Showing papers on "Noether's theorem published in 2023"

Noether's second theorem and covariant field theory of mechanical stresses in inhomogeneous ionic liquids.

Abstract: In this paper, we present a covariant approach that utilizes Noether's second theorem to derive a symmetric stress tensor from the grand thermodynamic potential functional. We focus on the practical case where the density of the grand thermodynamic potential is dependent on the first and second coordinate derivatives of the scalar order parameters. Our approach is applied to several models of inhomogeneous ionic liquids that consider electrostatic correlations of ions or short-range correlations related to packing effects. Specifically, we derive analytical expressions for the symmetric stress tensors of the Cahn-Hilliard-like model, Bazant-Storey-Kornyshev model, and Maggs-Podgornik-Blossey model. All of these expressions are found to be consistent with respective self-consistent field equations.

Traversable Wormhole Solutions Admitting Noether Symmetry in f(R,T2) Theory

Abstract: This paper uses the Noether symmetry approach to examine the viable and stable traversable wormhole solutions in the framework of the f(R,T2) theory, where R is the Ricci scalar and T2=TμνTμν is the self-contraction of the stress–energy tensor. For this purpose, we consider a specific model of this modified theory to obtain the exact solutions of the Noether equations. Further, we formulate the generators of the Noether symmetry and first integrals of motion. We analyze the presence of viable and stable traversable wormhole solutions corresponding to different redshift functions. In order to determine whether this theory provides physically viable and stable wormhole geometry or not, we check the graphical behavior of the null energy constraint, causality condition and adiabatic index for an effective stress–energy tensor. It is found that viable and stable traversable wormhole solutions exist in this modified theory.

Symmetry arguments and the totalitarian principle in the physics of liquid crystals and other condensed matter systems

Abstract: Symmetry arguments constitute one of the most powerful tools of physics. As shown by Emmy Noether, continuous symmetries correspond to physical conservation laws. In analogy with dimensional arguments, symmetry arguments can also be used to predict the behaviour of physical systems. Mistaking symmetries, on the other hand, can lead to grave errors. In this paper, we describe and give examples of certain types of symmetry arguments which can be useful in general and, in particular, in the field of liquid crystals. In using symmetry arguments to predict physical phenomena, the totalitarian principle, made popular by Murray Gell-Mann, is often used, either explicitly or implicitly. We discuss the totalitarian principle, its origins, history and philosophical implications. Our goal is to outline a simple symmetry-based procedure which may be useful in gaining insights into and describing the behaviour of complex physical systems such as liquid crystals.

Revisiting the Friedberg–Lee–Sirlin soliton model

Abstract: Non-topological solitons are localized classical field configurations stabilized by a Noether charge. Friedberg, Lee, and Sirlin proposed a simple renormalizable soliton model in their seminal 1976 paper, consisting of a complex scalar field that carries the Noether charge and a real-scalar mediator. We revisit this model, point out commonalities and differences with Q-ball solitons, and provide analytic approximations to the underlying differential equations.

Noether Symmetries and Conservation Laws in Static Cylindrically Symmetric Spacetimes via Rif Tree Approach

Abstract: A new approach is adopted to completely classify the Lagrangian associated with the static cylindrically symmetric spacetime metric via Noether symmetries. The determining equations representing Noether symmetries are analyzed using a Maple algorithm that imposes different conditions on metric coefficients under which static cylindrically symmetric spacetimes admit Noether symmetries of different dimensions. These conditions are used to solve the determining equations, giving the explicit form of vector fields representing Noether symmetries. The obtained Noether symmetry generators are used in Noether’s theorem to find the expressions for corresponding conservation laws. The singularity of the obtained metrics is discussed by finding their Kretschmann scalar.

Classification of the Lie and Noether Symmetries for the Klein-Gordon Equation in Anisotropic Cosmology

Abstract: We carried out a detailed group classification of the potential in Klein–Gordon equation in anisotropic Riemannian manifolds. Specifically, we consider the Klein–Gordon equations for the four-dimensional anisotropic and homogeneous spacetimes of Bianchi I, Bianchi III and Bianchi V. We derive all the closed-form expressions for the potential function where the equation admits Lie and Noether symmetries. We apply previous results which connect the Lie symmetries of the differential equation with the collineations of the Riemannian space which defines the Laplace operator, and we solve the classification problem in a systematic way.

More Insights into Symmetries in Multisymplectic Field Theories

Abstract: This work provides a general overview for the treatment of symmetries in classical field theories and (pre)multisymplectic geometry. The geometric characteristics of the relation between how symmetries are interpreted in theoretical physics and in the geometric formulation of these theories are clarified. Finally, a general discussion is given on the structure of symmetries in the presence of constraints appearing in singular field theories. Symmetries of some typical theories in theoretical physics are analyzed through the construction of the relevant multimomentum maps which are the conserved quantities (by Noether’s theorem) on the (pre)multisymplectic phase spaces.

Anomalies in quantum field theories

Abstract: Introduction:purpose: Noether's theorem connects symmetry of the Lagrangian to conserved quantities. Quantum effects cancel the conserved quantities. Methods: Triangle diagram, Path integral, Pauli-Villars regularisation. Results: Quantum effects that spoil conserved quantities of local gauge symmetries endager renormalisability. Conclusion: A careful treatment of anomalies is needed in order to obtain correct results. The p 0 → gg decay is perhaps the most notable "impossible" effect allowed by anomalies.

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Abstract: M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether's theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal's theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U(1)-action. When the limiting rotation is non-resonant, these maps admit formal U(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether's theorem. When the unperturbed U(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

Hierarchical proliferation of higher-rank symmetry defects in fractonic superfluids

Abstract: Symmetry defects, e.g., vortices in conventional superfluids, play a critical role in a complete description of symmetry breaking phases. In this paper, we develop the theory of symmetry defects in fractonic superfluids, i.e., spontaneously higher-rank symmetry (HRS) breaking phases initially proposed by Yuan et al. [Phys. Rev. Res. 2, 023267 (2020)] and by Chen et al. [Phys. Rev. Res. 3, 013226 (2021)]. By Noether's theorem, HRS is associated with the conservation law of higher moments, e.g., dipoles, quadrupoles, and angular moments. We establish finite-temperature phase diagrams by identifying a series of topological phase transitions via the renormalization group flow equations and Debye-H\"uckel approximation. Accordingly, a series of Kosterlitz-Thouless topological transitions are found to occur successively at different temperatures, which are triggered by proliferation of defects, defect bound states, and so on. Such a hierarchical proliferation brings rich phase structures. Meanwhile, a screening effect from sufficiently high density of defect bound states leads to instability and collapse of the intermediate temperature phases, which further enriches the phase diagrams. For concreteness, we consider a fractonic superfluid in which ``angular moments'' are conserved. We then present the general theory, in which other types of HRS can be analyzed in a similar manner. Further directions are present at the end of the paper.

On the electromagnetic symmetry producing fields charges at four bosons association

Abstract: Physics would like to know what electric charge is. As a matter property it generates EM fields, coupling constant,conserved current. Nevertheless Maxwell is uncomplete. It requires to be extended. An approach is supported byelectric charge transfer phenomenology. Consider on three flavours charges {+, 0, −} transmission.A generic electric charge is defined by the triad {+, 0, −}. It provides an exchange charge physics through thequadruplet {Aμ, Uμ, V ± μ }. An electromagnetic symmetry is constituted. It associates the four vectorial bosons. TheEM completeness of particles carrying three electric charges is found. A four photons EM is derived. It includes, Aμ asthe usual photon, Uμ massive photon, V±μ massive charged photons. A new electromagnetic physics is expressed through an enlarged abelian symmetry, Uq(1). Maxwell relationshipsbetween charge and fields are extended. Fields charges are more primitive than electric charge. Noether theoremimproves the attributes on electric charge conservation, symmetry equation and constraint. The symmetry equationwhich govern the electric charge dynamics shows a charge behaviour beyond matter, as fields flux. EM interaction isextended from fine structure constant to modulated and neutral charges.The four bosons electromagnetism introduces a non-univoque electromagnetic symmetry. A pluriformity of EM modelsis performed under similar abelian group, Uq(1). Opportunities for different EM models are constituted preservingcharge conservation law and sharing a common Lagrangian. Physical varieties on Noether theorem, fields strengths,Lagrangian coefficients, equations of motion, fields charges are expressed. Electric charge is englobed by fields charges.The simplest four bosons model is selected. That one which fields strengths are gauge invariants. Propitiatingmeasurable granular and collective fields strenghts. Four kinds of charges are expressed through equations of motion.Electric, modulate, neutral and Bianchi. Allowing to include new EM sectors. Extend Maxwell for nonintegers charges,nonlinearity, neutral EM, spintronics, weak interactions, photonics. A new EM energy emerges.

Relating a System’s Hamiltonian to Its Entropy Production Using a Complex Time Approach

Abstract: We exploit the properties of complex time to obtain an analytical relationship based on considerations of causality between the two Noether-conserved quantities of a system: its Hamiltonian and its entropy production. In natural units, when complexified, the one is simply the Wick-rotated complex conjugate of the other. A Hilbert transform relation is constructed in the formalism of quantitative geometrical thermodynamics, which enables system irreversibility to be handled analytically within a framework that unifies both the microscopic and macroscopic scales, and which also unifies the treatment of both reversibility and irreversibility as complementary parts of a single physical description. In particular, the thermodynamics of two unitary entities are considered: the alpha particle, which is absolutely stable (that is, trivially reversible with zero entropy production), and a black hole whose unconditional irreversibility is characterized by a non-zero entropy production, for which we show an alternate derivation, confirming our previous one. The thermodynamics of a canonical decaying harmonic oscillator are also considered. In this treatment, the complexification of time also enables a meaningful physical interpretation of both “imaginary time” and “imaginary energy”.

On the zero helicity condition for quantum vortex defects

Abstract: Abstract In this note we provide an analytical proof of the zero helicity condition for systems governed by the Gross–Pitaevskii equation (GPE). The proof is based on the hydrodynamic interpretation of the GPE, and the direct use of Noether's theorem by applying Kleinert's multi-valued gauge theory. As a by-product we also demonstrate the conservation and quantization of the circulation for the GPE.

Noether-Wald charge in supergravity: the fermionic contribution

Abstract: We study the invariance of N=1,d=4 supergravity solutions under diffeormophisms and show that, in order to obtain consistent conditions (``Killing equations'') invariant under local supersymmetry transformations, one has to perform supersymmetry transformations generated by the superpartner of the vector that generates standard diffeomorphisms, just as a superspace analysis indicates. Using these transformations, we construct a Noether-Wald charge of N=1,d=4 supergravity with fermionic contributions which is diff- Lorentz- and supersymmetry-invariant (up to a total derivative).

A viable form of the metric Teleparallel F(T) theory of gravity

Abstract: Unlike F(R) gravity, pure metric F(T) gravity in the vacuum dominated era, ends up with an imaginary action and is therefore not feasible. This eerie situation may only be circumvented by associating a scalar field, which can also drive inflation in the very early universe. We show that, despite diverse claims, F(T) theory admits Noether symmetry only in the pressure-less dust era in the form F(T) proportional to the nth power of T, n being odd integers. A suitable form of F(T), admitting a viable Friedmann-like radiation dominated era, together with early deceleration and late-time accelerated expansion in the pressure-less dust era, has been proposed.

Symmetry analysis, laws of conservation, and numerical and approximate analysis of Burger’s fractional order differential equation

Abstract: The computation of fractional order differential and integration equations is highly utilized in numerous fields, such as mathematics in biology, ecology, physics, chemistry, and so on. This article aims at presenting methods for analytical and numerical solutions, Lie’s symmetry analysis, computing conservation laws for Burger’s fractional order differential equation (FODE), and reducing time-fractional cylindrical-Burgers equation order. Hence, we have employed a generalized (G′/G) expansion method for analytical solutions and a non-standard finite difference scheme for numerically solving Burger’s FODE. Additionally, the fractional derivative generalization of Noether's theorem has been utilized to compute the equation’s conservation laws. Numerical results have been reported here to approve theoretical results acquired in non-standard finite difference schemes. The proposed generalized (G′/G) expansion method is a simple, accurate, and practical approach to problem solving. Additionally, this method can be employed to execute non-linear wave equations. Programs including MATLAB and Maple have been utilized to simplify the process of solving complex equations.

Conservation laws of mean field games equations

Abstract: Mean field games equations are examined for conservation laws. The system of mean field games equations consists of two partial differential equations: the Hamilton-Jacobi-Bellman equation for the value function and the forward Kolmogorov equation for the probability density. For separable Hamiltonians, this system has a variational structure, i.e., the equations of the system are Euler-Lagrange equations for some Lagrangian functions. Therefore, one can use the Noether theorem to derive the conservation laws using variational and divergence symmetries. In order to find such symmetries, we find symmetries of the PDE system and select variational and divergence ones. The paper considers separable, state-independent Hamiltonians in one-dimensional state space. It examines the most general form of the mean field games system for symmetries and conservation laws and identifies particular cases of the system which lead to additional symmetries and conservation laws.

Algebraically Closed Fields

Abstract: We establish a simple algebraic elimination of quantifiers procedure for the theory of algebraically closed fields. This theory is model complete (Corollary 10.3.2). Among the applications are Hilbert’s Nullstellensatz and the Bertini–Noether theorem.

An Invariance and Closed Form Analysis of the Nonlinear Biharmonic Beam Equation

Abstract: In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx+2uxxyy+uyyyy=f(u) . To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE.

Brill-Noether theory of Hilbert schemes of points on surfaces

Abstract: We show that Brill--Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive, and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$ where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino's descriptionof the Hilbert--Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill--Noether loci given by nested Hilbert schemes.

The energy–momentum complex in non-local gravity

Abstract: In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a specific volume. Here, we generalize the Einstein gravitational energy–momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator [Formula: see text] are taken into account. We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy–momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations. Furthermore, the energy–momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy–momentum complex, taking into account both gravitational and matter components, can be derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications.

Curves on Brill-Noether special K3 surfaces

Abstract: Mukai showed that projective models of Brill-Noether general polarized K3 surfaces of genus $6-10$ and $12$ are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill-Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill-Noether generality of any polarized K3 surface $(S,H)$ is equivalent to the Brill-Noether generality of smooth curves $C$ in the linear system $|H|$, is still open. Using Lazarsfeld-Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus $\leq 19$, which provides a new and unified proof even in the genera where Mukai models exist.

Conserved quantities in the Ekman layer

Abstract: The classical Ekman (1905)'s theory of wind-driven currents at the surface boundary layer is a well-known mathematical model that describes transport phenomena in coastal processes, (e.g.) upwellings and downwellings, and represents an essential part of modern oceanography. Understanding the Ekman layer is important to quantify deviations to observed magnitudes from the classical behavior. In this theoretical work, the Ekman layer is revisited through Noether's theorem. This theorem plays a central role in theoretical physics and Lie group theory showing a direct connection between symmetries and conservation laws. Therefore, the goal of the work is to determine what quantities are conserved in the Ekman layer.

Research on the Symmetry of the Hamiltonian System under Generalized Operators

Abstract: Generalized operators have recently been proposed with great potential applications. Here, we present research carried out on Noether figury and perturbation to Noether symmetry for Hamiltonian systems within generalized operators. There are four parts, and each part contains two kinds of generalized operator. Firstly, Hamilton equations are established. Secondly, the Noether symmetry method is used for finding the solutions to the differential equations of motion, and conserved quantities are obtained. Thirdly, perturbation to Noether symmetry and adiabatic invariants are further explored. In the end, two examples are given to illustrate the methods and results.

Similarity Solution for a System of Fractional-Order Coupled Nonlinear Hirota Equations with Conservation Laws

Abstract: The analysis of differential equations using Lie symmetry has been proved a very robust tool. It is also a powerful technique for reducing the order and nonlinearity of differential equations. Lie symmetry of a differential equation allows a dynamic framework for the establishment of invariant solutions of initial value and boundary value problems, and for the deduction of laws of conservations. This article is aimed at applying Lie symmetry to the fractional-order coupled nonlinear complex Hirota system of partial differential equations. This system is reduced to nonlinear fractional ordinary differential equations (FODEs) by using symmetries and explicit solutions. The reduced equations are exhibited in the form of an Erdelyi–Kober fractional (E-K) operator. The series solution of the fractional-order system and its convergence is investigated. Noether’s theorem is used to devise conservation laws.

The Attractor Flow for AdS5 Black Holes in $\mathcal{N} = 2$ Gauged Supergravity

Abstract: We study the flow equations for BPS black holes in $\mathcal{N} = 2$ five-dimensional gauged supergravity coupled to any number of vector multiplets via FI couplings. We develop the Noether-Wald procedure in this context and exhibit the conserved charges as explicit integrals of motion, in the sense that they can be computed at any radius on the rotating spacetime. The boundary conditions needed to solve the first order differential equations are discussed in great detail. We extremize the entropy function that controls the near horizon geometry and give explicit formulae for all geometric variables at their supersymmetric extrema. We have also considered a complexification of the near-horizon variables that elucidates some features of the theory from the near-horizon perspective.