Invariance Symmetry Research Articles

1D diffeomorphism invariant model of a free scalar relativistic particle: Supervariable and BRST approaches

We apply the supervariable approach to derive the proper quantum Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetries for the 1D diffeomorphism invariant model of a free scalar relativistic particle by exploiting the infinitesimal classical reparameterization (i.e. 1D diffeomorphism) symmetry of this theory. We derive the conserved and off-shell nilpotent (anti-)BRST charges and prove their absolute anticommutativity property by using the virtues of Curci–Ferrari (CF)-type restriction of our present theory. We establish the sanctity of the existence of CF-type restriction (i) by considering the (anti-)BRST symmetry transformations of the coupled (but equivalent) Lagrangians and (ii) by proving the symmetry invariance of the Lagrangians within the framework of supervariable approach. We capture the nilpotency and absolute anticommutativity of the conserved (anti-)BRST charges within the framework of (anti-)chiral supervariable approach (ACSA) to BRST formalism. One of the novel observations of our present endeavor is the derivation of CF-type restriction by using the modified Bonora–Tonin (BT) supervariable approach (while deriving the (anti-)BRST symmetries for the target space–time and/or momenta variables) and by symmetry considerations of the Lagrangians of the theory. The rest of the (anti-)BRST symmetries, for the other variables, are derived by using the newly proposed ACSA. We also demonstrate the existence of CF-type restriction in the proof of absolute anticommutativity of the (anti-)BRST charges.

Explicit form of Berry phase for time dependent harmonic oscillators in noncommutative space

In this paper we look for the existence of Berry phase in time dependent harmonic oscillators in noncommutative space. Two systems are considered in our study in the noncommutative framework. The first one is a system in which a scale invariant term is present in the Hamiltonian from the beginning, and the second one is a system in which a scale invariant term emerges due to a change of variables from the noncommutative to the commutative ones. We first compute the eigenstates of both the systems using the Lewis invariant approach. We then employ the Lewis invariant technique to obtain the geometric phase under adiabatic approximation. We also calculate the Berry phase explicitly by choosing appropriate forms of the time dependent parameters appearing in the Hamiltonian. Our analysis surprisingly reveals that a scale invariant time reversal symmetry breaking term may not always lead to a non trivial Berry phase.

Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-set Method

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-C\'ardenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/j.jcp.2022.111291] that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting to harness curvature symmetry invariance in favor of precision and stability. The core of this solver is a multilayer perceptron trained on circular- and sinusoidal-interface samples. Its role is to quantify the error in numerical curvature approximations and emit corrected estimates for select grid vertices along the free boundary. These corrections arise in response to preprocessed context level-set, curvature, and gradient data. To promote neural capacity, we have adopted sample negative-curvature normalization, reorientation, and reflection-based augmentation. In the same manner, our system incorporates dimensionality reduction, well-balancedness, and regularization to minimize outlying effects. Our training approach is likewise scalable across mesh sizes. For this purpose, we have introduced dimensionless parametrization and probabilistic subsampling during data production. Together, all these elements have improved the accuracy and efficiency of curvature calculations around under-resolved regions. In most experiments, our strategy has outperformed the numerical baseline at twice the number of redistancing steps while requiring only a fraction of the cost.

Superfield Approaches to a Model of Bosonic String: Curci-Ferrari-Type Restrictions

Exploiting the theoretical potential of the modified Bonora-Tonin superfield approach (MBTSA) as well as the (anti-)chiral superfield approach (ACSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the complete set of off-shell nilpotent (anti-)BRST symmetry transformations corresponding to the classical two- 1 + 1 dimensional (2D) diffeomorphism symmetry transformations on the worldsheet (that is traced out by the motion of a model of bosonic string). Only the BRST symmetry transformations for this model have been discussed in the earlier literature. We derive the (anti-)BRST invariant Curci-Ferrari- (CF-) type restrictions (using MBTSA) which turn out to be the root cause behind the absolute anticommutativity of the above (anti-)BRST symmetry transformations. We capture the symmetry invariance of the (anti-)BRST invariant Lagrangian densities within the ambit of ACSA. The derivation of the proper anti-BRST transformations (corresponding to the already-known BRST transformations) and the (anti-)BRST invariant CF-type restrictions are the novel results in our present endeavor.

Supersymmetry and shape invariance of exceptional orthogonal polynomials

We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schrödinger equation to define “Hamiltonians” enables us to study the EOPs in the framework of the SUSYQM and to realize the underlying shape invariance associated with such systems. We show that the underlying shape invariance symmetry is responsible for the solubility of the differential equations associated with these polynomials.

Soft pion scattering in infrared-conformal gauge-fermion theories

We consider the problem of soft scattering for the analogue of pion states in gauge-fermion theories which approach a conformal fixed point in the infrared limit. Introducing a fermion mass into such a theory will explicitly break both scale invariance and chiral symmetry, leading to confinement and a spectrum of bound states. We argue that in such a theory, the pion scattering length diverges in the limit of zero fermion mass, in sharp contrast to QCD-like theories where the chiral Lagrangian predicts a vanishing scattering length. We demonstrate this effect both with a simple dimensional argument, and in a generalized linear sigma model which we argue can be used to describe the interactions of light scalar and pseudoscalar bound states in the soft limit of a mass-deformed infrared-conformal theory. As a result, lattice calculations of pion scattering lengths could be a sensitive probe for infrared scale invariance in gauge-fermion theories.

Hidden scale invariance in the Gay-Berne model.

This paper presents a numerical study of the Gay-Berne liquid crystal model with parameters corresponding to calamitic (rod-shaped) molecules. The focus is on the isotropic and nematic phases at temperatures above unity, where we find strong correlations between the virial and potential-energy thermal fluctuations, reflecting the hidden scale invariance symmetry. This implies the existence of isomorphs, which are curves in the thermodynamic phase diagram of approximately invariant physics. We study numerically one isomorph in the isotropic phase and one in the nematic phase. In both cases, good invariance of the dynamics is demonstrated via data for the mean-square displacement and the reduced-unit time-autocorrelation functions of the velocity, angular velocity, force, torque, and first- and second-order Legendre polynomial orientational order parameters. Deviations from isomorph invariance are observed at short times for the orientational time-autocorrelation functions, which reflects the fact that the moment of inertia is assumed to be constant and thus not isomorph-invariant in reduced units. Structural isomorph invariance is demonstrated from data for the radial distribution functions of the molecules and their orientations. For comparison, all quantities were also simulated along an isochore of similar temperature variation, in which case invariance is not observed. We conclude that the thermodynamic phase diagram of the calamitic Gay-Berne model is essentially one-dimensional in the studied regions as predicted by isomorph theory, a fact that potentially allows for simplifications of future theories and numerical studies.

Variational λ‐symmetries and exact solutions to Euler‐Lagrange equations lacking standard symmetries

Variational ‐symmetries are used to find exact solutions to second‐ and fourth‐order Euler‐Lagrange equations associated to variational problems for which standard procedures fail. A one‐parameter family of exact solutions in terms of Bessel functions is obtained for a first‐order variational problem whose Euler‐Lagrange equation does not admit Lie symmetries. A family of second‐order equations, involving arbitrary functions and parameters, is first written in variational form. The variational ‐symmetry method successes in finding one‐parameter families of exact solutions, despite the lack of Lie point and variational symmetries. A three‐parameter family of exact solutions for a fourth‐order equation with absence of Lie point symmetries is also deduced.

Dyadic Helmholtz Green’s Function for Electromagnetic Wave Transmission/Diffraction through a Subwavelength Nano-Hole in a 2D Quantum Plasmonic Layer: An Exact Solution Using “Contact Potential”-like Dirac Delta Functions

The dyadic Helmholtz Green’s function for electromagnetic (EM) wave transmission/ diffraction through a subwavelength nano-hole in a two-dimensional (2D) plasmonic layer is discussed here analytically and numerically, employing “contact potential”-like Dirac delta functions in 1 and 2 dimensions (δ(z) and δ(x)δ(y)≡δ(2)(r→)). This analysis is carried out employing a succession of two coupled integral equations. The first integral equation determines the dyadic electromagnetic Green’s function G^fs for the full non-perforated 2D quantum plasma layer in terms of the bulk 3D infinite-space dyadic electromagnetic Green’s function G^3D, with δ(z) representing the confinement of finite quantum plasma conductivity to the plane of the plasma layer at z=0. The second integral equation determines the dyadic electromagnetic “hole” Green’s function G^hole for the perforated 2D quantum plasma layer (containing the nano-hole) in terms of the dyadic electromagnetic Green’s function G^fs for the full non-perforated 2D plasma layer, with δ(2)(r→) describing the exclusion of the quantum plasma layer conductivity properties from the nano-hole region in the vicinity of r→=0 on the plane. Taking the radius of the subwavelength nano-hole to be the smallest length scale of the system in conjunction with the 2D Dirac delta function representation of the excluded nano-hole plasma conductivity, both of the successive coupled integral equations are solved exactly, and we present a thorough numerical analysis (based on the exact analytic solution) for the resulting dyadic “hole” Green’s function G^hole in full detail in both 3D and density plots. This result has been successfully applied to the determination of electromagnetic wave transmission/diffraction through the nano-hole of the perforated quantum plasmonic layer, jointly with the EM wave transmission through the rest of the plasma layer. This success necessarily involves spatial translational asymmetry induced by the use of spatial Dirac delta functions confining finite conductivity to the 2D quantum plasma sheet and the excision at a bit of it about the origin to represent the nano-hole perforation, thus breaking spatial translational invariance symmetry.

Symmetry and spectral invariance for topologically graded C∗$C^*$‐algebras and partial action systems

A discrete group G ${\sf G}$ is called rigidly symmetric if the projective tensor product between the convolution algebra ℓ 1 ( G ) $\ell ^1({\sf G})$ and any C ∗ $C^*$ -algebra A $\mathcal {A}$ is symmetric. We show that in each topologically graded C ∗ $C^*$ -algebra over a rigidly symmetric group there is a ℓ 1 $\ell ^1$ -type symmetric Banach ∗ $^*$ -algebra, which is inverse closed in the C ∗ $C^*$ -algebra. This includes new general classes, as algebras admitting dual actions and partial crossed products. Results including convolution dominated kernels, inverse closedness with respect with ideals or weighted versions of the ℓ 1 $\ell ^1$ -decay are included. Various concrete examples are presented.

On the Lie symmetries of characteristic function hierarchy in compressible turbulence

We compute the Lie symmetries of characteristic function (CF) hierarchy of compressible turbulence, ignoring the effects of viscosity and heat conductivity. In the probability density function (PDF) hierarchy, a typical non-local nature is observed, which is naturally eliminated in the CF hierarchy. We observe that the CF hierarchy retains all the symmetries satisfied by compressible Euler equations. Broadly speaking, four types of symmetries can be discerned in the CF hierarchy: (i) symmetries corresponding to coordinate system invariance, (ii) scaling/dilation groups, (iii) projective groups and (iv) statistical symmetries, where the latter define measures of intermittency and non-gaussianity. As the multi-point CFs need to satisfy additional constraints such as the reduction condition, the projective symmetries are only valid for monatomic gases, that is, the specific heat ratio, $\gamma = 5/3$ . The linearity of the CF hierarchy results in the statistical symmetries due to the superposition principle. For all of the symmetries, the global transformations of the CF and various key compressible statistics are also presented.

Transformations, symmetries and Noether theorems for differential-difference equations

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether’s theorem. We state and prove the differential-difference version of Noether’s second theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether’s two theorems. These results are applied to various equations from physics.

Variational symmetries and superintegrability in multifield cosmology

We consider a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker background space with an ideal gas and a multifield Lagrangian consisting of two minimally coupled scalar fields which evolve in a field space of constant curvature. For this cosmological model we classify the potential function for the scalar fields such that variational point symmetries exist. The corresponding conservation laws are calculated. Finally, analytic solutions are presented for specific functional forms of the scalar field potential in which the cosmological field equations are characterized as a Liouville integrable system by point symmetries. The free parameters of the cosmological model are constrained in order to describe analytic solutions for an inflationary epoch. Finally, stability properties of exact closed-form solutions are investigated. These solutions are scaling solutions with important physical properties for the cosmological model.

Variational existence proof for multiple periodic orbits in the planar circular restricted three-body problem

The restricted three-body problem (R3BP) is an important research area that deals with significant issues in celestial mechanics, such as analyzing asteroid movement behavior and orbit design for space probes. We aim to show the existence of periodic orbits in the planar circular R3BP. To find these orbits, we adapt a variational approach and symmetry.

A modified formal Lagrangian formulation for general differential equations

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler--Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether's Theorem and in particular, a nontrivial conservation law of the Fornberg--Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.

Massive 4D Abelian 2-form theory: Nilpotent symmetries from the (anti-)chiral superfield approach

The off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries are obtained by using the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism for the four [Formula: see text]-dimensional (4D) Stückelberg-modified massive Abelian 2-form gauge theory. We perform exactly similar kind of exercise for the derivation of the off-shell nilpotent (anti-)co-BRST symmetry transformations, too. In the above derivations, the symmetry invariant restrictions on the superfields play very important and decisive roles. To prove the sanctity of the above nilpotent symmetries, we generalize our 4D ordinary theory (defined on the 4D flat Minkowskian space–time manifold) to its counterparts [Formula: see text]-dimensional (anti-)chiral super-submanifolds of the [Formula: see text]-dimensional supermanifold which is parametrized by the superspace coordinates [Formula: see text] where [Formula: see text] [Formula: see text] are the bosonic coordinates and a pair of Grassmannian variables [Formula: see text] are fermionic: [Formula: see text], [Formula: see text] in nature. One of the novel observations of our present endeavor is the derivation of the Curci–Ferrari (CF)-type restrictions from the requirement of the symmetry invariance of the coupled (but equivalent) Lagrangian densities of our theory within the framework of ACSA to BRST formalism. We also exploit the standard techniques of ACSA to capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST as well as the conserved (anti-)co-BRST charges. In a subtle manner, the proof of the absolute anticommutativity of the above conserved charges also implies the existence of the appropriate CF-type restrictions on our theory. This proof is also a novel result.

Searches for Violation of CPT Symmetry and Lorentz Invariance with Astrophysical Neutrinos

Neutrinos are a powerful tool for searching physics beyond the standard model of elementary particles. In this review, we present the status of the research on charge-parity-time (CPT) symmetry and Lorentz invariance violations using neutrinos emitted from the collapse of stars such as supernovae and other astrophysical environments, such as gamma-ray bursts. Particularly, supernova neutrino fluxes may provide precious information because all neutrino and antineutrino flavors are emitted during a burst of tens of seconds. Models of quantum gravity may allow the violation of Lorentz invariance and possibly of CPT symmetry. Violation of Lorentz invariance may cause a modification of the dispersion relation and, therefore, in the neutrino group velocity as well in the neutrino wave packet. These changes can affect the arrival time signal registered in astrophysical neutrino detectors. Direction or time-dependent oscillation probabilities and anisotropy of the neutrino velocity are manifestations of the same kind of new physics. CPT violation, on the other hand, may be responsible for different oscillation patterns for neutrino and antineutrino and unconventional energy dependency of the oscillation phase or of the mixing angles. Future perspectives for possible CPT and Lorentz violating systems are also presented.

Confirming the trilinear form of the optical magnetoelectric effect in the polar honeycomb antiferromagnet Co2Mo3O8

Magnetoelectric phenomena are intimately linked to relativistic effects and also require the material to break spatial inversion symmetry and time-reversal invariance. Magnetoelectric coupling can substantially affect light–matter interaction and lead to non-reciprocal light propagation. Here, we confirm on a fully experimental basis, without invoking either symmetry-based or material-specific assumptions, that the optical magnetoelectric effect in materials with non-parallel magnetization (M) and electric polarization (P) generates a trilinear term in the refractive index, δn ∝ k ⋅ (P × M), where k is the propagation vector of light. Its sharp magnetoelectric resonances in the terahertz regime, which are simultaneously electric and magnetic dipole active excitations, make Co2Mo3O8 an ideal compound to demonstrate this fundamental relation via independent variation of M, P, and k. Remarkably, the material shows almost perfect one-way transparency in moderate magnetic fields for one of these magnetoelectric resonances.

Bi-Variationality, Symmetries and Approximate Solutions

By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator S to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples.

Lie Algebra Representation, Conservation Laws and Some Invariant Solutions for a Generalized Emden-Fowler Equation

All generators of the optimal algebra associated with a generalization of the Endem-Fowler equation are showed; some of them allow to give invariant solutions. Variational symmetries and the respective conservation laws are also showed. Finally, a representation of Lie symmetry algebra is showed by groups of matrices.