnár 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 Molnár's theorems, namely, we prove a characterisation of all (not necessarily bijective) transformations on the Grassmann space which fix the quantity Tr P Q (ie the sum of the squares of cosines of principal angles) for every pair of rank-n projections P and Q.