Dimensional Minkowski Space Research Articles

Matter relative to quantum hypersurfaces

We explore the canonical description of a scalar field in 1+1 dimensional Minkowski space as a parametrized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces X relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals |ψϕ[X]⟩ interpreted as the state of the field relative to the hypersurface X, thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparametrization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect. Published by the American Physical Society 2024

Partner Curves in Three Dimensional Degenerated Space Form

Using elementary and effective methods we study cone curves and their associated curves or partner curves in the three dimensional lightlike cone $\Q^3$, which is called three dimensional degenerated space form of the four dimensional Minkowski space $\E^4_1$. We define the associated curve of the cone curve and also partner curves of some special curves, such as a Bertrand curve and a Mannheim curve. We consider the properties and relations of a curve and its associated curve or partner curve. Some geometric characterizations of these curves are also given.

Global Nonlinear Stability of Traveling Wave Solution to Time-Like Extremal Hypersurface in Minkowski Space

Abstract There are a few results about the global nonlinear stability of nontrivial large solution to quasilinear wave equations. Time-like extremal surface in Minkowski space is an important model of quasilinear wave equation. There are two folds in this paper. Firstly, we get the existence of traveling wave solution to the time-like extremal hypersurface in $\mathbb {R}^{1+(n+1)}$, which can be considered as the generalized Bernstein theorem. For $n=2$, we are also concerned with global stability of traveling wave solutions with speed of light to time-like extremal hypersurface equation in $1+(2+1)$ dimensional Minkowski space, which is corresponding with quasilinear wave equation in two space dimensions.

Ricci Solitons of Three-Dimensional Lorentzian Bianchi-Cartan-Vranceanu Spaces

In this paper, we obtain explicit formulae for homogenous Ricci solitons on three-dimensional Lorentzian Bianchi-Cartan-Vranceanu spaces. We also give a result about Ricci solitons on a three dimensional Minkowski space.

Effective orientifolds from broken supersymmetry

We recently proposed a class of type IIB vacua that yield, at low energies, four–dimensional Minkowski spaces with broken supersymmetry and a constant string coupling. They are compactifications with an internal five-torus bearing a five–form flux Φ and warp factors depending on a single coordinate. The breaking of supersymmetry occurs when the internal space includes a finite interval. A probe-brane analysis revealed a gravitational repulsion and a charge attraction of equal magnitude from the left end of the interval, together with a singularity at the other end. Here we complete the analysis revealing the presence, at one end, of an effective of negative tension and positive five–form charge. We also determine the values of these quantities, showing that , and characterize the singularity present at the other end of the interval, which hosts an opposite charge. Finally, we discuss various forms of the gravity action in the presence of a boundary and identify a self–adjoint form for its fluctuations. Invited contribution to the special issue of Journal of Physics A: Mathematical and Theoretical on ‘Fields, Gravity, Strings and Beyond: In Memory of Stanley Deser’

Optimal blowup stability for supercritical wave maps

We study corotational wave maps from (1+4)-dimensional Minkowski space into the 4-sphere. We prove the stability of an explicitly known self-similar wave map under perturbations that are small in the critical Sobolev space.

Global Stability for Charged Scalar Fields in an Asymptotically Flat Metric in Harmonic Gauge

We prove global stability for the charged scalar field system on a background spacetime, which is close to 1+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1+3$$\\end{document}-dimensional Minkowski space and whose outward light cones converge to those for the Schwarzs-child metric at null infinity. The key technique to this proof is the use of a modified null frame, depending only on the mass M of the metric, which captures the asymptotic behavior of the metric at future null infinity. Our results are analogous to results obtained in Minkowski space by Lindblad and Sterbenz in (IMRP Int Math Res Pap 2006:52976, 2006) up to a change in coordinates, and will in the sequel be used to prove the full structure of the Einstein–charge scalar field system in these modified harmonic coordinates.

Wigner rotation and Euler angle parametrization

Analogous to the famous Euler angle parametrization in three-dimensional Euclidean space, a reflection-free Lorentz transformation in (2 + 1)-dimensional Minkowski space can be decomposed into three simple parts. Applying this decomposition to the Wigner rotation problem, we are able to show that the related mathematics becomes much simpler and the physical meanings more comprehensible and enlightening.

Towards the Structure of a Cubic Interaction Vertex for Massless Integer Higher Spin Fields

The structure of a cubic Lagrangian vertex is clarified for irreducible fields of helicities $s_1, s_2, s_3$ in a $d$-dimensional Minkowski space. An explicit form of the operator $\mathcal{Z}_j$ entering the vertex in a non-multiplicative way (examined in arXiv:2105.12030[hep-th] for $j=1$) is obtained. The solution is found within the BRST approach with complete BRST operators, which contain all constraints corresponding to the conditions that extract the irreducible fields, including trace operators.

Celestial Klein spaces

We consider the analytic continuation of ($p+q$)-dimensional Minkowski space (with $p$ and $q$ even) to $(p,q)$ signature, and study the conformal boundary of the resulting ``Klein space.'' Unlike the familiar $(\ensuremath{-}+++\ensuremath{\cdots})$ signature, now the null infinity $\mathcal{I}$ has only one connected component. The spatial and timelike infinities (${i}^{0}$ and ${i}^{\ensuremath{'}}$) are quotients of generalizations of AdS spaces to nonstandard signature. Together, $\mathcal{I}$, ${i}^{0}$, and ${i}^{\ensuremath{'}}$ combine to produce the topological boundary ${S}^{p+q\ensuremath{-}1}$ as an ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ fibration over a null segment. The highest weight states (the $L$-primaries) and descendants of $SO(p,q)$ with integral weights give rise to natural scattering states. One can also define $H$-primaries which are highest weight with respect to a signature-mixing version of the Cartan-Weyl generators that leave a point on the celestial ${S}^{p\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{S}^{q\ensuremath{-}1}$ fixed. These correspond to massless particles that emerge at that point and are Mellin transforms of plane wave states.

Yang-Baxter deformations of the flat space string

We define integrability preserving Yang-Baxter deformations of symmetric space sigma models with non-semi-simple symmetry group, in particular the flat space string, using only the essential structures of a symmetric space sigma model. For homogeneous deformations, the Lax connection is of the same form as the semi-simple case, although the R operator no longer satisfies a freestanding operator equation. For inhomogeneous deformations, the form of the Lax connection needs to be relaxed, by modifying the underlying algebra. We illustrate the construction by discussing nonabelian deformations of three dimensional Minkowski space.

Constant Mean Curvature Surfaces Along a Spacelike Curve

We construct constant mean curvature surfaces along a given spacelike curve in 3 dimensional Minkowski space. We parametrically present these surfaces using the famous Frenet frame of the curve in question. We give the sufficient conditions for the so called marching scale functions, which are the coefficients of the Frenet frame fields. We show that it is possible to obtain such surfaces for any given spacelike curve. Finally, the validity of the presented method is supported with illustrative examples.

Global Invariants of Objects in two-Dimensional Minkowski Space

In this paper, ”the concept of T-object” is defined in two dimensional Minkowski space. By this concept, the complete systems of invariants of m-point, paths and vector fields are investigated as a single theory. Using this approach, we obtain the complete systems of invariants of T-object and the equivalence conditions of two T-objects are given in terms of their global invariants.

New optical radial direction for optical modeling with Minkowski extended frame

Numerous geometric applications have been improved to functionalize optical fibers and geometric fabrics. Optical uniform motions are designed by electromagnetic directions. In this article, we manufacture optical important conditions for extended fields. Also, we design new uniformly optical approach for unchanged direction optical motion, uniformly accelerated optical motion, and uniformly circular optical motion. Also, we study magnetic curves and their optical modeling and give optical relations between Fermi–Walker parallel transportation and uniformly optical modeling in four dimensional Minkowski space. Finally, the results also indicated that applied geometric explanation for radial direction of uniformly motion in Minkowski 4-space.

Evolutions of the Ruled Surfaces along a Spacelike Space Curve

In this paper, we work on the ruled surfaces obtained by a quasi normal and quasi binormal vectors along a spacelike space curve in three dimensional Minkowski space. Time evolution equations depending on quasi curvatures are obtained. Studying directional evolutions of both quasi normal and quasi binormal ruled surfaces by using their directrices, we investigate some geometric properties such as inextensibilty, developability, flatness and minimality of these ruled surfaces.

An asymptotic expansion of two-bubble wave maps in high equivariance classes

This is the first part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the two-sphere valued equivariant energy critical wave maps equation on 1+2 dimensional Minkowski space, with equivariance class k>3. It is known that every topologically trivial wave map with energy less than twice that of the unique k-equivariant harmonic map Q scatters in both time directions. We study maps with precisely the threshold energy, i.e., twice the energy of Q. In this paper, we give a refined construction of a wave map with threshold energy that converges to a superposition of two harmonic maps (bubbles), asymptotically decoupling in scale. We show that this two-bubble solution possesses H^2 regularity. We give a precise dynamical description of the modulation parameters as well as an expansion of the map into profiles. In the next paper in the series, we show that this solution is unique (up to the natural invariances of the equation) relying crucially on the detailed properties of the solution constructed here. Combined with our earlier work, we can now give an exact description of every threshold wave map.

A geometrical model of dishwasher spray arm for CornerWash

<abstract><p>Many dishwasher manufacturers are in search of innovative solutions for issues such as washing performance, energy efficiency, water consumption and noise. The most critical place to make these improvements is in the spray arm design. With a good spray arm design, it can be reduced water consumption, energy consumption, and noise. In this study, we focus on a new spray arm design (studied by Arçelik) that increases efficiency. Also, we have obtained a geometrical interpretation of the new spray arm motion. We investigate the path followed by the new spray arm in $ 3- $dimensional Minkowski space and its projection on a plane. Finally, we present its three-dimensional and two-dimensional images.</p></abstract>

The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

We consider the Enneper family of real maximal surfaces via Weierstrass data (1,ζm) for ζ∈C, m∈Z≥1. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space E2,1. Moreover, we propose that the family has degree (2m+1)2 (resp., class 2m(2m+1)) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Topological Bihyperbolic Modules

The aim of this article is introducing and researching hyperbolic modules, bihyperbolic modules, topological hyperbolic modules, and topological bihyperbolic modules. In this regard, we define balanced, convex and absorbing sets in hyperbolic and bihyperbolic modules. In particular, we investigate convex sets in hyperbolic numbers set (it is a hyperbolic module over itself) by considering the isomorphic relation of this set with 2−2−dimensional Minkowski space. Moreover, bihyperbolic numbers set is a bihyperbolic module over itself, too. So, we define convex sets in this module by considering hypersurfaces of 4−4−dimensional semi Euclidean space that are isomorphic to some subsets of bihyperbolic numbers set. We also study the interior and closure of some special sets and neighbourhoods of the unit element of the module in the introduced topological bihyperbolic modules. In the light of obtained results, new relationships are presented for idempotent representations in topological bihyperbolic modules

Extension of Fermat\u2019s last theorem in Minkowski natural spaces

Minkowski natural (N + 1)-dimensional spaces constitute the framework where the extension of Fermat’s last theorem is discussed. Based on empirical experience obtained via computational results, some hints about the extension of Fermat’s theorem from (2 + 1)-dimensional Minkowski spaces to (N + 1)-dimensional ones. Previous experience permits to conjecture that the theorem can be extended in (3 + 1) spaces, new results allow to do the same in (4 + 1) spaces, with an anomaly present here but difficult to find in higher dimensions. In (N + 1) dimensions with N > 4 there appears an increased difficulty to find Fermat vectors, there is discussed a possible source of such an obstacle, separately of the combinatorial explosion associated to the generation of natural vectors of high dimension.