Wigner-type theorem on transition probability preserving maps in semifinite factors

Abstract

The Wigner's theorem, which is one of the cornerstones of the mathematical formulation of quantum mechanics, asserts that every symmetry of quantum system is unitary or anti-unitary. This classical result was first given by Wigner in 1931. Thereafter it has been proved and generalized in various ways by many authors. Recently, G. P. Gehér extended Wigner's and Molnár's theorems and characterized the transformations on the Grassmann space of all rank-n projections which preserve the transition probability. The aim of this paper is to provide a new approach to describe the general form of the transition probability preserving (not necessarily bijective) maps between Grassmann spaces. As a byproduct, we are able to generalize the results of Molnár and G. P. Gehér.

Book Review: Wigner-type theorems for Hilbert Grassmannians

“…It turns out that, despite the significantly weakened assumption on our map, the structure of these transformations is exactly the same as given by Molnár's theorem from the previous paragraph. We also note that in a recent interesting paper [16], Qian, Wang, Wu, and Yuan further generalized this result to the setting of semifinite factors.…”

Surjective $L^p$-isometries of Grassmann spaces

“…As for the isometries with respect to the L 2 -norm, Gehér generalized the early work of Molnár [15] and proved that every (not necessarily surjective) L 2 -isometry on Grassmann spaces in B(H) is induced by a Jordan *-homomorphisms of B(H) [6]. This result was later generalized to the case of L 2 -isometry on Grassmann spaces in semifinite factors [20].…”

“…Assume that p = 2. Proceed as in the proof of Lemma 1 in [15] (or see the proof of Lemma 2.2 in [20]), we can show that φ has a unique linear extension Φ on the linear span of P(M 1 ). More explicitly, Φ is defined as follows…”

“…for every trace 1/2 projection P in N. Recall that the linear span of the set of trace 1/2 projections is dense in N (see Lemma 3.7 in [20]). Therefore, ϕ ′ is the identity *-isomorphism, and the lemma is proved.…”

Isometries between projection lattices of von Neumann algebras

“…They made use of the idea of geodesics between two projections, which is also essential in our proof of Theorem 2.1. See also [16], [3], [13] and [10], [15], in which mappings between projection lattices with an assumption which is different from ours are studied.…”