Abstract
Wigner theorem is the cornerstone of the mathematical formula of quan-tum mechanics, it has promoted the research of basic theory of quantum mechanics. In this article, we give a certain pair of functional equations between two real spaces s or two real sn(H), that we called “phase isometry”. It is obtained that all such solutions are phase equivalent to real linear isometries in the space s and the space sn(H).
Highlights
Mazur and Ulam in [1] proved that every surjective isometry U between X and Y is a affine, states that the mapping with U (0) = 0, U is linear
It is obtained that all such solutions are phase equivalent to real linear isometries in the space s and the space sn ( H )
If the two spaces are Hilbert spaces, Rätz proved that the phase isometries V : X → Y are precisely the solutions of functional equation in [7]
Summary
It is obtained that all such solutions are phase equivalent to real linear isometries in the space s and the space sn ( H ) . If the two spaces are Hilbert spaces, Rätz proved that the phase isometries V : X → Y are precisely the solutions of functional equation in [7]. Suppose that V0 : Sr0 ( s) → Sr0 ( s) is a mapping satisfying Equation (1).
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