Wigner's Theorem In Normed Spaces

The Wigner's theorem is the cornerstone of the mathematical formulation of quantum mechanics,which promotes the research of the basic theory of quantum mechanics.In this paper,we review the introduction and status of the Wigner's theorem in Hilbert spaces.And we introduce the results of Wigner's theorem in normed spaces.That is,let X and Y be normed spaces,and let f:X?Y be an operator satisfying the following equation:{?f(x)+f(y)?,?f(x)-f(y)?}={?x+y?,?x-y?},{x,y?X).This paper studies whether f that satisfying the above equation is phase equivalent to a linear isometric operator in lp(?)(0<p<?)and L?(?)-type spaces,and the affirmative answer of this question is obtained.In the first chapter,we mainly introduce the background and the status of the Wigner's theorem and the main problems that need to be discussed in this paper are given.In the second chapter,we first give the basic knowledge needed in this paper,and introduce the basic contents and some equivalent descriptions of Wigner's theorem.In addition,in order to better understand Wigner's theorem,we introduce the various proofs and conclusions of Wigner's theorem in Hilbert space.In the third chapter,we mainly study the Wigner-type's theorem in some normed spaces.In particular,we discuss the Wigner's theorem in the L?(?)-type spaces,and answer the questions raised in the first chapter.This generalizes the Wigner's theorem.