Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Computational Mathematics

Wigner’s Theorem in s* and sn(H) Spaces

Meimei Song, Shunxin Zhao

Published Jan 1, 2018• 0 citations
Publisher

Abstract

Wigner theorem is the cornerstone of the mathematical formula of quantum 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 spaces ( ) n s 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 ( ) n s H .

DOI
10.4236/ajcm.2018.83017

Related papers

Additive isometries on a quaternionic Hilbert space

C. S. Sharma, Dennis Almeida

A systematic study of additive isometries on a quaternionic Hilbert space is presented. A number of new results describing the properties of such operators are proved. The work culminates in the first mathematical proof of Wigner’s theorem for quaternionic Hilbert spaces of dimension other than 2 which asserts that any operator which preserves the absolute value of the inner product on a quaternionic Hilbert space is equivalent, in the sense of differing pointwise by a mere phase factor, to a linear isometry. A complete and concise description of the exceptional situation in a two-dimensional quaternionic Hilbert space is given.

Journal of Mathematical Physics•May 1, 1990•8 citations

REPRESENTATION OF SYMMETRY TRANSFORMATIONS IN QUANTUM MECHANICS

U. Uhlhorn

A generalization of Wigner's representation theorem is presented that may be applied when superselection rules (of a specified character) must be taken into account. At the same time some gaps in existing proofs of Wigner's theorem are eliminated. The connection between Wigner,s theorem and the so called fundamental theorem of projective geometry is pointed out, and a new, short proof of the latter theorem (valid for an arbitrary field of numbers) is presented. The equivalent possibilities of characterizing symmetry transformations as mappings leaving invariant the logical structure of quantum mechanics, as isometric linear or conjugated linear transformations connecting two complex Hilbert spaces, or as mappings leaving invariant the algebraic structure of quantum mechanics (or equivalently the structure of the equations of motion) are discussed and lead to an invariant (coordinate free) characterization of reversal of the direction of motion (time inversion). An analogue of Wigner's theorem for symmetry transformations in the recently discussed quaternion quantum mechanics is proved in an appendix, and it is pointed out how quaternion quantum mechanics, as well as ordinary complex quantum mechanics, may be consistently represented in the framework of real quantum mechanics, which in this sense appears as a super theory. The symmetry transformationsmore » of classical mechanics are discussed in relation to their quantum mechanical analogues in another appendix. They are intimately connected with the canonical formalism. (auth)« less

Arkiv Fysik•Jan 1, 1963•155 citations

Quantum mechanics in phase space

Philip Watson

The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.

•Jan 1, 1991•0 citations

Wigner's Theorem and Spectrum Shrinking Maps

Allen, Meaghan

Wigner's Theorem, an influential result in quantum mechanics established by Eugene Wigner in 1931, states that every symmetry transformation on a ray space is induced by either a unitary or anti-unitary transformation. In this work, we generalize Wigner's Theorem in two directions: first, to maps between different Hilbert spaces, and second, to maps between factors of type II. In both cases, we focus on weakening the assumptions on the map, specifically by utilizing spectrum shrinking conditions. For the first generalization, let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces, and let $k$ be a positive integer such that $2\leq k < \frac{1}{2} \dim(\mathcal{H}_1)$. We show that if an injective map $\psi: \mathcal{P}_k(\mathcal{H}_1) \to \mathcal{P}_k(\mathcal{H}_2)$ satisfies two spectrum shrinking conditions, then it is induced by a linear or conjugate-linear isometry. For the second generalization, let $\mathcal{M}$ be a type II$_1$ or type II$_\infty$ factor, let $\tau_\mathcal{M}$ be a faithful normal semi-finite tracial weight of $\mathcal M$, and let $0

•May 1, 2025•0 citations

On quantum mechanical phase-space wave functions

Joachim J. Wl odarz

An approach to quantum mechanics based on the notion of a phase-space wave function is proposed within the Weyl–Wigner–Moyal representation. It is shown that the Schrödinger equation for the phase-space wave function is equivalent to the quantum Liouville equation for the Wigner distribution function. The relationship to the recent results by Torres-Vega and Frederick [J. Chem. Phys. 98, 3103 (1993)] is also discussed.

The Journal of Chemical Physics•May 15, 1994•23 citations

Wigner-Type Theorems for Hilbert Grassmannians

Mark Pankov

Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.

Cambridge University Press eBooks•Jan 6, 2020•23 citations
Wigner theorem is the cornerstone of the mathematical formula of quantum 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 spaces ( ) n s 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 ( ) n s H .
A systematic study of additive isometries on a quaternionic Hilbert space is presented. A number of new results describing the properties of such operators are proved. The work culminates in the first mathematical proof of Wigner’s theorem for quaternionic Hilbert spaces of dimension other than 2 which asserts that any operator which preserves the absolute value of the inner product on a quaternionic Hilbert space is equivalent, in the sense of differing pointwise by a mere phase factor, to a linear isometry. A complete and concise description of the exceptional situation in a two-dimensional quaternionic Hilbert space is given.
A generalization of Wigner's representation theorem is presented that may be applied when superselection rules (of a specified character) must be taken into account. At the same time some gaps in existing proofs of Wigner's theorem are eliminated. The connection between Wigner,s theorem and the so called fundamental theorem of projective geometry is pointed out, and a new, short proof of the latter theorem (valid for an arbitrary field of numbers) is presented. The equivalent possibilities of characterizing symmetry transformations as mappings leaving invariant the logical structure of quantum mechanics, as isometric linear or conjugated linear transformations connecting two complex Hilbert spaces, or as mappings leaving invariant the algebraic structure of quantum mechanics (or equivalently the structure of the equations of motion) are discussed and lead to an invariant (coordinate free) characterization of reversal of the direction of motion (time inversion). An analogue of Wigner's theorem for symmetry transformations in the recently discussed quaternion quantum mechanics is proved in an appendix, and it is pointed out how quaternion quantum mechanics, as well as ordinary complex quantum mechanics, may be consistently represented in the framework of real quantum mechanics, which in this sense appears as a super theory. The symmetry transformationsmore » of classical mechanics are discussed in relation to their quantum mechanical analogues in another appendix. They are intimately connected with the canonical formalism. (auth)« less
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
Wigner's Theorem, an influential result in quantum mechanics established by Eugene Wigner in 1931, states that every symmetry transformation on a ray space is induced by either a unitary or anti-unitary transformation. In this work, we generalize Wigner's Theorem in two directions: first, to maps between different Hilbert spaces, and second, to maps between factors of type II. In both cases, we focus on weakening the assumptions on the map, specifically by utilizing spectrum shrinking conditions. For the first generalization, let H1\mathcal{H}_1H1​ and H2\mathcal{H}_2H2​ be Hilbert spaces, and let kkk be a positive integer such that 2≤k<12dim⁡(H1)2\leq k < \frac{1}{2} \dim(\mathcal{H}_1)2≤k<21​dim(H1​). We show that if an injective map ψ:Pk(H1)→Pk(H2)\psi: \mathcal{P}_k(\mathcal{H}_1) \to \mathcal{P}_k(\mathcal{H}_2)ψ:Pk​(H1​)→Pk​(H2​) satisfies two spectrum shrinking conditions, then it is induced by a linear or conjugate-linear isometry. For the second generalization, let M\mathcal{M}M be a type II1_11​ or type II∞_\infty∞​ factor, let τM\tau_\mathcal{M}τM​ be a faithful normal semi-finite tracial weight of M\mathcal MM, and let $0
Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.