Two-dimensional Scalar Field Research Articles

Two-dimensional vector field topology and scalar fields in viscous flows: Reconstruction methods

Two-dimensional vector field topology and scalar fields in viscous flows: Reconstruction methods

Generalized magnetostatic target field method for shielded magnetic field coils in a separable coordinate system.

A theoretical method is described to analytically calculate a pair of surface current densities, which produce a desired static magnetic field in one region of the space and zero magnetic field in another. The analysis is based on the known relationship between a surface current density and a stream function, the equivalence of stream functions and surface magnetic dipole density, and the scalar potential representation of the associated magnetic field in free space. From these relations, we formulate the magnetostatic problem, which is often treated as a vector field problem, as a scalar field problem in which a two-dimensional scalar field (stream function) is related to a three-dimensional one (magnetic scalar potential) via the differentiation of the electrostatic Green's function 1/|r-rs|. It is shown that, in a coordinate system in which a separated form of the Green's function exists (separable coordinate system), there exists a simple relationship between a harmonic component of a stream function and a harmonic component of the magnetic scalar potential. The method is applied to calculate idealized surface current patterns for actively shielded, linear gradient field coils in the Cartesian, cylindrical, and spherical coordinates.

Infinite conformal symmetry and emergent chiral fields of topologically nontrivial configurations: From Yang-Mills-Higgs theory to the Skyrme model

In this manuscript, we discuss a remarkable phenomenon concerning nonlinear and nonintegrable field theories in ($3+1$) dimensions, living at finite density and possessing nontrivial topological charges and non-Abelian internal symmetries (both local and global). With suitable types of Ans\"atze, one can construct infinite-dimensional families of analytic solutions with nonvanishing topological charges (representing the baryonic number) labeled by both two integer numbers and by free scalar fields in ($1+1$) dimensions. These exact configurations represent ($3+1$)-dimensional topological solitons hosting ($1+1$)-dimensional chiral modes localized at the energy density peaks. First, we analyze the Yang-Mills-Higgs model, in which the fields depend on all the space-time coordinates (to keep alive the topological Chern-Simons charge), but in such a way to reduce the equations system to the field equations of two-dimensional free massless chiral scalar fields. Then, we move to the nonlinear sigma model, showing that a suitable Ansatz reduces the field equations to the one of a two-dimensional free massless scalar field. Then, we discuss the Skyrme model concluding that the inclusion of the Skyrme term gives rise to a chiral two-dimensional free massless scalar field (instead of a free massless field in two dimensions as in the nonlinear sigma model) describing analytically spatially modulated hadronic layers and tubes. The comparison of the present approach both with the instanton-dyon liquid approach and with lattice QCD is shortly outlined.

Marginal TT¯ -like deformation and modified Maxwell theories in two dimensions

Recently, the ModMax theory has been proposed as a unique conformal nonlinear extension of electrodynamics. We have shown in [1] that this modification can be reproduced a marginal $T\bar{T}$-like deformation from pure Maxwell theory. Further, this deformation is solved by using a perturbative approach. In this letter, we will investigate another ModMax-like deformation for a two-dimensional (2D) scalar field theory. In this regard, we first find a marginal $T\bar{T}$-like deformation in two dimensions and then reproduce the MM-like Lagrangian from a multiple 2D scalar field theory.

Non-linear composition and infinite conformal symmetry of topologically non-trivial solutions in (3+1)-dimensional Yang\u2013Mills theory

An infinite-dimensional family of analytic solutions in pure SU(2) Yang–Mills theory at finite density in (3+1) dimensions is constructed. It is labelled by two integeres (p and q) as well as by a two-dimensional free massless scalar field. The gauge field depends on all the 4 coordinates (to keep alive the topological charge) but in such a way to reduce the (3+1)-dimensional Yang–Mills field equations to the field equation of a 2D free massless scalar field. For each p and q, both the on-shell action and the energy-density reduce to the action and Hamiltonian of the corresponding 2D CFT. The topological charge density associated to the non-Abelian Chern–Simons current is non-zero. It is possible to define a non-linear composition within this family as if these configurations were “Lego blocks”. The non-linear effects of Yang–Mills theory manifest themselves since the topological charge density of the composition of two solutions is not the sum of the charge densities of the components. This leads to an upper bound on the amplitudes in order for the topological charge density to be well-defined. This suggests that if the temperature and/or the energy is/are high enough, the topological density of these configurations is not well-defined anymore. Semiclassically, one can show that (depending on whether the topological charge is even or odd) some of the operators appearing in the 2D CFT should be quantized as Fermions (despite the Bosonic nature of the classical field).

Efficient two-dimensional scalar fields reconstruction of laminar flames from infrared hyperspectral measurements with a machine learning approach

The latest hyperspectral measurements of combustion flames by Rhoby et al. (2014) provided extensive spatially and spectrally resolved information of flame radiation, which has been explored to retrieve two-dimensional, multi-scalar values of these flames with the conventional gradient-based optimization method. The drawback of that method is that the inverse radiation problem was solved through iterations with computationally intensive radiative heat transfer calculations and high-resolution wide-spectrum modeling, making the retrieving process very time-consuming. In the present study, we propose a machine learning based efficient inverse radiation model to retrieve two-dimensional temperature, CO2, H2O, and CO mole fractions of laminar flames from hyperspectral measurements. The model is trained with synthetic numerical data and is tested against previously made OH-laser absorption measurements and chemical equilibrium calculations for ethylene laminar flames with different equivalence ratios. The training data generation process, machine learning model architecture, model training, and validations are discussed in detail. Results have shown that the proposed machine learning based inverse radiation model is both accurate and efficient.

A field theory study of entanglement wedge cross section: odd entropy

We study odd entanglement entropy (odd entropy in short), a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories. Our study is restricted to Gaussian states of scale-invariant theories as well as their finite temperature generalizations, for which we show that odd entropy is a well-defined measure for mixed states. Motivated from holographic results, the difference between odd and von Neumann entropy is also studied. In particular, we show that large amounts of quantum correlations ensure the odd entropy to be larger than von Neumann entropy, which is qualitatively consistent with the holographic CFT. In general cases, we also find that this difference is not even a monotonic function with respect to size of (and distance between) subsystems.

Classical instanton solutions in quantum field theory

Instantons are non-trivial solutions of classical Euclidean equations of motion with a finite action. They provide stationary phase points in the path integral for tunnel amplitude between two topologically distinct vacua. It make them useful in many applications of quantum theory, especially for describing the wave function of systems with a degenerate vacua in the framework of the path integrals formalism. Our goal is to introduce the current situation about research on instantons and prepare for experiments. In this paper we give a review of instanton effects in quantum theory. We find in stanton solutions in some quantum mechanical problems, namely, in the problems of the one-dimensional motion of a particle in two-well and periodic potentials. We describe known instantons in quantum field theory that arise, in particular, in the two-dimensional Abelian Higgs model and in SU(2) Yang – Mills gauge fields. We find instanton solutions of two-dimensional scalar field models with sine-Gordon and double-well potentials in a limited spatial volume. We show that accounting of instantons significantly changes the form of the Yukawa potential for the sine-Gordon model in two dimensions.

Configurational entropic study of the enhanced longevity in resonant oscillons

We use an information-theoretic measure of shape complexity known as configurational entropy (CE) to investigate numerically the remarkably long lifetimes of spherically-symmetric “resonant oscillons” in three-dimensional and of azimuthally-symmetric oscillons in two-dimensional relativistic scalar field theories, which have been conjectured to be infinite. In 3d, we obtain a power law relating a stability measure derived from CE to the oscillons' lifetimes that, upon extrapolation to large times, offers support to this conjecture. In 2d, we obtain a three-way relation between the oscillons' energies, a CE-derived measure of their stability, and their radiation rates to support the conjecture that they asymptotically tend toward a classically-stable attractor solution.

Self-gravitating kinks in two-dimensional pseudo-Riemannian universes

Two-dimensional scalar field theories with spontaneous symmetry breaking subject to the action of Jackiw-Teitelboim gravity are studied. Solutions for the $\phi^4$ and sine-Gordon self-gravitating kinks are presented, both for general gravitational coupling and in the perturbative regime. The analysis is extended to deal with a hierarchy of kinks related to transparent P\"{o}schl-Teller potentials

Regressive and generative neural networks for scalar field theory

We explore the perspectives of machine learning techniques in the context of quantum field theories. In particular, we discuss two-dimensional complex scalar field theory at nonzero temperature and chemical potential -- a theory with a nontrivial phase diagram. A neural network is successfully trained to recognize the different phases of this system and to predict the value of various observables, based on the field configurations. We analyze a broad range of chemical potentials and find that the network is robust and able to recognize patterns far away from the point where it was trained. Aside from the regressive analysis, which belongs to supervised learning, an unsupervised generative network is proposed to produce new quantum field configurations that follow a specific distribution. An implicit local constraint fulfilled by the physical configurations was found to be automatically captured by our generative model. We elaborate on potential uses of such a generative approach for sampling outside the training region.

The equivalence principle at work in radiation from unaccelerated atoms and mirrors

The equivalence principle is a perennial subject of controversy, especially in connection with radiation by a uniformly accelerated classical charge, or a freely falling charge observed by a supported detector. Recently, related issues have been raised in connection with the Unruh radiation associated with accelerated detectors (including two-level atoms and resonant cavities). A third type of system, very easy to analyze because of conformal invariance, is a two-dimensional scalar field interacting with perfectly reflecting boundaries (mirrors). After reviewing the issues for atoms and cavities, we investigate a stationary mirror from the point of view of an accelerated detector in ‘Rindler space’. In keeping with the conclusions of earlier authors about the electromagnetic problem, we find that a radiative effect is indeed observed; from an inertial point of view, the process arises from a collision of the negative vacuum energy of Rindler space with the mirror. There is a qualitative symmetry under interchange of accelerated and inertial subsystems (a vindication of the equivalence principle), but it hinges on the accelerated detector’s being initially in its own ‘Rindler vacuum’. This observation is consistent with the recent work on the Unruh problem.

Kinks in higher derivative scalar field theory

We study static kink configurations in a type of two-dimensional higher derivative scalar field theory whose Lagrangian contains second-order derivative terms of the field. The linear fluctuation around arbitrary static kink solutions is analyzed. We find that, the linear spectrum can be described by a supersymmetric quantum mechanics problem, and the criteria for stable static solutions can be given analytically. We also construct a superpotential formalism for finding analytical static kink solutions. Using this formalism we first reproduce some existed solutions and then offer a new solution. The properties of our solution is studied and compared with those preexisted. We also show the possibility in constructing twinlike model in the higher derivative theory, and give the consistency conditions for twinlike models corresponding to the canonical scalar field theory.

Nodal-line dynamics via exact polynomial solutions for coherent waves traversing aberrated imaging systems.

We obtain exact polynomial solutions for two-dimensional coherent complex scalar fields propagating through arbitrary aberrated shift-invariant linear imaging systems. These solutions are used to model nodal-line dynamics of coherent fields output by such systems.

Cooling stochastic quantization with colored noise

Smoothing of field configurations is highly important for precision calculations of physical quantities on the lattice. We present a cooling method based on Stochastic Quantization with a built-in UV momentum cutoff. The latter is implemented via a UV-regularized, hence colored, noise term. Our method is tested in a two-dimensional scalar field theory. We show, that UV modes can be removed systematically without altering the physics content of the theory. The approach has an interpretation in terms of the non-perturbative (Wilsonian) renormalization group that facilitates the physics interpretation of the cutoff procedure. It also can be used to define the maximal colored cooling applicable without changing the theory.

Geomorphometric analysis of cave ceiling channels mapped with 3-D terrestrial laser scanning

Abstract. The change of hydrological conditions during the evolution of caves in carbonate rocks often results in a complex subterranean geomorphology, which comprises specific landforms such as ceiling channels, anastomosing half tubes, or speleothems organized vertically in different levels. Studying such complex environments traditionally requires tedious mapping; however, this is being replaced with terrestrial laser scanning technology. Laser scanning overcomes the problem of reaching high ceilings, providing new options to map underground landscapes with unprecedented level of detail and accuracy. The acquired point cloud can be handled conveniently with dedicated software, but applying traditional geomorphometry to analyse the cave surface is limited. This is because geomorphometry has been focused on parameterization and analysis of surficial terrain. The theoretical and methodological concept has been based on two-dimensional (2-D) scalar fields, which are sufficient for most cases of the surficial terrain. The terrain surface is modelled with a bivariate function of altitude (elevation) and represented by a raster digital elevation model. However, the cave is a 3-D entity; therefore, a different approach is required for geomorphometric analysis. In this paper, we demonstrate the benefits of high-resolution cave mapping and 3-D modelling to better understand the palaeohydrography of the Domica cave in Slovakia. This methodological approach adopted traditional geomorphometric methods in a unique manner and also new methods used in 3-D computer graphics, which can be applied to study other 3-D geomorphological forms.

Highly nonlinear wave solutions in a dual to the chiral model

We consider a two-dimensional scalar field theory with a nilpotent current algebra, which is dual to the Principal Chiral Model. The quantum theory is renormalizable and not asymptotically free: the theory is strongly coupled at short distances (encountering a Landau pole). We suggest it can serve as a toy model for $\lambda\phi^{4}$ theory in four dimensions, just as the principal chiral model is a useful toy model for Yang-Mills theory. We find some classical wave solutions that survive the strong coupling limit and quantize them by the collective variable method. They describe excitations with an unusual dispersion relation $\omega\propto|k|^{\frac{2}{3}}$ . Perhaps they are the "preons" at strong coupling, whose bound states form massless particles over long distances.

Obtaining a closed-form representation for the dual bosonic thermal Green function by using methods of integration on the complex plane

We derive an exact closed-form representation for the Euclidean thermal Green function of the two-dimensional (2D) free massless scalar field in coordinate space. This can be interpreted as the real part of a complex analytic function of a variable that conformally maps the infinite strip -∞ < x < ∞ (0 < τ < β) of the z = x + iτ (τ: imaginary time) plane into the upper-half-plane. Use of the Cauchy-Riemann conditions, then allows us to identify the dual thermal Green function as the imaginary part of that function.

Thermal Soliton Correlation Functions in Theories with a <i>Z</i>(N) Symmetry

We show that the quantum solitons occurring in theories describing a complex scalar field in (1 + 1)-dimensions with a Z(N) symmetry may be identified with sine-Gordon quantum solitons in the phase of this field. Then using both the Euclidean thermal Green function of the two-dimensional free massless scalar field in coordinate space and its dual, we obtain an explicit series expression for the corresponding solitonic correlation function at finite temperature.

NOTE ON TWO-DIMENSIONAL GAUGED LIFSHITZ MODELS

We fermionize the two-dimensional free Lifshitz scalar field in order to identify what the gauge-covariant couplings are, and then they are bosonized back to get the gauged Lifshitz scalar field theories. We show that they give the same physical modes with those of the corresponding Lorentz invariant gauged scalar theories, although the dispersion relations are different.