Classical Field Theory Research Articles

Lie groupoids in classical field theory II: Gauge theories, minimal coupling and Utiyama's theorem

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations.In this second part, we shall adapt the formalism developed in the first paper to the context of gauge theories and deal with minimal coupling and Utiyama's theorem.

Constructing abelian extensions with prescribed norms

Given a number field K K , a finite abelian group G G and finitely many elements α 1 , … , α t ∈ K \alpha _1,\ldots ,\alpha _t\in K , we construct abelian extensions L / K L/K with Galois group G G that realise all of the elements α 1 , … , α t \alpha _1,\ldots ,\alpha _t as norms of elements in L L . In particular, this shows existence of such extensions for any given parameters. Our approach relies on class field theory and a recent formulation of Tate’s characterisation of the Hasse norm principle, a local-global principle for norms. The constructions are sufficiently explicit to be implemented on a computer, and we illustrate them with concrete examples.

Gravity Field (Low Frequency) of Planet Earth Described by Theory of New Axioms and Laws

The theory of new axioms and laws is published by the same author. It describes nonparametric and nonlinear processes and contains 2 new axioms and 8 new laws. Unlike Classical field theory, it describes longitudinal or transverse non-uniform motions which are accelerating or decelerating. According to the Axiom 1 every unevenly rotation of one vector forms open vortex which can be transverse or longitudinal and accelerating or decelerating. From the planetary model of Rutherford it is known that there is analogy between the electrons and an planets including the planet Earth. By analogy - the electrons and the internal planets are similar Gravitational bodies. According the Law 1 the model of the electron or the Earth represents a decelerating transverse vortex rolled into a plane (2D) and generating in its center accelerating longitudinal Gravity Funnel in (3D), perpendicular to the same plane. Inside- the primary accelerating longitudinal Gravity vectors are with decreasing dimensions and forms the decelerating Magnetic Field as a Back wave passing through the center of Earth. Outside- because of resistance of environment in periphery, is formed Back wave or decelerating Gravity vortex that passes outside the body of Earth. According Axiom 2 the reason for the creation of the electron is the generation by the corresponding proton. By analogythe reason for the creation of internal planets, including the planet Earth is in the generation of a specific vortex inside the corresponding for Earth resonator in the volume of the Sun. It is Low Frequency vortex which is formed in the third cylindrical resonator that corresponds to Earth. According Law 2 the proton or the resonator inside Sun is generated by a decelerating longitudinal vortex with direction from outside to inside which creates an accelerating transverse vortex from inside to outside in the perpendicular plane. As it is strongly accelerated this vortex from inside to outside it shoots itself into space in direction to the Earth. Due to the friction it decelerates and according to the previous Law 1 it winds into as a decelerating transverse vortex generating the body of Earth. According Law 5 decelerating vortex emits decelerating cross vortices from itself to outside. This decelerating vortices in periphery of the Earth emit energy and warm the center of the Earth. That is why the periphery of Earth is cool ,but the center of Earth is hot . According Law 6 the accelerating cross vortex sucks accelerating cross vortices to itself . This accelerating cross vortex at center of the Sun sucks energy and warm from center and emits them to periphery of Sun. That is why the center of Sun is cool but the periphery is extremely accelerated and hot. The described generating mechanism only applies to the inner planets. For the outer planets, the generation algorithm is orthogonal and will be described further.

Optimal Entanglement Witnesses: A Scalable Data-Driven Approach.

Multipartite entanglement is a key resource allowing quantum devices to outperform their classical counterparts, and entanglement certification is fundamental to assess any quantum advantage. The only scalable certification scheme relies on entanglement witnessing, typically effective only for special entangled states. Here, we focus on finite sets of measurements on quantum states (hereafter called quantum data), and we propose an approach which, given a particular spatial partitioning of the system of interest, can effectively ascertain whether or not the dataset is compatible with a separable state. When compatibility is disproven, the approach produces the optimal entanglement witness for the quantum data at hand. Our approach is based on mapping separable states onto equilibrium classical field theories on a lattice and on mapping the compatibility problem onto an inverse statistical problem, whose solution is reached in polynomial time whenever the classical field theory does not describe a glassy system. Our results pave the way for systematic entanglement certification in quantum devices, optimized with respect to the accessible observables.

Density perturbations in axionlike particles: Classical versus quantum field treatment

Axions and axion-like particles are bosonic quantum fields. They are often assumed to follow classical field equations due to their high degeneracy in the phase space. In this work, we explore the disparity between classical and quantum field treatments in the context of density and velocity fields of axions. Once the initial density and velocity field are specified, the evolution of the axion fluid is unique in the classical field treatment. However, in the quantum field treatment, there are many quantum states consistent with the given initial density and velocity field. We show that evolutions of the density perturbations for these quantum states are not necessarily identical and, in general, differ from the unique classical evolution. To illustrate the underlying physics, we consider a system of large number of bosons in a one-dimensional box, moving under the gravitational potential of a heavy static point-mass. We ignore the self-interactions between the bosons here. Starting with homogeneous number density and zero velocity field, we determine the density perturbations in the linear regime in both quantum and classical field theories. We find that classical and quantum evolutions are identical in the linear regime if only one single-particle state is occupied by all the bosons and the self-interaction is absent. If more than one single-particle states are occupied, the density perturbations in quantum evolutions differ from the classical prediction after a certain time which depends upon the parameters of the system.

Is negative kinetic energy metastable?

We explore the possibility that theories with negative kinetic energy (ghosts) can be meta-stable up to cosmologically long times. In classical mechanics, ghosts undergo spontaneous lockdown rather than run-away if weakly-coupled and non-resonant. Physical examples of this phenomenon are shown. In quantum mechanics this leads to meta-stability similar to vacuum decay. In classical field theory, lockdown is broken by resonances and ghosts behave statistically, drifting towards infinite entropy as no thermal equilibrium exists. We analytically and numerically compute the run-away rate finding that it is cosmologically slow in 4-derivative gravity, where ghosts have gravitational interactions only. In quantum field theory the ghost run-away rate is naively infinite in perturbation theory, analogously to what found in early attempts to compute vacuum tunnelling; we do not know the true rate.

Instabilities in liquid crystal elastomers

Stability is an important and fruitful avenue of research for liquid crystal elastomers. At constant temperature, upon stretching, the homogeneous state of a nematic body becomes unstable, and alternating shear stripes develop at very low stress. Moreover, these materials can experience classical mechanical effects, such as necking, void nucleation and cavitation, and inflation instability, which are inherited from their polymeric network. We investigate the following two problems: First, how do instabilities in nematic bodies change from those found in purely elastic solids? Second, how are these phenomena modified if the material constants fluctuate? To answer these questions, we present a systematic study of instabilities occurring in nematic liquid crystal elastomers, and examine the contribution of the nematic component and of fluctuating model parameters that follow probability laws. This combined analysis may lead to more realistic estimations of subsequent mechanical damage in nematic solid materials. Because of their complex material responses in the presence of external stimuli, liquid crystal elastomers have many potential applications in science, manufacturing, and medical research. The modeling of these materials requires a multiphysics approach, linking traditional continuum mechanics with liquid crystal theory, and has led to the discovery of intriguing mechanical effects. An important problem for both applications and our fundamental understanding of nematic elastomers is their instability under large strains, as this can be harnessed for actuation, sensing, or patterning. The goal is then to identify parameter values at which a bifurcation emerges, and how these values change with external stimuli, such as temperature or loads. However, constitutive parameters of real manufactured materials have an inherent variation that should also be taken into account, thus the need to quantify uncertainties in physical responses, which can be done by combining the classical field theories with stochastic methods that enable the propagation of uncertainties from input data to output quantities of interest. The present study demonstrates how to characterize instabilities found in nematic liquid crystal elastomers with probabilistic material parameters at the macroscopic scale, and paves the way for a systematic theoretical and experimental study of these fascinating materials.

Unified Extended Irreversible Thermodynamics and the Stability of Relativistic Theories for Dissipation

In a relativistic context, the main purpose of Extended Irreversible Thermodynamics (EIT) is to generalize the principles of non-equilibrium thermodynamics to the domain of fluid dynamics. In particular, the theory aims at modeling any diffusion-type process (like heat as diffusion of energy, viscosity as diffusion of momentum, charge-conductivity as diffusion of particles) directly from thermodynamic laws. Although in Newtonian physics this task can be achieved with a first-order approach to dissipation (i.e. Navier–Stokes–Fourier like equations), in a relativistic framework the relativity of simultaneity poses serious challenges to the first-order methodology, originating instabilities which are, instead, naturally eliminated within EIT. The first part of this work is dedicated to reviewing the most recent progress made in understanding the mathematical origin of this instability problem. In the second part, we present the formalism that arises by promoting non-equilibrium thermodynamics to a classical effective field theory. We call this approach Unified Extended Irreversible Thermodynamics (UEIT), because it contains, as particular cases, EIT itself, in particular the Israel-Stewart theory and the divergence-type theories, plus Carter’s approach and most branches of non-equilibrium thermodynamics, such as relativistic chemistry and radiation hydrodynamics. We use this formalism to explain why all these theories are stable by construction (provided that the microscopic input is correct), showing that their (Lyapunov) stability is a direct consequence of the second law of thermodynamics.

A review on geometric formulations for classical field theory: the Bonzom–Livine model for gravity

Motivated by the study of physical models associated with general relativity, we review some finite-dimensional, geometric and covariant formulations that allow us to characterize in a simple manner the symmetries for classical field theory by implementing an appropriate fibre-bundle structure, either at the Lagrangian, the Hamiltonian multisymplectic or the polysymplectic levels. In particular, we are able to formulate Noether’s theorems by means of the covariant momentum maps and to systematically introduce a covariant Poisson–Hamiltonian framework. Also, by focusing on the space plus time decomposition for a generic classical field theory and its relation to these geometric formulations, we are able to successfully recover the gauge content and the true local degrees of freedom for the theory. In order to illustrate the relevance of these geometric frameworks, we center our attention to the analysis of a model for three-dimensional theory of general relativity that involves an arbitrary Immirzi-like parameter. At the Lagrangian level, we reproduce the field equations of the system which for this model turn out to be equivalent to the vanishing torsion condition and the Einstein equations. We also concentrate on the analysis of the gauge symmetries of the system in order to obtain the Lagrangian covariant momentum map associated with the theory and, consequently, its corresponding Noether currents. Next, within the Hamiltonian multisymplectic approach, we aim our attention to describing how the gauge symmetries of the model yield covariant canonical transformations on the covariant multimomenta phase-space, thus giving rise to the existence of a covariant momentum map. Besides, we analyze the physical system under consideration within the De Donder–Weyl canonical theory implemented at the polysymplectic level, thus establishing a relation from the covariant momentum map to the conserved currents of the theory within this covariant Hamiltonian approach. Finally, after performing the space plus time decomposition of the space-time manifold, and taking as a starting point the Hamiltonian multisymplectic formulation, we are able to recover both the extended Hamiltonian and the gauge structure content that characterize the gravity model of our interest within the instantaneous Dirac–Hamiltonian formulation.

People & Appointments

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations.In this second part, we shall adapt the formalism developed in the first paper to the context of gauge theories and deal with minimal coupling and Utiyama's theorem.

Gauged double field theory as an L\u221e algebra

L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the {L}_3^{mathrm{gauge}} structure of (Bosonic) Enhanced Double Field Theory.

Multiform description of the AKNS hierarchy and classical r-matrix

In recent years, new properties of space-time duality in the Hamiltonian formalism of certain integrable classical field theories have been discovered and have led to their reformulation using ideas from covariant Hamiltonian field theory: in this sense, the covariant nature of their classical r-matrix structure was unravelled. Here, we solve the open question of extending these results to a whole hierarchy. We choose the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. To do so, we introduce for the first time a Lagrangian multiform for the entire AKNS hierarchy. We use it to construct explicitly the necessary objects introduced previously by us: a symplectic multiform, a multi-time Poisson bracket and a Hamiltonian multiform. Equipped with these, we prove the following results: (i) the Lax form containing the whole sequence of Lax matrices of the hierarchy possesses the rational classical r-matrix structure; (ii) the zero curvature equations of the AKNS hierarchy are multiform Hamilton equations associated to our Hamiltonian multiform and multi-time Poisson bracket; (iii) the Hamiltonian multiform provides a way to characterise the infinite set of conservation laws of the hierarchy reminiscent of the familiar criterion {I, H} = 0 for a first integral I.

Entanglement transfer from quantum matter to classical geometry in an emergent holographic dual description of a scalar field theory

Applying recursive renormalization group transformations to a scalar field theory, we obtain an effective quantum gravity theory with an emergent extra dimension, described by a dual holographic Einstein-Klein-Gordon type action. Here, the dynamics of both the dual order-parameter field and the metric tensor field originate from density-density and energy-momentum tensor-tensor effective interactions, respectively, in the recursive renormalization group transformation, performed approximately in the Gaussian level. This linear approximation in the recursive renormalization group transformation for the gravity sector gives rise to a linearized quantum Einstein-scalar theory along the z-directional emergent space. In the large N limit, where N is the flavor number of the original scalar fields, quantum fluctuations of both dynamical metric and dual scalar fields are suppressed, leading to a classical field theory of the Einstein-scalar type in (D+1)-spacetime dimensions. We show that this emergent background gravity describes the renormalization group flows of coupling functions in the UV quantum field theory through the extra dimension. More precisely, the IR boundary conditions of the gravity equations correspond to the renormalization group β-functions of the quantum field theory, where the infinitesimal distance in the extra-dimensional space is identified with an energy scale for the renormalization group transformation. Finally, we also show that this dual holographic formulation describes quantum entanglement in a geometrical way, encoding the transfer of quantum entanglement from quantum matter to classical gravity in the large N limit. We claim that this entanglement transfer serves as a microscopic foundation for the emergent holographic duality description.

Application of cross vortices, according the new Axioms and Laws

This article reveals of an application of a theory of nonparametric and nonlinear processes. This theory is described by new axioms and laws which include 2 new axioms and 8 new laws. They were explained in previous reports by the same author. This new theory expands the Classic Field Theory which is about parametric and linear processes. The theory of new axiom and laws is a more general theory because it consists new philosophy as nonparametric decoding, new objects as an accelerating or a decelerating field and new forms of movement as transverse and longitudinal motion. In present report are used 2 axioms and 5 laws only. It is known that Maxwell’s laws (1864) are based on a single axiom [1]. It states that the movement in a closed loop leads to evenly movement (with constant speed) of a vector E: div rot E = 0. The author changes this classic axiom with a new Axiom 1. According to the new Axiom1 the movement in an open loop (div rot E ≠ 0) or vortex (div Vor E ≠ 0) leads to unevenly movement (with variable speed) of a vector E [2]. The subsequent results are: the evenly movement is replaced with unevenly movement which can be decelerating or accelerating. According to subsequent laws: the cross vortex in 2D is transformed to a longitudinal vortex in 3D (transformation Δ1) and inversely (transformation Δ2); decelerating vortex emits free cross vortices to the environment; accelerating vortex sucks the same ones and so on. The electron is a model that contains a decelerating cross vortex from outside to inside and emits free cross vortices into the environment. The proton is a model that is generated by an accelerating cross vortex from inside to outside and sucks these the same free cross vortices from the environment. According to the Axiom 2 the decelerating (electron) and accelerating (proton) vortices form a resonant circle. In this circle they exchange their energy (accelerating and decelerating) along the real connection and exchange the mass of free vortices (emitted and sucked) along feedback. These free cross vortices then self-organize into something like primary dipoles which resemble electrons but in very smaller scale. The passive primary dipoles resemble electrons from the inner orbits of the atom which are contracted balls with a minimal polarization. This is the reason that these primary dipoles do not react to the amplitude of an applied external Electromagnetic Field. But they react instantly at high acceleration of the EM field. The reason is that the high acceleration strongly polarizes the passive primary dipole (ball), turns it to an active dipole (toroid) and directs it to the active pole of the EM field. In this article is described Nikola Tesla's approach about using of free cross vortices called” free energy”.

Localizable Particles in the Classical Limit of Quantum Field Theory

A number of arguments purport to show that quantum field theory cannot be given an interpretation in terms of localizable particles. We show, in light of such arguments, that the classical $\hbar\to 0$ limit can aid our understanding of the particle content of quantum field theories. In particular, we demonstrate that for the massive Klein-Gordon field, the classical limits of number operators can be understood to encode local information about particles in the corresponding classical field theory.

Spin angular momentum of gravitational wave interference

Spin angular momentum (SAM) is an important feature for wave systems, prominent in various properties like spin-momentum locking of wave propagations. Here, we study the SAM of gravitational waves in general relativity based on the Noether theorem in classical field theory. We demonstrate gravitational wave spin in various interference cases and evanescent waves, which is manifested as locally rotating metric perturbations, driving particles on geodesic spheroid locally deformed in elliptical trajectories. For non-polarized gravitational wave modes of zero SAM, their superpositions can induce nonzero density of SAM with interfered patterns. The evanescent gravitational wave shows clear SAM, which is also a consequence of wave interference between propagating and transverse evanescent components. The spin-momentum locking relations are clearly present for all different interference cases of wave modes and components based on general relativity.

Quantum stabilizer codes, lattices, and CFTs

There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.

Congruence and metaplectic covariance: Rational biquadratic reciprocity and quantum entanglement

The purpose of the paper is to elucidate the cyclotomographic applications of the coadjoint orbit methodology to the Legendre-Hilbert-Artin symbolic tower of class field theory in the sense of the theories of Chevalley, Hasse, Weil and Witt. The Witt arithmetics concludes with the law of rational biquadratic reciprocity and quantum entanglement. The purpose of the paper is to elucidate the cyclotomographic applications of the coadjoint orbit methodology to the Legendre-Hilbert-Artin symbolic tower of class field theory in the sense of the theories of Chevalley, Hasse, Weil and Witt. The Witt arithmetics concludes with the law of rational biquadratic reciprocity and quantum entanglement.

4D Chern\u2013Simons theory and affine Gaudin models

We relate two formalisms recently proposed for describing classical integrable field theories. The first (Costello and Yamazaki in Gauge Theory and Integrability, III, 2019) is based on the action of four-dimensional Chern–Simons theory introduced and studied by Costello, Witten and Yamazaki. The second (Costello and Yamazaki, in Gauge Theory and Integrability, III, 2017) makes use of classical generalised Gaudin models associated with untwisted affine Kac–Moody algebras.

4d F(4) gauged supergravity and black holes of class \u2131

We perform a consistent reduction of 6d matter-coupled F(4) supergravity on a compact Riemann surface {Sigma}_{mathfrak{g}} of genus mathfrak{g} , at the level of the bosonic action. The result is an mathcal{N} = 2 gauged supergravity coupled to two vector multiplets and a single hypermultiplet. The four-dimensional model is holographically dual to the 3d superconformal field theories of class ℱ, describing different brane systems in massive type IIA and IIB wrapped on {Sigma}_{mathfrak{g}} . The naive reduction leads to a non-standard 4d mixed duality frame with both electric and magnetic gauge fields, as well as a massive tensor, that can be only described in the embedding tensor formalism. Upon a chain of electromagnetic dualities, we are able to determine the scalar manifolds and electric gaugings that uniquely specify the model in the standard supergravity frame. We then use the result to construct the first examples of static dyonic black holes in AdS6 and perform a microscopic counting of their entropy via the 5d topologically twisted index. Finally, we show the existence of further subtruncations to the massless sector of the 4d theory, such as the Fayet-Iliopoulos gauged T3 model and minimal gauged supergravity. We are in turn able to find new asymptotically AdS4 solutions, providing predictions for the squashed S3 partition functions and the superconformal and refined twisted indices of class ℱ theories.