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. Geh\'{e}r extended Wigner's and Moln\'{a}r'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 Moln\'{a}r and G. P. Geh\'{e}r.