Special Issue "Mathematical physics"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 May 2015)

Special Issue Editor

Guest Editor
Prof. Dr. Palle E.T. Jorgensen

Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA 52242-1419, USA
E-Mail
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

Special Issue Information

Dear Colleagues,

Mathematical physics dates back a long time, but in the 20th century it has come to center around mathematical methods used in quantum theory, both relativistic and non-relativistic; and including such major areas as quantum field theory, and quantum statistical mechanics. Each of these areas in turn entails notions of dynamics, dynamical systems, scattering theory, the study of interactions, and rigorous theories of phase transition. Because of the foundations of quantum theory, and the framework suggested by John von Neumann and Paul Dirac, the problems and the models studied in mathematical physics rely on the theory linear operators in Hilbert space, and on algebras of operators. Of more recent vintage is a set of advances in quantum information theory, and associated qubit (or quantum bit) algorithms. Related to this is the theory of quantum measurement problems. Within quantum field theories, there are the axiom systems of Wightman fields, as well of Euclidean fields; and both areas relying on measures in path-space; as well as consideration of unitary representations and their harmonic analysis and spectral theory; as well as on analytic continuation tools. Only a year ago, we saw a solution to the Kadison-Singer problem, which also has its roots in Dirac's formulation of quantum theory; although it turns out to also have other striking implications, even outside mathematical physics proper.

Prof. Dr. Palle Jorgensen
Guest Editor

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Keywords

  • Quantum fields
  • Euclidean fields
  • statistical mechanics
  • mathematical methods in physics
  • measures on function spaces
  • use of algebras of operators in math phys
  • use of representations of Lie groups in physics
  • reflection positivity and representations
  • Schrödinger operators
  • approaches via Feynman and Kac
  • spectral analysis
  • approximation
  • Monte-Carlo methods

Published Papers (23 papers)

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Research

Jump to: Review, Other

Open AccessArticle Two Dimensional Temperature Distributions in Plate Heat Exchangers: An Analytical Approach
Mathematics 2015, 3(4), 1255-1273; doi:10.3390/math3041255
Received: 20 June 2015 / Revised: 26 November 2015 / Accepted: 30 November 2015 / Published: 16 December 2015
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Abstract
Analytical solutions are developed to work out the two-dimensional (2D) temperature changes of flow in the passages of a plate heat exchanger in parallel flow and counter flow arrangements. Two different flow regimes, namely, the plug flow and the turbulent flow are considered.
[...] Read more.
Analytical solutions are developed to work out the two-dimensional (2D) temperature changes of flow in the passages of a plate heat exchanger in parallel flow and counter flow arrangements. Two different flow regimes, namely, the plug flow and the turbulent flow are considered. The mathematical formulation of problems coupled at boundary conditions are presented, the solution procedure is then obtained as a special case of the two region Sturm-Liouville problem. The results obtained for two different flow regimes are then compared with experimental results and with each other. The agreement between the analytical and experimental results is an indication of the accuracy of solution method. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Free W*-Dynamical Systems From p-Adic Number Fields and the Euler Totient Function
Mathematics 2015, 3(4), 1095-1138; doi:10.3390/math3041095
Received: 26 May 2015 / Revised: 18 September 2015 / Accepted: 23 September 2015 / Published: 2 December 2015
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Abstract
In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚp (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ,
[...] Read more.
In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚp (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ, of the arithmetic algebra A , consisting of all arithmetic functions. In particular, we apply such free probability to consider operator-theoretic and operator-algebraic properties of W * -dynamical systems induced by ℚp under free-probabilistic (and hence, spectral-theoretic) techniques. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Gauge Invariance and Symmetry Breaking by Topology and Energy Gap
Mathematics 2015, 3(4), 984-1000; doi:10.3390/math3040984
Received: 16 July 2015 / Revised: 8 October 2015 / Accepted: 12 October 2015 / Published: 22 October 2015
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Abstract
For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners
[...] Read more.
For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Photon Localization Revisited
Mathematics 2015, 3(3), 897-912; doi:10.3390/math3030897
Received: 2 July 2015 / Accepted: 11 September 2015 / Published: 23 September 2015
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Abstract
In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this
[...] Read more.
In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as quantum field ontology associated with the eventualization principle, which enables us to explain the mutual relations, back and forth, between quantum fields and various forms of particles in the localized form of the former. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Chern-Simons Path Integrals in S2 × S1
Mathematics 2015, 3(3), 843-879; doi:10.3390/math3030843
Received: 30 May 2015 / Revised: 11 August 2015 / Accepted: 12 August 2015 / Published: 21 August 2015
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Abstract
Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of
[...] Read more.
Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\). Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics
Mathematics 2015, 3(3), 781-842; doi:10.3390/math3030781
Received: 14 March 2015 / Revised: 28 July 2015 / Accepted: 14 August 2015 / Published: 20 August 2015
Cited by 1 | PDF Full-text (523 KB) | HTML Full-text | XML Full-text
Abstract
A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in
[...] Read more.
A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle The Segal–Bargmann Transform for Odd-Dimensional Hyperbolic Spaces
Mathematics 2015, 3(3), 758-780; doi:10.3390/math3030758
Received: 7 July 2015 / Revised: 7 August 2015 / Accepted: 10 August 2015 / Published: 18 August 2015
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Abstract We develop isometry and inversion formulas for the Segal–Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle From Classical to Discrete Gravity through Exponential Non-Standard Lagrangians in General Relativity
Mathematics 2015, 3(3), 727-745; doi:10.3390/math3030727
Received: 24 May 2015 / Revised: 30 July 2015 / Accepted: 4 August 2015 / Published: 14 August 2015
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Abstract
Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this
[...] Read more.
Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this paper is to fill the gap. Though non-standard Lagrangians may be defined by a multitude form, in this paper, we considered the exponential type. One basic feature of exponential non-standard Lagrangians concerns the modified Euler-Lagrange equation obtained from the standard variational analysis. Accordingly, when applied to spacetime geometries, one unsurprisingly expects modified geodesic equations. However, when taking into account the time-like paths parameterization constraint, remarkably, it was observed that mutually discrete gravity and discrete spacetime emerge in the theory. Two different independent cases were obtained: A geometrical manifold with new spacetime coordinates augmented by a metric signature change and a geometrical manifold characterized by a discretized spacetime metric. Both cases give raise to Einstein’s field equations yet the gravity is discretized and originated from “spacetime discreteness”. A number of mathematical and physical implications of these results were discussed though this paper and perspectives are given accordingly. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Zeta Function Expression of Spin Partition Functions on Thermal AdS3
Mathematics 2015, 3(3), 653-665; doi:10.3390/math3030653
Received: 12 May 2015 / Accepted: 16 July 2015 / Published: 28 July 2015
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Abstract
We find a Selberg zeta function expression of certain one-loop spin partition functions on three-dimensional thermal anti-de Sitter space. Of particular interest is the partition function of higher spin fermionic particles. We also set up, in the presence of spin, a Patterson-type formula
[...] Read more.
We find a Selberg zeta function expression of certain one-loop spin partition functions on three-dimensional thermal anti-de Sitter space. Of particular interest is the partition function of higher spin fermionic particles. We also set up, in the presence of spin, a Patterson-type formula involving the logarithmic derivative of zeta. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle On the Nature of the Tsallis–Fourier Transform
Mathematics 2015, 3(3), 644-652; doi:10.3390/math3030644
Received: 2 May 2015 / Revised: 11 July 2015 / Accepted: 13 July 2015 / Published: 21 July 2015
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Abstract
By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis’ q-statistics may revolve around equivalence classes of distributions and
[...] Read more.
By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis’ q-statistics may revolve around equivalence classes of distributions and not individual ones, as orthodox statistics does. We solve here the qFT’s non-invertibility issue, but discover a problem that remains open. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Singular Bilinear Integrals in Quantum Physics
Mathematics 2015, 3(3), 563-603; doi:10.3390/math3030563
Received: 6 April 2015 / Revised: 12 June 2015 / Accepted: 17 June 2015 / Published: 29 June 2015
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Abstract
Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to
[...] Read more.
Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle The Schwartz Space: Tools for Quantum Mechanics and Infinite Dimensional Analysis
Mathematics 2015, 3(2), 527-562; doi:10.3390/math3020527
Received: 17 April 2015 / Accepted: 3 June 2015 / Published: 16 June 2015
Cited by 1 | PDF Full-text (335 KB) | HTML Full-text | XML Full-text
Abstract An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Effective Summation and Interpolation of Series by Self-Similar Root Approximants
Mathematics 2015, 3(2), 510-526; doi:10.3390/math3020510
Received: 25 March 2015 / Accepted: 9 June 2015 / Published: 15 June 2015
Cited by 1 | PDF Full-text (209 KB) | HTML Full-text | XML Full-text
Abstract
We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by
[...] Read more.
We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that of Padé approximants, when the latter can be defined. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle The 1st Law of Thermodynamics for the Mean Energy of a Closed Quantum System in the Aharonov-Vaidman Gauge
Mathematics 2015, 3(2), 428-443; doi:10.3390/math3020428
Received: 18 February 2015 / Accepted: 19 May 2015 / Published: 1 June 2015
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Abstract
The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law
[...] Read more.
The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law of Thermodynamics that is applicable to the mean energy of a closed quantum system when the mean energy is expressed in the Aharonov-Vaidman gauge, i.e., when the system’s energy is weak valued. This is achieved by identifying the generalized heat and work exchange terms that appear in the equation of motion for weak valued energy. The complex valued contributions of the additive gauge term to these generalized exchange terms are discussed and this extended 1st Law is shown to subsume the usual 1st Law that is applicable for the mean energy of a closed quantum system. It is found that the gauge transformation introduces an additional energy uncertainty exchange term that—while it is neither a heat nor a work exchange term—is necessary for the conservation of weak valued energy. A spin-1/2 particle in a uniform magnetic field is used to illustrate aspects of the theory. It is demonstrated for this case that the extended 1st Law implies the existence of a gauge potential ω and that it generates a non-vanishing gauge field F. It is also shown for this case that the energy uncertainty exchange accumulated during the evolution of the system along a closed evolutionary cycle C in an associated parameter space is a geometric phase. This phase is equal to both the path integral of ω along C and the integral of the flux of F through the area enclosed by C. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle High-Precision Arithmetic in Mathematical Physics
Mathematics 2015, 3(2), 337-367; doi:10.3390/math3020337
Received: 19 January 2015 / Accepted: 16 April 2015 / Published: 12 May 2015
Cited by 2 | PDF Full-text (904 KB) | HTML Full-text | XML Full-text
Abstract
For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This
[...] Read more.
For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Action at a Distance in Quantum Theory
Mathematics 2015, 3(2), 329-336; doi:10.3390/math3020329
Received: 8 March 2015 / Accepted: 22 April 2015 / Published: 6 May 2015
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Abstract
The purpose of this paper is to present a consistent mathematical framework that shows how the EPR (Einstein. Podolsky, Rosen) phenomenon fits into our view of space time. To resolve the differences between the Hilbert space structure of quantum theory and the manifold
[...] Read more.
The purpose of this paper is to present a consistent mathematical framework that shows how the EPR (Einstein. Podolsky, Rosen) phenomenon fits into our view of space time. To resolve the differences between the Hilbert space structure of quantum theory and the manifold structure of classical physics, the manifold is taken as a partial representation of the Hilbert space. It is the partial nature of the representation that allows for action at a distance and the failure of the manifold picture. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle There Are Quantum Jumps
Mathematics 2015, 3(2), 319-328; doi:10.3390/math3020319
Received: 10 March 2015 / Accepted: 20 April 2015 / Published: 5 May 2015
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Abstract
In this communication we take up the age-old problem of the possibility to incorporate quantum jumps. Unusually, we investigate quantum jumps in an extended quantum setting, but one of rigorous mathematical significance. The general background for this formulation originates in the Balslev-Combes theorem
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In this communication we take up the age-old problem of the possibility to incorporate quantum jumps. Unusually, we investigate quantum jumps in an extended quantum setting, but one of rigorous mathematical significance. The general background for this formulation originates in the Balslev-Combes theorem for dilatation analytic Hamiltonians and associated complex symmetric representations. The actual jump is mapped into a Jordan block of order two and a detailed derivation is discussed for the case of the emission of a photon by an atom. The result can be easily reassigned to analogous cases as well as generalized to Segrè characteristics of arbitrary order. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Maxwell–Lorentz Electrodynamics Revisited via the Lagrangian Formalism and Feynman Proper Time Paradigm
Mathematics 2015, 3(2), 190-257; doi:10.3390/math3020190
Received: 25 January 2015 / Accepted: 27 March 2015 / Published: 17 April 2015
Cited by 2 | PDF Full-text (498 KB) | HTML Full-text | XML Full-text
Abstract
We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field
[...] Read more.
We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field theory approach to the Lagrangian and Hamiltonian, the formulations of alternative classical electrodynamics models are analyzed in detail and their Dirac type quantization is suggested. Problems closely related to the radiation reaction force and electron mass inertia are analyzed. The validity of the Abraham-Lorentz electromagnetic electron mass origin hypothesis is argued. The related electromagnetic Dirac–Fock–Podolsky problem and symplectic properties of the Maxwell and Yang–Mills type dynamical systems are analyzed. The crucial importance of the remaining reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Multiple q-Zeta Brackets
Mathematics 2015, 3(1), 119-130; doi:10.3390/math3010119
Received: 27 January 2015 / Accepted: 13 March 2015 / Published: 20 March 2015
Cited by 1 | PDF Full-text (221 KB) | HTML Full-text | XML Full-text
Abstract
The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, Bachmann constructed a q-analogue of the MZVs—the
[...] Read more.
The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, Bachmann constructed a q-analogue of the MZVs—the so-called bi-brackets—for which the two products are dual to each other, in a very natural way. We overview Bachmann’s construction and discuss the radial asymptotics of the bi-brackets, its links to the MZVs, and related linear (in)dependence questions of the q-analogue. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessArticle Quantum Measurements of Scattered Particles
Mathematics 2015, 3(1), 92-118; doi:10.3390/math3010092
Received: 18 January 2015 / Accepted: 11 March 2015 / Published: 19 March 2015
Cited by 1 | PDF Full-text (373 KB) | HTML Full-text | XML Full-text
Abstract
We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random variables.
[...] Read more.
We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random variables. We link the asymptotic properties of this process to spectral characteristics of the dynamics. We show that the process has decaying time correlations and that a zero-one law holds. We deduce that if the incoming probes are not sharply localized with respect to the spectrum of the measurement operator, then the process does not converge. Nevertheless, the scattering modifies the measurement outcome frequencies, which are shown to be the average of the measurement projection operator, evolved for one interaction period, in an asymptotic state. We illustrate the results on a truncated Jaynes–Cummings model. Full article
(This article belongs to the Special Issue Mathematical physics)

Review

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Open AccessReview Twistor Interpretation of Harmonic Spheres and Yang–Mills Fields
Mathematics 2015, 3(1), 47-75; doi:10.3390/math3010047
Received: 24 January 2015 / Accepted: 4 March 2015 / Published: 16 March 2015
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Abstract
We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective
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We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space \(\Omega G\) of a compact Lie group \(G\) and the moduli space of Yang–Mills \(G\)-fields on \(\mathbb R^4\). Full article
(This article belongs to the Special Issue Mathematical physics)

Other

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Open AccessEssay A Moonshine Dialogue in Mathematical Physics
Mathematics 2015, 3(3), 746-757; doi:10.3390/math3030746
Received: 18 June 2015 / Revised: 29 July 2015 / Accepted: 30 July 2015 / Published: 14 August 2015
Cited by 2 | PDF Full-text (905 KB) | HTML Full-text | XML Full-text
Abstract
Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive
[...] Read more.
Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive at dessins d’enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell’s theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincaré would have been happy to see it. Full article
(This article belongs to the Special Issue Mathematical physics)
Open AccessLetter The Complement of Binary Klein Quadric as a Combinatorial Grassmannian
Mathematics 2015, 3(2), 481-486; doi:10.3390/math3020481
Received: 9 May 2015 / Accepted: 5 June 2015 / Published: 8 June 2015
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Abstract
Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It
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Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286; 563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888). Full article
(This article belongs to the Special Issue Mathematical physics)

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