On normed spaces with the Wigner Property

Ruidong Wang,Dariusz Bugajewski

doi:10.1007/s43034-019-00035-y

Ruidong Wang, Dariusz Bugajewski

Open Access

https://doi.org/10.1007/s43034-019-00035-y

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Abstract

The aim of this paper is to generalize the Wigner Theorem to real normed spaces. A normed space is said to have the Wigner Property if the Wigner Theorem holds for it. We prove that every two-dimensional real normed space has the Wigner Property. We also study the Wigner Property of real normed spaces of dimension at least three. It is also shown that strictly convex real normed spaces possess the Wigner Property.

Highlights

The well-known Wigner Theorem, which plays an important role in quantum mechanics, states that any transformations of the states of a physical system which preserve the transition probability associated to any pair of states are induced either by a unitary or by an anti-unitary operator on the Hilbert space associated with the physical system
Theorem 6 is important in the study of surjective phase isometry operators between two dimensional real normed spaces
By Lemma 8 and Theorem 6, we infer that T is a phase equivalent to a linear isometry

Summary

Introduction

The well-known Wigner Theorem, which plays an important role in quantum mechanics, states that any transformations of the states of a physical system which preserve the transition probability associated to any pair of states are induced either by a unitary or by an anti-unitary operator on the Hilbert space associated with the physical system (see [5,23]). We would like to draw the readers’ attention to the very recent paper [16] in which the author establishes a Wigner’s type theorem for linear operators which map projections of a fixed rank to projections of other fixed rank. Notice that by the Wigner Theorem, this problem is solved when both E and F are real inner product spaces. E is said to have the Wigner Property if for any real normed space F, and any surjective phase isometry T : E → F, T is phase equivalent to a linear isometry from E to F. Tan and Huang [20] proved that smooth real normed spaces have the Wigner Property. By |A| we will denote the cardinality of the set A and span{A} will denote the linear subspace generated by the set A

Preliminary results

Two-dimensional normed spaces with the Wigner Property

Wigner Property on real Banach spaces of dimension at least three