Nonlocal Field Research Articles

The Nicolai-map approach to supersymmetry

In 1980 Hermann Nicolai proposed a characterization of supersymmetric theories that became known as the Nicolai map. This is a particular nonlocal and nonlinear field transformation, whose perturbative expansion is given by fermion-line trees with bosonic leaves. Quantum correlation functions can by evaluated using the inversely transformed fields in the free theory. After initial promise and excitement (fuelling the author’s PhD work!), the subject all but fell dormant for 35 years. Recently however, technical progress in the construction as well as a deeper insight into the nature of the map have been achieved, from quantum mechanics to super Yang–Mills in various dimensions. I will present the Nicolai map from this modern perspective and touch on some of the current developments.

A proposed renormalization scheme for non-local QFTs and application to the hierarchy problem

We propose a renormalization scheme for non-local Quantum Field Theories (QFTs) with infinite derivatives inspired by string theory. Our Non-locality Renormalization Scheme (NRS) is inspired by Dimensional Regularization (DR) in local QFTs and is shown to significantly improve the UV behavior of non-local QFTs. We illustrate the scheme using simple examples from the ϕ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^{3}$$\\end{document} and ϕ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^{4}$$\\end{document} theories, then we evaluate the viability of NRS-enhanced non-local QFTs to solve the hierarchy problem using a simplified toy model. We find that non-locality protects the mass of a light scalar from receiving large corrections from any UV sector to which it couples, as long as the non-locality scale Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is sufficiently smaller than the scale of the UV sector. We also find that NRS eliminates any large threshold corrections from the IR sector.

Differential geometry-based thermodynamics derivation of isotropic and anisotropic eikonal non-local gradient (ENLG) damage models using a micromorphic media framework

The Eikonal non-local damage (ENL) approach models damage as a space-deforming phenomenon that affects the interaction distance between material points. As damage increases, interactions between points decrease, ultimately resulting in no interaction. In the integral version of such an approach, non-local interaction distances between material points are computed by solving a stationary Hamilton–Jacobi equation with a damage-dependent Riemannian metric. In the implicit gradient version of ENL models, the Riemannian metric figures in the Helmholtz equation to be solved for computing the non-local field controlling damage evolution. However, one of the main criticisms of such formulation is the lack of thermodynamics basis in its derivation. This paper presents a thermodynamics derivation of the Eikonal implicit gradient formulation based on differential geometry concepts to overcome this issue. A free-energy potential is defined considering the non-local strain as a morphological descriptor belonging to the abstract differentiable manifold (where the Riemannian metric is defined). Following a micromorphic media framework, the balance equations of the model are obtained. It is shown that the stress tensor and thermodynamic force associated with damage are the sum of standard and additional non-local contributions. It is also shown that the resulting energy dissipation is always positive, thus verifying the Clausius–Duhem inequality. After presenting all development considering second-order anisotropic continuum damage, the isotropic formulation is obtained as a particular case. Finally, a two-dimensional numerical implementation of an isotropic implicit Eikonal non-local gradient damage model is illustrated. Test cases are simulated to show the relocalization features of the considered formulation and its natural capability of naturally representing damage-to-fracture transition for high damage levels.

General Fractional Noether Theorem and Non-Holonomic Action Principle

Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows us to take into account a wide class of nonlocalities in space and time compared to the usual fractional calculus. The use of non-holonomic variation equations allows us to consider field equations and equations of motion for a wide class of irreversible processes, dissipative and open systems, non-Lagrangian and non-Hamiltonian field theories and systems. In addition, the proposed GF action principle and the GF Noether theorem are generalized to equations containing general fractional integrals (GFI) in addition to general fractional derivatives (GFD). Examples of field equations with GFDs and GFIs are suggested. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for general fractional non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory.

A half de Sitter holography

A long-standing and intriguing question is: does the holographic principle apply to cosmologies like de Sitter spacetime? In this work, we consider a half dS spacetime wherein a timelike boundary encloses the bulk spacetime, presenting a version of de Sitter holography. By analyzing the holographic entanglement entropy in this space and comparing it with that in AdS/CFT, we argue that gravity on a half dSd+1 is dual to a highly non-local field theory residing on dSd boundary. This non-locality induces a breach in the subadditivity of holographic entanglement entropy. Remarkably, this observation can be linked to another argument that time slices in global de Sitter space overestimate the degrees of freedom by redundantly counting the same Hilbert space multiple times.

Nonlocal Static and Dynamical Vacuum Field Correlations and Casimir-Polder Interactions.

In this review, we investigate several aspects and features of spatial field correlations for the massless scalar field and the electromagnetic field, both in stationary and nonstationary conditions, and show how they manifest in two- and many-body static and dynamic dispersion interactions (van der Waals and Casimir-Polder). We initially analyze the spatial field correlations for noninteracting fields, stressing their nonlocal behavior, and their relation to two-body dispersion interactions. We then consider how field correlations are modified by the presence of a field source, such as an atom or in general a polarizable body, firstly in a stationary condition and then in a dynamical condition, starting from a nonstationary state. We first evaluate the spatial field correlation for the electric field in the stationary case, in the presence of a ground-state or excited-state atom, and then we consider its time evolution in the case of an initially nonstationary state. We discuss in detail their nonlocal features, in both stationary and nonstationary conditions. We then explicitly show how the nonlocality of field correlations can manifest itself in van der Waals and Casimir-Polder interactions between atoms, both in static and dynamic situations. We discuss how this can allow us to indirectly probe the existence and the properties of nonlocal vacuum field correlations of the electromagnetic field, a research subject of strong actual interest, also in consequence of recent measurements of spatial field correlations exploiting electro-optical sampling techniques. The subtle and intriguing relation between nonlocality and causality is also discussed.

Nonlocal Fractional Quantum Field Theory and Converging Perturbation Series

The main purpose of this paper is to derive a new perturbation theory (PT) that has converging series. Such series arise in the nonlocal scalar quantum field theory (QFT) with fractional power potential. We construct a PT for the generating functional (GF) of complete Green functions (including disconnected parts of functions) Zj as well as for the GF of connected Green functions Gj=lnZj in powers of coupling constant g. It has infrared (IR)-finite terms. We prove that the obtained series, which has the form of a grand canonical partition function (GCPF), is dominated by a convergent series—in other words, has majorant, which allows for expansion beyond the weak coupling g limit. Vacuum energy density in second order in g is calculated and researched for different types of Gaussian part S0[ϕ] of the action S[ϕ]. Further in the paper, using the polynomial expansion, the general calculable series for Gj is derived. We provide, compare and research simplifications in cases of second-degree polynomial and hard-sphere gas (HSG) approximations. The developed formalism allows us to research the physical properties of the considered system across the entire range of coupling constant g, in particular, the vacuum energy density.

Perspective on an Emerging Frontier of Nanoscience Opened up by Dressed Photon Studies

The core parts of developing dressed photon (DP) research that require advanced knowledge of highly mathematical quantum field theory and their potentially important impacts on the wide spectrum of long-term scientific activities in general, not necessarily restricted to those in the natural science sector, are succinctly explained in this article. Although a considerable number of remarkable technological achievements in the field of nanophotonics have been attained by utilizing DP phenomena, from the theoretical viewpoint, they remain enigmatic, as in the case of dark matter and energy in cosmology. Under such circumstances, an intriguing working hypothesis (WH) for DPs is proposed by the authors of this article through a combination of Ojima’s micro-macro duality theory and the Greenberg-Robinson theorem, claiming that the space-like momentum contribution is an inevitable element for quantum field interactions to occur. Note that, as the Schrödinger’s cat thought experiment clearly shows, the widespread common quantum mechanics knowledge is incapable of explaining how the invisible quantum world is connected to our familiar visible classical world. In the above-mentioned WH, the main reason why we cannot explain either DPs or dark entities in cosmology is shown to have roots in the fact that the prevailing theories have not revealed an important role of spacelike momentum in connecting the quantum and classical worlds. Our new WH further shows that the entire universe is connected by an instantaneous spacelike entropic spin network, as in the case of quantum spin entanglement explained in mainstream physics. Since such a network may have a close relation with the nonlocal consciousness field, which seems to be the final frontier of physics, our perspective on such a possibility is briefly given in the final section of this paper.

Fine-Grained and Multiple Classification for Alzheimer's Disease With Wavelet Convolution Unit Network.

In this article, we propose a novel wavelet convolution unit for the image-oriented neural network to integrate wavelet analysis with a vanilla convolution operator to extract deep abstract features more efficiently. On one hand, in order to acquire non-local receptive fields and avoid information loss, we define a new convolution operation by composing a traditional convolution function and approximate and detailed representations after single-scale wavelet decomposition of source images. On the other hand, multi-scale wavelet decomposition is introduced to obtain more comprehensive multi-scale feature information. Then, we fuse all these cross-scale features to improve the problem of inaccurate localization of singular points. Given the novel wavelet convolution unit, we further design a network based on it for fine-grained Alzheimer's disease classifications (i.e., Alzheimer's disease, Normal controls, early mild cognitive impairment, late mild cognitive impairment). Up to now, only a few methods have studied one or several fine-grained classifications, and even fewer methods can achieve both fine-grained and multi-class classifications. We adopt the novel network and diffuse tensor images to achieve fine-grained classifications, which achieved state-of-the-art accuracy for all eight kinds of fine-grained classifications, up to 97.30%, 95.78%, 95.00%, 94.00%, 97.89%, 95.71%, 95.07%, 93.79%. In order to build a reference standard for Alzheimer's disease classifications, we actually implemented all twelve coarse-grained and fine-grained classifications. The results show that the proposed method achieves solidly high accuracy for them. Its classification ability greatly exceeds any kind of existing Alzheimer's disease classification method.

A modified damage and fracture phase field model considering heterogeneity for rock‐like materials

AbstractDamage and fracture are the most extensive failure modes of rock materials, which may easily induce disaster and instability of engineering structures. This study developed a nonlocal damage fracture phase field model for rocks considering the heterogeneity of rocks. The modified phase field model introduced the heterogeneity of fracture parameters and modified the governing equations. Meanwhile, the free energy was constructed by the elastic strain energy sphere‐bias decomposition and the plastic strain energy. As for the numerical implementation, the three layers finite elements method structure was used in the frame of the finite element method. The ability of the modified phase field model has been illustrated by reproducing the experiment results of rock samples with pre‐existing cracks under compression.

Existence for a nonlocal Penrose-Fife type phase field system with inertial term

This article presents a nonlocal Penrose-Fife type phase field system with inertial term. We do not know whether we can prove the existence of solutions to the problem as in Colli-Grasselli-Ito [3] or not. In this article we introduce a time discretization scheme, then pass to the limit as the time step h approaches 0, and obtain an error estimate for the difference between the continuous solution and the discrete solution.

On a class of conformal mathcal{E} -models and their chiral Poisson algebras

In this paper, we study conformal points among the class of mathcal{E} -models. The latter are σ-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate mathcal{E} -models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a σ-model. The case of degenerate mathcal{E} -models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical mathcal{W} -algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal mathcal{E} -models using the standard operator formalism of two-dimensional CFT.

Existence and multiplicity of positive solutions of certain nonlocal scalar field equations

AbstractWe study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: where , , , and is a nonnegative functional, that is, whenever u is a nonnegative function in . We establish Palais–Smale decomposition of the functional associated with the above equation. Using the decomposition, we establish existence of three positive solutions to (), under the condition that with as and is small enough (but ). Further, we prove that () admits at least two positive solutions when , as and is small enough (but ). Finally, we prove the existence of a positive solution when under certain asymptotic behavior on the function a.

Asymptotically nonlocal gravity

Asymptotically nonlocal field theories interpolate between Lee-Wick theories with multiple propagator poles, and ghost-free nonlocal theories. Previous work on asymp- totically nonlocal scalar, Abelian, and non-Abelian gauge theories has demonstrated the existence of an emergent regulator scale that is hierarchically smaller than the lightest Lee-Wick partner, in a limit where the Lee-Wick spectrum becomes dense and decoupled. We generalize this construction to linearized gravity, and demonstrate the emergent regula- tor scale in three examples: by studying the resolution of the singularity (i) at the origin in the classical solution for the metric of a point particle, and (ii) in the nonrelativistic gravitational potential computed via a one-graviton exchange amplitude; (iii) we also show how this derived scale regulates the one-loop graviton contribution to the self energy of a real scalar field. We comment briefly on the generalization of our approach to the full, nonlinear theory of gravity.

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

Confining complex ghost degrees of freedom

We show a theorem proving that a non-local bosonic field upon a covariant interaction with a confining gauge field undergoes the confinement of its degrees of freedom present in the free theory changing completely the physical mass spectrum following Kugo-Ojima criterion. This is applicable to an infinite number of excitations of the bosonic field including ghosts whereas we pay special attention to the modes with the complex conjugate masses, states appearing in the string field theory motivated infinite-derivative models. The same recipe will obviously work for the Lee-Wick models.

On the curvature dependence of gradient damage models: Control and opportunities

Curvature dependence of gradient damage models is an often undesired side effect. It originates from a change of the orientation of the non-local gradient term. The resulting curvature dependence then implicitly affects the loading behaviour, the predicted fracture energy and also the damage distribution. Two extended approaches are presented here for curvature control and applied to a micromorphic gradient damage model and to a phase-field model for fracture. Both allow for a better control of the crack path and consider the Hessian of the non-local damage field by means of an additional micromorphic field, since curvature is connected to second derivatives. The first approach is based on an additional curvature-dependent potential within the Helmholtz energy and automatically fulfils the second law of thermodynamics. The second approach modifies the micro forces of the balance law that is responsible for the curvature dependence directly. Unlike the first approach, it allows to eliminate the curvature dependence completely. However, thermodynamic consistency has to be checked explicitly.

Massive to massless by applying a nonlocal field redefinition

The invariance of physical observables like particle's mass under a local field redefinition is a well-known and important property of quantum field theory. In this paper, on the other hand, we investigate nonlocal field redefinitions eliminating a physical observable from a theory. Indeed, by giving explicit examples, we present invertible nonlocal field redefinitions reducing a massive theory to the massless one, and vice versa. The studied examples are: classical harmonic oscillator, the scalar field theory, and the Dirac field theory.

Extra Dimensions and Fuzzy Branes in String-inspired Nonlocal Field Theory

Extra Dimensions and Fuzzy Branes in String-inspired Nonlocal Field Theory

Contour prescriptions in string-inspired nonlocal field theories

In quantum field theory, a consistent prescription to define and deform integration contours in the complex energy plane is needed to evaluate loop integrals and compute scattering amplitudes. In some nonlocal field theories, including string field theory, interaction vertices contain transcendental functions of momenta that can diverge along certain complex directions, thus making it impossible to use standard techniques, such as Wick rotation, to perform loop integrals. The aim of this paper is to investigate the viability of several contour prescriptions in the presence of nonlocal vertices. We consider three ``different'' prescriptions, and establish their (in)equivalence in local and nonlocal theories. In particular, we prove that all these prescriptions turn out to be equivalent in standard local theories, while this is not the case for nonlocal theories where amplitudes must be defined first in Euclidean space, and then analytically continued to Minkowski. We work at one-loop level and focus on the bubble diagram. In addition to proving general results for a large class of nonlocal theories, we show explicit calculations in a string inspired nonlocal scalar model.