Special Issue "Mathematical Physics and Symmetry"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (30 April 2019).

Special Issue Editor

Prof. Dr. Palle E.T. Jorgensen
E-Mail Website
Guest Editor
Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA 52242-1419, USA
Fax: +1 319 335 0627
Interests: mathematical physics; Euclidean field theory; reflection positivity; representation theory; operators in Hilbert space; harmonic analysis; fractals; wavelets; stochastic processes; financial mathematics
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Special Issue Information

Dear Colleagues,

Starting with key ideas on reflection positivity in quantum physics, the subject has moved in a number of other directions, including the study of stochastic processes that appear in the representation theory of Lie groups. Motivated by reflection symmetries in Lie groups, there is a new trend in the study of representation theoretic aspects of reflection positivity: reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant Gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations. Since early work in mathematical physics, starting in the 1970s, and initiated by A. Jaffe, and by K. Osterwalder and R. Schrader, the subject of reflection positivity has had an increasing influence on both non-commutative harmonic analysis, and on duality theories for spectrum and geometry. In its original form, the Osterwalder-Schrader idea served to link Euclidean field theory to relativistic quantum field theory. It has been remarkably successful, especially in view of the abelian property of the Euclidean setting, contrasted with the non-commutativity of quantum fields. Osterwalder-Schrader and reflection positivity have also become a powerful tool in the theory of unitary representations of Lie groups.

Prof. Palle E.T. Jorgensen
Guest Editor

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Keywords

  • Reflection positivity
  • Quantum physics
  • Representations
  • Lie groups
  • Gaussian measures
  • Osterwalder-Schrader

Published Papers (10 papers)

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Research

Open AccessArticle
The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space
Symmetry 2019, 11(2), 223; https://doi.org/10.3390/sym11020223 - 14 Feb 2019
Abstract
The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only [...] Read more.
The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrödinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrödinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space. Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
Open AccessArticle
The Existence of Symmetric Positive Solutions of Fourth-Order Elastic Beam Equations
Symmetry 2019, 11(1), 121; https://doi.org/10.3390/sym11010121 - 20 Jan 2019
Abstract
In this study, we consider the eigenvalue problems of fourth-order elastic beam equations. By using Avery and Peterson’s fixed point theory, we prove the existence of symmetric positive solutions for four-point boundary value problem (BVP). After this, we show that there is at [...] Read more.
In this study, we consider the eigenvalue problems of fourth-order elastic beam equations. By using Avery and Peterson’s fixed point theory, we prove the existence of symmetric positive solutions for four-point boundary value problem (BVP). After this, we show that there is at least one positive solution by applying the fixed point theorem of Guo-Krasnosel’skii. Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
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Open AccessArticle
Some Similarity Solutions and Numerical Solutions to the Time-Fractional Burgers System
Symmetry 2019, 11(1), 112; https://doi.org/10.3390/sym11010112 - 18 Jan 2019
Cited by 1
Abstract
In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the [...] Read more.
In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the time-fractional derivative is the Riemann–Liouville type. The FODS can be approximated by some integer-order ordinary differential equations; here, we present three such integer-order ordinary differential equations (called IODE-1, IODE-2, and IODE-3, respectively). For IODE-1, we obtain its similarity solutions and numerical solutions, which approximate the similarity solutions and the numerical solutions of the TFBS. Secondly, we apply the numerical analysis method to obtain the numerical solutions of IODE-2 and IODE-3. Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
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Open AccessArticle
On the Statistical Convergence of Order α in Paranormed Space
Symmetry 2018, 10(10), 483; https://doi.org/10.3390/sym10100483 - 11 Oct 2018
Cited by 1
Abstract
The aim of the present work is to introduce notions of statistical convergence, strongly p-Cesàro summability and the statistically Cauchy sequence of order α in paranormed spaces. Some certain topological properties of these new concepts are examined. Furthermore, we introduce the some [...] Read more.
The aim of the present work is to introduce notions of statistical convergence, strongly p-Cesàro summability and the statistically Cauchy sequence of order α in paranormed spaces. Some certain topological properties of these new concepts are examined. Furthermore, we introduce the some inclusion relations among them. Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
Open AccessArticle
Ground State Representations of Some Non-Rational Conformal Nets
Symmetry 2018, 10(9), 415; https://doi.org/10.3390/sym10090415 - 19 Sep 2018
Cited by 1
Abstract
We construct families of ground state representations of the U ( 1 ) -current net and of the Virasoro nets Vir c with central charge c 1 . We show that these representations are not covariant with respect to the original dilations, [...] Read more.
We construct families of ground state representations of the U ( 1 ) -current net and of the Virasoro nets Vir c with central charge c 1 . We show that these representations are not covariant with respect to the original dilations, and those on the U ( 1 ) -current net are not solitonic. Furthermore, by going to the dual net with respect to the ground state representations of Vir c , one obtains possibly new family of Möbius covariant nets on S 1 . Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
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Open AccessArticle
On Fitting Ideals of Kahler Differential Module
Symmetry 2018, 10(9), 413; https://doi.org/10.3390/sym10090413 - 19 Sep 2018
Abstract
Let k be an algebraically closed field of characteristic zero, and R / I and S / J be algebras over k . 1 ( R / I ) and 1 ( S / J ) denote universal module of first [...] Read more.
Let k be an algebraically closed field of characteristic zero, and R / I and S / J be algebras over k . 1 ( R / I ) and 1 ( S / J ) denote universal module of first order derivation over k . The main result of this paper asserts that the first nonzero Fitting ideal 1 ( R / I k S / J ) is an invertible ideal, if the first nonzero Fitting ideals 1 ( R / I ) and 1 ( S / J ) are invertible ideals. Then using this result, we conclude that the projective dimension of 1 ( R / I k S / J ) is less than or equal to one. Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
Open AccessArticle
On Connection between Second-Degree Exterior and Symmetric Derivations of Kähler Modules
Symmetry 2018, 10(9), 365; https://doi.org/10.3390/sym10090365 - 27 Aug 2018
Abstract
Mathematical physics looks for ways to apply mathematical ideas to problems in physics. In differential forms, the tensor form is first defined, and the definitions of exterior and symmetric differential forms are made accordingly. For instance, M is an R-module, M R [...] Read more.
Mathematical physics looks for ways to apply mathematical ideas to problems in physics. In differential forms, the tensor form is first defined, and the definitions of exterior and symmetric differential forms are made accordingly. For instance, M is an R-module, M R M the tensor product of M with itself and H a submodule of M R M generated by x y y x , where x , y in M. Then, 2 ( M ) = M R M / H is called the second symmetric power of M. A role of the exterior differential forms in field theory is related to the conservation laws for physical fields, etc. In this study, I present a new approach to emphasize the properties of second exterior and symmetric derivations on Kahler modules, and I find a connection between them. I constitute exact sequences of 2 ( Ω 1 ( S ) ) and Λ 2 ( Ω 1 ( S ) ) , and I describe and prove a new isomorphism in the following: Let S be an affine algebra presented by R / I , where R = k [ x 1 , x s ] is a polynomial algebra and I = ( f 1 , f m ) an ideal of R. Then, I have J 1 Ω 1 ( S ) Ω 1 ( S ) 2 ( Ω 1 ( S ) ) Λ 2 ( Ω 1 ( S ) . Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
Open AccessArticle
A Different Study on the Spaces of Generalized Fibonacci Difference bs and cs Spaces Sequence
Symmetry 2018, 10(7), 274; https://doi.org/10.3390/sym10070274 - 11 Jul 2018
Cited by 2
Abstract
The main topic in this article is to define and examine new sequence spaces bs(F^(s,r)) and cs(F^(s,r))), where F^( [...] Read more.
The main topic in this article is to define and examine new sequence spaces bs(F^(s,r)) and cs(F^(s,r))), where F^(s,r) is generalized difference Fibonacci matrix in which s,rR\0. Some algebric properties including some inclusion relations, linearly isomorphism and norms defined over them are given. In addition, it is shown that they are Banach spaces. Finally, the α-, β- and γ-duals of the spaces bs(F^(s,r)) and cs(F^(s,r)) are appointed and some matrix transformations of them are given. Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
Open AccessFeature PaperArticle
Reflection Negative Kernels and Fractional Brownian Motion
Symmetry 2018, 10(6), 191; https://doi.org/10.3390/sym10060191 - 01 Jun 2018
Abstract
In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space E and show in particular that [...] Read more.
In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space E and show in particular that fractional Brownian motion for Hurst index 0 < H 1 / 2 is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if 0 < H < 1 / 2 . We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of GL 2 ( R ) . We relate this to a measure preserving action on a Gaussian L 2 -Hilbert space L 2 ( E ) . Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
Open AccessArticle
On the Second-Degree Exterior Derivation of Kahler Modules on XY
Symmetry 2018, 10(5), 166; https://doi.org/10.3390/sym10050166 - 16 May 2018
Cited by 1
Abstract
This article presents a new approach to stress the properties of Kahler modules. In this paper, we construct the Kahler modules of second-degree exterior derivations and we constitute an exact sequence of X Y -modules. Particularly, we examine Kahler modules on X [...] Read more.
This article presents a new approach to stress the properties of Kahler modules. In this paper, we construct the Kahler modules of second-degree exterior derivations and we constitute an exact sequence of X Y -modules. Particularly, we examine Kahler modules on X Y , and search for the homological size of Λ 2 ( Ω 1 ( X Y ) ) . Full article
(This article belongs to the Special Issue Mathematical Physics and Symmetry)
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