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We revisit the use of Meta-Generalized Gradient Approximations (mGGAs) in time-dependent density functional theory, reviewing conceptual questions and solving the generalized Kohn–Sham equations by real-time propagation. After discussing the technical aspects of using mGGAs in combination with pseudopotentials and comparing real-space and basis set results, we focus on investigating the importance of the current-density based gauge invariance correction. For the two modern mGGAs that we investigate in this work, TASK and r2SCAN, we observe that for some systems, the current density correction leads to negligible changes, but for others, it changes excitation energies by up to 40% and more than 0.8 eV. In the cases that we study, the agreement with the reference data is improved by the current density correction. [ABSTRACT FROM AUTHOR]

Ever since new tetrads in four-dimensional Lorentzian curved spacetimes were introduced, many outstanding properties and results have been proven regarding Riemannian geometry, gauge theory or group theory. These tetrads might carry spacetime-kinematic information about particle multiplets. Among other properties, these new tetrads, locally and covariantly diagonalize stress–energy tensors. In the case where particles have an electromagnetic field, the local planes of gauge symmetry served as a tool in order to propose a new gravitational-kinematic interpretation of particle multiplets. The core reason stems from the mathematical fact that all local gauge symmetries associated to the Standard Model have been proven to be isomorphic to local groups of tetrad transformations in four-dimensional Lorentzian spacetimes. Therefore, the different stress–energy tensors have been proven to determine the local gauge geometry as a natural part of Riemannian geometry. The stress–energy tensors determine locally the orthogonal planes such that the rotation on either of them of the local tetrad vectors that span these planes is isomorphic to local tetrad gauge transformations. All these properties put together allow for a reinterpretation presented previously of particle states as particle gravitational-kinematic-spacetime tetrad states in the asymptotically flat limit. We proceed in this paper to study the case where there are T , U and V isospin subalgebras or submultiplets within the context of this new Riemannian interpretation of gauge symmetries. [ABSTRACT FROM AUTHOR]

After a brief recount of small and large gauge transformations and the nature of observables, we discuss superselection sectors in gauge theories. There are an infinity of them, classified by large gauge transformations. Gauge theory sectors are labeled by the eigenvalues of a complete commuting set (CCS) of these transformations. In QED, the standard chemical potential is one such operator generating global U(1). There are many more given by the moments of the electric field on the sphere at infinity. In QCD, the CCS are constructed from the two commuting generators spanning a Cartan subalgebra. Large gauge transformations commute with the Hamiltonian and preserve the equations of motion. They form an infinite number of ‘classical symmetries’. But most of them are anamolous changing the superselection sectors. We show that any element of a large gauge transformation can be added to the standard Hamiltonian as a generalized chemical potential without changing field equations and that in QCD, they lead to confined and deconfined phases. A speculation about the physical meaning of these chemical potentials is also made. [ABSTRACT FROM AUTHOR]

We study the Maxwell field with a general gauge fixing (GF) term in the radiation-dominant (RD) and matter-dominant (MD) stages of expanding Universe, as a continuation to the previous work in the de Sitter space. We derive the exact solutions, perform the covariant canonical quantization and obtain the stress tensor in the Gupta–Bleuler (GB) physical states, which is independent of the GF constant and is also invariant under the quantum residual gauge transformation. The transverse stress tensor is similar in all flat Robertson–Walker spacetimes, and its vacuum part is ∝ k 4 and becomes zero after the 0th-order adiabatic regularization. The longitudinal-temporal stress tensor, in both the RD and MD stages, is zero due to a cancelation between the longitudinal and temporal parts in the GB states, and so is the particle part of the GF stress tensor. The vacuum GF stress tensor, in the RD stage, contains k 4 , k 2 divergences and becomes zero by the 2nd-order regularization, however, in the MD stage, contains k 4 , k 2 , k 0 divergences and becomes zero by the 4th-order regularization. So, the order of adequate regularization depends not only upon the type of fields, but also upon the background spacetimes. In summary, in both the RD and MD stages, as in the de Sitter space, the total regularized vacuum stress tensor is zero, independent of the GF constant, only the transverse photon part remains, there is no trace anomaly, and the vanishing GF stress tensor cannot be a candidate for the dark energy. [ABSTRACT FROM AUTHOR]

We apply the gauge transformations (differential type) and (integral type) to study the discrete mKP hierarchies. We prove that and can be commutative and the product of and satisfies the Sato equation. By means of gauge transformations, we arrive at the necessary and sufficient condition for reducing the generalized Wronskian solutions to constrained hierarchies. Finally, we give an example in the Appendix. [ABSTRACT FROM AUTHOR]

Non-Kerr media, twin-core nonlinear optical fibers and waveguides are widely applied in optical fiber communications. In order to model the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide, a coupled variable-coefficient cubic-quintic nonlinear Schrödinger system is investigated in this paper. For the two components of the electromagnetic fields, on the basis of the existing Lax pair, we take a gauge transformation to convert a Lax pair into an Ablowitz-Kaup-Newell-Segur system. Then we acquire a system, which is equal to the original system. The N-fold Darboux transformation is derived based on our Lax pair, where N is a positive integer. Applying Taylor series, we derive the N-fold generalized Darboux transformation. The second- and third-order vector breather solutions are obtained via the generalized Darboux transformation method. All our results depend on the variable coefficients of the original system. Our results might be of some use in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide system. [ABSTRACT FROM AUTHOR]

Flux tube solutions within non-Abelian SU(3) Proca theory with external sources are obtained. It is shown that such tubes have a longitudinal chromoelectric field possessing two components (nonlinear and gradient), as well as a transverse chromomagnetic field whose force lines create concentric circles with the center on the axis of the tube. The scenario of a possible relationship between non-Abelian Proca theory and quantum chromodynamics is considered. In such scenario: (a) the components of color fields have different behavior: those which are almost classical, and those which are purely quantum; (b) the second components create a gluon condensate that is a source of the field for the almost classical components of the Proca field; (c) Proca masses may appear as a result of an approximate description of the gluon condensate; (d) the question of gauge invariance is considered. It is shown that the results obtained are in good agreement with the results of lattice calculations. We make an assumption that an approximate description of a flux tube in quantum chromodynamics can be carried out using classical Proca equations but with a mandatory account of a gluon condensate. [ABSTRACT FROM AUTHOR]

For the Stückelberg-modified massive Abelian 3-form theory in any arbitrary D-dimension of spacetime, we show that its classical gauge symmetry transformations are generated by the first-class constraints. We establish that the Noether conserved charge (corresponding to the local gauge symmetry transformations) is same as the standard form of the generator for the underlying local gauge symmetry transformations (expressed in terms of the first-class constraints). We promote these classical local, continuous and infinitesimal gauge symmetry transformations to their quantum counterparts Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations which are respected by the coupled (but equivalent) Lagrangian densities. We derive the conserved (anti-)BRST charges by exploiting the theoretical potential of Noether's theorem. However, these charges turn out to be non-nilpotent. Some of the highlights of our present investigation are (i) the derivation of the off-shell nilpotent versions of the (anti-)BRST charges from the standard non-nilpotent Noether conserved (anti-)BRST charges, (ii) the appearance of the operator forms of the first-class constraints at the quantum level through the physicality criteria with respect to the nilpotent versions of the (anti-)BRST charges, and (iii) the deduction of the Curci–Ferrari-type restrictions from the straightforward equality of the coupled (anti-)BRST invariant Lagrangian densities as well as from the requirement of the absolute anticommutativity of the off-shell nilpotent versions of the conserved (anti-)BRST charges. [ABSTRACT FROM AUTHOR]

We examine the gauge transformations between the three-component Kadomtsev–Petviashvili (KP) hierarchy and the three-component modified Kadomtsev–Petviashvili (mKP) hierarchy. We introduce the auto-Bäcklund transformation of the three-component KP hierarchy and describe the Miura transformation of the three-component KP and mKP hierarchies, that is, . Additionally, the auto-Bäcklund transformation of the three-component mKP hierarchy is also given. This provides more powerful evidence that the gauge transformations generate Miura and auto-Bäcklund transformation on the eigenfunctions of the three-component KP and mKP hierarchies. [ABSTRACT FROM AUTHOR]

Some characteristics of stationary flows of the Sawada–Kotera system lend themselves to generalization, producing a large class of separable Lagrangian systems with two degrees of freedom. All of these systems come in couples that have the same equations of motion, although they are not related by a gauge transform. Some nonpolynomial examples are provided. [ABSTRACT FROM AUTHOR]

This work investigates the non‐global existence of solutions for the inhomogeneous nonlinear Schrödinger problem without gauge invariance. Indeed, initially, we establish the non‐global existence of certain mass‐subcritical solutions. Subsequently, we demonstrate the non‐global existence of energy‐subcritical solutions characterized by small initial data. These results are are acknowledged to be unsuccessful for the classical gauge invariant inhomogeneous nonlinear Schrödinger equation (INLS). [ABSTRACT FROM AUTHOR]

A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such 'gauge invariance' is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems. [ABSTRACT FROM AUTHOR]

In this paper, we will discuss an alternative theory for the origin of dark matter and dark energy based on the new concept of tetrad gauge states of spacetime. The new tetrads already introduced new physics since it has been proved that local electromagnetic gauge transformations can boost the local tetrad fields in a four-dimensional curved Lorentz spacetime. It is within this context that we will present a different qualitative explanation for the so-called dark matter and dark energy and new possible topological solutions of a galactic nature for Einstein–Maxwell spacetimes in its relation to dark matter and new possible topological solutions of a cosmological nature for Einstein–Maxwell spacetimes in its relation to dark energy. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem of the one dimensional Maxwell-Schrödinger (MS) system under the Lorenz gauge condition. Different from the classical case, we consider the electromagnetic and electrostatic potentials which are growing at space infinity. More precisely, the electrostatic potential is allowed to grow linearly, while for the electromagnetic potential the growth is sublinear. Based on the energy estimates and the gauge transformation, we prove the global existence and the uniqueness of the weak solutions to this system. [ABSTRACT FROM AUTHOR]

We generalize the U(1) gauge transformations of electrodynamics by means of an analytical extension of their parameter space and observe that this leads naturally to two gauge potentials, one electric, one magnetic, which permit the writing of local Lagrangians describing elementary particles with electric and magnetic charges. Gauge invariance requires a conformal transformation of the metric tensor. We apply this approach, which borrows from Utiyama's methodology, to a model with a massless scalar field and a model with a massless spinor field. We observed that for spinor models non-symmetrized Lagrangians can enable the existence of magnetic monopoles, but this is not possible with symmetrized Lagrangian. Such restrictions do not occur for spinless fields, but the model does not allow spin-one fields interacting with monopoles. [ABSTRACT FROM AUTHOR]

We study how the symmetry operator of the axial |$U(1)$| non-invertible symmetry acts on the 't Hooft line operator in the |$U(1)$| gauge theory by employing the modified Villain-type lattice formulation. We model the axial anomaly by a compact scalar boson, the "QED axion". For the gauge invariance, the simple 't Hooft line operator, which is defined by a line integral of the dual |$U(1)$| gauge potential, must be "dressed" by the scalar and |$U(1)$| gauge fields. A careful consideration on the basis of the anomalous Ward–Takahashi identity containing the 't Hooft operator with the dressing factor and a precise definition of the symmetry operator on the lattice shows that the symmetry operator leaves no effect when it sweeps out a 't Hooft loop operator. This result appears inequivalent with the phenomenon concluded in the continuum theory. In an appendix, we demonstrate that the half-space gauging of the magnetic |$\mathbb {Z}_N$| 1-form symmetry, when formulated in an appropriate lattice framework, leads to the same conclusion as above. A similar result is obtained for the axion string operator. [ABSTRACT FROM AUTHOR]

Under an appropriate Miura-type transformation, a super modified Korteweg–de Vries (mKdV) equation, introduced by Hu in 1997, is shown to be a modification of the generalised super Korteweg–de Vries (KdV) equation. By correspondingly changing bi-Hamiltonian formulations of the latter, bi-Hamiltonian structures are established for the super mKdV equation. Moreover, from an ansatz about gauge transformations of its linear spectral problem, elementary Darboux transformations are constructed, and suitably composing them yields a binary Darboux transformation, as well as the fourth one. As a reduction of the fourth Darboux transformation, a proper Darboux transformation is obtained for Kupershmidt's super mKdV equation. [ABSTRACT FROM AUTHOR]

The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces H s for s > − 1 2 . The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard's explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy. [ABSTRACT FROM AUTHOR]

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Gauge theory compactified on an orbifold is defined by gauge symmetry, matter contents, and boundary conditions (BCs). There are equivalence classes (ECs), each of which consists of physically equivalent BCs. We propose the powerful necessary conditions, trace conservation laws (TCLs), which achieve a sufficient classification of ECs in U(N) and SU(N) gauge theories on T 2/ Zm orbifolds (m = 2, 3, 4, 6). The TCLs yield the equivalent relations between the diagonal BCs without relying on an explicit form of gauge transformations. The TCLs also show the existence of off-diagonal ECs, which consist only of off-diagonal matrices, on T 2/ Z 4 and T 2/ Z 6. After the sufficient classification, the exact numbers of ECs are obtained. [ABSTRACT FROM AUTHOR]

Recently, lattice formulations of Abelian chiral gauge theory in two dimensions have been devised on the basis of the Abelian bosonization. A salient feature of these 2D lattice formulations is that the gauge invariance is exactly preserved for anomaly-free theories and thus is completely free from the question of the gauge mode decoupling. In the present paper, we propose yet another lattice formulation sharing this desired property. A particularly unique point in our formulation is that the vertex operator of the dual scalar field, which carries the vector charge of the fermion and the "magnetic charge" in the bosonization, is represented by a "hole" excised from the lattice; this is the excision method formulated recently by Abe et al. in a somewhat different context. [ABSTRACT FROM AUTHOR]

It has been known for more than a decade that the gauge variance of the kinetic energy density τ leads to additional terms in the magnetic orbital rotation Hessian used in linear-response time-dependent density functional theory (TDDFT), affecting excitation energies obtained with τ-dependent exchange–correlation functionals. While previous investigations found that a correction scheme based on the paramagnetic current density has a small effect on benchmark results, we report more pronounced effects here, in particular, for the popular M06-2X functional and for some other meta-generalized gradient approximations (mGGAs). In the first part of this communication, this is shown by a reassessment of a set of five Ni(II) complexes for which a previous benchmark study that did not impose gauge invariance has found surprisingly large errors for excitation energies obtained with M06-2X. These errors are more than halved by restoring gauge invariance. The variable importance of imposing gauge invariance for different mGGA-based functionals can be rationalized by the derivative of the mGGA exchange energy integrand with respect to τ. In the second part, a large set of valence excitations in small main-group molecules is analyzed. For M06-2X, several selected n → π* and π → π ⊥ * excitations are heavily gauge-dependent with average changes of −0.17 and −0.28 eV, respectively, while π → π ‖ * excitations are marginally affected (−0.04 eV). Similar patterns, but of the opposite signs, are found for SCAN0. The results suggest that reevaluation of previous gauge variant TDDFT results based on M06-2X and other mGGA functionals is warranted. [ABSTRACT FROM AUTHOR]

A new (2 + 1)-dimensional generalized KP-type equation and its Lax pair are established. Based on gauge transformation between spectral problems, a Darboux transformation for the (2 + 1)-dimensional generalized KP-type equation is obtained. Some exact solutions are derived by utilizing the formula of Darboux transformation, including soliton solutions, trigonometric function solution, lump solutions, lump-soliton solution. In order to analyze the dynamical behavior of the solutions, the plots of solutions are depicted by Mathematical software. [ABSTRACT FROM AUTHOR]

We examine the model of two-dimensional quadratic gravity as a consequence of symmetry breaking within the framework of background field (BF) theory. This theory is essentially an extension of BF theory, introducing an additional polynomial term that operates on both the gauge and background fields. We analyze the theory using the Dirac and Faddeev–Jackiw procedures, determining the form of the gauge transformation, the full structure of the constraints, the counting of degrees of freedom, and the generalized Faddeev–Jackiw brackets. Additionally, we demonstrate the coincidence of the Faddeev–Jackiw and Dirac's brackets. Finally, we provide some remarks and discuss prospects. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two types of U(1)-gauge field theories on G 2 -manifolds. One is the U(1)-Yang–Mills theory which admits the classical instanton solutions. We show that G 2 -manifolds emerge from the anti-self-dual U(1) instantons, which is an analogy of Yang's result for Calabi–Yau manifolds. The other one is the higher-order U(1)-Chern–Simons theory as a generalization of Kähler–Chern–Simons theory. We introduce the notion of higher-order U(1)-instanton, as the vacuum configurations of higher-order U(1)-Chern–Simons theory. By suitable choice of gauge and regularization technique, we calculate the partition function under semiclassical approximation. Finally, to make sure of the invariance at quantum level under the large gauge transformations, we use Deligne–Beilinson cohomology theory to give the higher-order U(1)-Chern–Simons actions (U(1)-BF-type actions) for nontrivial U(1)-principle bundles. [ABSTRACT FROM AUTHOR]

Gradient Flow Exact Renormalization Group (GF-ERG) is a framework to define the renormalization group flow of Wilsonian effective action utilizing coarse-graining along the diffusion equations. We apply it for Scalar Quantum Electrodynamics and derive flow equations for the Wilsonian effective action with the perturbative expansion in the gauge coupling. We focus on the quantum corrections to the correlation functions up to the second order of the gauge coupling and discuss the gauge invariance of the GF-ERG flow. We demonstrate that the anomalous dimension of the gauge field agrees with the standard perturbative computation and that the mass of the photon keeps vanishing in general spacetime dimensions. The latter is a noteworthy fact that contrasts with the conventional Exact Renormalization Group formalism in which an artificial photon mass proportional to a cutoff scale is induced. Our results imply that the GF-ERG can give a gauge-invariant renormalization group flow in a non-perturbative way. [ABSTRACT FROM AUTHOR]

Poisson–Lie T-duality/plurality was recently generalized to Jacobi–Lie T-plurality formulated in terms of double field theory and based on Leibniz algebras given by the structure coefficients fab c , fc ab , and Za , Z a . We investigate three- and four-dimensional sigma models corresponding to six-dimensional Leibniz algebras with fb ba ≠ 0, Z a = 0. We show that these algebras are plural one to another and, moreover, to an algebra with fb ba = 0, Z a = 0. These pluralities are used for construction of Jacobi–Lie models. It was conjectured that plural models should satisfy generalized supergravity equations. We have found examples of models satisfying "true" generalized supergravity equations where no trivialization to usual supergravity equations is possible. On the other hand, we show that there are also models corresponding to algebras with fb ba ≠ 0, Z a = 0 where the Killing vector appearing in generalized supergravity equations either vanishes or can be removed by suitable gauge transformation. Such models then satisfy usual supergravity equations, i.e. vanishing beta-function equations. [ABSTRACT FROM AUTHOR]

In this paper, we will analyze the relationship between both potentials in vacuum electrodynamics and their local gauge transformations. [ABSTRACT FROM AUTHOR]

U (n) is a semi-direct product group characterized by nontrivial homomorphisms mapping U (1) into the automorphism group of S U (n) . For U (3) , there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of U (3) Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a mod 3 permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same U (3) quantum numbers. One can interpret this as three species of matter fields in the defining representation. [ABSTRACT FROM AUTHOR]

Sen's action for a p-form gauge field with self-dual field strength coupled to a spacetime metric g involves an explicit Minkowski metric and the presence of this raises questions as to whether the action is coordinate independent and whether it can be used on a general spacetime manifold. A natural generalisation of Sen's action is presented in which the Minkowski metric is replaced by a second metric on spacetime. The theory is covariant and can be formulated on any spacetime. The theory describes a physical sector, consisting of the chiral p-form gauge field coupled to the dynamical metric g, plus a shadow sector consisting of a second chiral p-form and the second metric . The fields in this shadow sector only couple to each other and have no interactions with the physical sector, so that they decouple from the physical sector. The resulting theory is covariant and can be formulated on any spacetime. Explicit expressions are found for the interactions and extensions to include interactions with other physical fields or higher-derivative field equations are given. A spacetime with two metrics has some interesting geometry and some of this is explored here and used in the construction of the interactions. The action has two diffeomorphism-like symmetries, one acting only on the physical sector and one acting only on the shadow sector, with the spacetime diffeomorphism symmetry arising as the diagonal subgroup. This allows a further generalisation in which is not a tensor field but is instead a gauge field whose transition functions involve the usual coordinate transformation together with a shadow sector gauge transformation. [ABSTRACT FROM AUTHOR]

We study the Hamiltonian structure of unimodular-like theories, where the cosmological constant (or other supposed constants of nature) are demoted from fixed parameters to classical constants of motion. No new local degrees of freedom are present as a result of a U(1) gauge invariance of the theory. Hamiltonian analysis of the action reveals that the only possible gauge fixing that can be enforced is setting the spatial components of the four-volume time vector T i ≈ 0 . As a consequence of this, the gauge-fixed unimodular path integral is equivalent to the minisuperspace unimodular path integral. However, should we break the U(1) gauge invariance, two things happen: a massless propagating degree of freedom appears, and the (gauge-invariant) zero-mode receives modified dynamics. The implications are investigated, with the phenomenology depending crucially on the target "constant". [ABSTRACT FROM AUTHOR]

It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant. [ABSTRACT FROM AUTHOR]

In U (1) lattice gauge theory with compact U (1) variables, we construct the symmetry operator, i.e. the topological defect, for the axial U (1) noninvertible symmetry. This requires a lattice formulation of chiral gauge theory with an anomalous matter content and we employ the lattice formulation on the basis of the Ginsparg–Wilson relation. The invariance of the symmetry operator under the gauge transformation of the gauge field on the defect is realized, imitating the prescription by Karasik in continuum theory, by integrating the lattice Chern–Simons term on the defect over smooth lattice gauge transformations. The projection operator for allowed magnetic fluxes on the defect then emerges with lattice regularization. The resulting symmetry operator is manifestly invariant under lattice gauge transformations. In an appendix, we give another way of constructing the symmetry operator on the basis of a 3D |$\mathbb {Z}_N$| topological quantum field theory, the level- N BF theory on the lattice. [ABSTRACT FROM AUTHOR]

This is the first in a series of papers on projective positive energy representations of gauge groups. Let Ξ→M be a principal fiber bundle, and let Γc(M,Ad(Ξ)) be the group of compactly supported (local) gauge transformations. If P is a group of 'space-time symmetries' acting on Ξ→M, then a projective unitary representation of Γc(M,Ad(Ξ))⋊P is of positive energy if every 'timelike generator' p0∈p gives rise to a Hamiltonian H(p0) whose spectrum is bounded from below. Our main result shows that in the absence of fixed points for the cone of timelike generators, the projective positive energy representations of the connected component Γc(M,Ad(Ξ))0 come from 1-dimensional P-orbits. For compact M this yields a complete classification of the projective positive energy representations in terms of lowest weight representations of affine Kac-Moody algebras. For noncompact M, it yields a classification under further restrictions on the space of ground states. In the second part of this series we consider larger groups of gauge transformations, which contain also global transformations. The present results are used to localize the positive energy representations at (conformal) infinity. [ABSTRACT FROM AUTHOR]

In this paper, it has been proven that locally the inertial frames and gauge states of the electromagnetic field are equivalent. This proof is valid for Einstein–Maxwell theories in four-dimensional Lorentzian spacetimes. Theorems proved in a previous paper will be used. These theorems state that locally the group of electromagnetic gauge transformations is isomorphic to the local group of Lorentz transformations of a special set of tetrad vectors. The tetrad that locally and covariantly diagonalizes any non-null electromagnetic stress–energy tensor. There are in total two isomorphisms, one for each orthogonal plane of stress–energy eigenvectors. We discuss the opposite problem in this paper. What happens with local electromagnetic gauge when the test object under study is boosted by any mechanical means? We will prove that boosting matter is indistinguishable from introducing an appropriate local electromagnetic gauge transformation. [ABSTRACT FROM AUTHOR]

Phenomenological Poincaré invariant quantum mechanical models can provide an efficient description of the dynamics of strongly interacting particles that is frame independent and consistent with spectral and scattering observables. These models are representation dependent and in order to apply them to reactions with electromagnetic probes it is necessary to use a consistent electromagnetic current operator. The purpose of this work is to use local gauge invariance to construct consistent strong current operators. Current operators are constructed from a model Hamiltonian by replacing momentum operators in the Weyl representation by gauge covariant derivatives. The construction provides a systematic method to construct expressions for current operators that are consistent with relativistic models of strong interaction dynamics. [ABSTRACT FROM AUTHOR]

We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative U(1) gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena. [ABSTRACT FROM AUTHOR]

In 5D and 6D U (N) and SU (N) gauge theories compactified on S 1/ Z 2 and T 2/ Z 3 orbifolds, we propose a new method to classify the equivalence classes (ECs) of boundary conditions (BCs) without depending on the structure of gauge transformations. Some of the BCs are connected through gauge transformations and constitute ECs, each of which contains physically equivalent BCs. Previous methods for classifying ECs have been used for specific gauge transformations. In this paper, we show that a geometric property of orbifolds significantly narrows down the possibilities of connecting BCs and completes the classification of ECs. [ABSTRACT FROM AUTHOR]

Meta-generalized gradient approximations (meta-GGAs) and local hybrid functionals generally depend on the kinetic energy density τ. For magnetic properties, this necessitates generalizations to ensure gauge invariance. In most implementations, τ is generalized by incorporating the external magnetic field. However, this introduces artifacts in the response of the density matrix and does not satisfy the iso-orbital constraint. Here, we extend previous approaches based on the current density to paramagnetic nuclear magnetic resonance (NMR) shieldings and electron paramagnetic resonance (EPR) g-tensors. The impact is assessed for main-group compounds and transition-metal complexes considering 25 density functional approximations. It is shown that the current density leads to substantial improvements—especially for the popular Minnesota and strongly constrained and appropriately normed (SCAN) functional families. Thus, we strongly recommend to use the current density generalized τ in paramagnetic NMR and EPR calculations with meta-GGAs. [ABSTRACT FROM AUTHOR]

Chemical relaxation phenomena, including photochemistry and electron transfer processes, form a vigorous area of research in which nonadiabatic dynamics plays a fundamental role. However, for electronic systems with spin degrees of freedom, there are few if any applicable and practical quasiclassical methods. Here, we show that for nonadiabatic dynamics with two electronic states and a complex-valued Hamiltonian that does not obey time-reversal symmetry (as relevant to many coupled nuclear-electronic-spin systems), the optimal semiclassical approach is to generalize Tully's surface hopping dynamics from coordinate space to phase space. In order to generate the relevant phase-space adiabatic surfaces, one isolates a proper set of diabats, applies a phase gauge transformation, and then diagonalizes the total Hamiltonian (which is now parameterized by both R and P). The resulting algorithm is simple and valid in both the adiabatic and nonadiabatic limits, incorporating all Berry curvature effects. Most importantly, the resulting algorithm allows for the study of semiclassical nonadiabatic dynamics in the presence of spin–orbit coupling and/or external magnetic fields. One expects many simulations to follow as far as modeling cutting-edge experiments with entangled nuclear, electronic, and spin degrees of freedom, e.g., experiments displaying chiral-induced spin selectivity. [ABSTRACT FROM AUTHOR]

The covariance of the one-electron Schrödinger equation in a gauge transformation of electromagnetic potentials is customarily discussed, introducing the relationships that describe their changes, induced by a generating function of position coordinates and time, together with a phase transformation of the wavefunction, according to a proposal by Fock, i.e.,2 relying on three distinct hypotheses. The present note shows that (i) a single similarity transformation employing the Fock phase factor is sufficient to prove the covariance of the wave equation and (ii) the Fock transformations of the nonperturbed Hamiltonian h ̂ (0) and of the time-derivative operator determine the changes appearing in first and second-order Hamiltonians, h ̂ (1) and h ̂ (2) , a single term being obtained by Fock-transforming h ̂ (1) . [ABSTRACT FROM AUTHOR]

After some years of controversy, it was recently demonstrated how to obtain the correct long-distance limit [point-dipole approximation (PDA)] of pseudo-contact nuclear magnetic resonance chemical shifts from rigorous first-principles quantum mechanics [Lang et al., J. Phys. Chem. Lett. 11, 8735 (2020)]. This result confirmed the classical Kurland–McGarvey theory. In the present contribution, we elaborate on these results. In particular, we provide a detailed derivation of the PDA both from the Van den Heuvel–Soncini equation for the chemical shielding tensor and from a spin Hamiltonian approximation. Furthermore, we discuss in detail the PDA within the approximate density functional theory and Hartree–Fock theories. In our previous work, we assumed a relatively crude effective nuclear charge approximation for the spin–orbit coupling operator. Here, we overcome this assumption by demonstrating that the derivation is also possible within the fully relativistic Dirac equation and even without the assumption of a specific form for the Hamiltonian. Crucial ingredients for the general derivation are a Hamiltonian that respects gauge invariance, the multipolar gauge, and functional derivatives of the Hamiltonian, where it is possible to identify the first functional derivative with the electron number current density operator. The present work forms an important foundation for future extensions of the Kurland–McGarvey theory beyond the PDA, including induced magnetic quadrupole and higher moments to describe the magnetic hyperfine field. [ABSTRACT FROM AUTHOR]

For the Lax-type Hamiltonian flows in the space dual to the Lie algebra of fractional integrodifferential operators, we develop a rational factorization method, which enables one to get new integrable hierarchies of nonlinear fractional-differential dynamical systems on the Lie algebra of ordinary integrodifferential operators and obtain infinite sequences of their conservation laws. By using the Bäcklund transformation, we show that the system of two flows of this kind for a pair of fractional integrodifferential operators related by a gauge transformation is equivalent to a system of two evolutionary equations for fractional differential operators, which determine the corresponding rational factorization, and find the Hamiltonian representation for this system of evolutionary equations. We show that its quasiclassical approximation is a system of two evolutionary equations for polynomials of fractional degree in a certain complex parameter. The proposed method is used to construct a new integrable hierarchy of nonlinear fractionaldifferential systems on the Lie algebra of ordinary integrodifferential operators and an infinite sequence of its conservation laws, as well as a new integrable hierarchy of Benney-type hydrodynamic systems, which is its quasiclassical approximation. [ABSTRACT FROM AUTHOR]

We consider a ferromagnetic nanowire, with an energy functional |$E$| with easy-axis in the direction |$e_{1}$|⁠ , and which takes into account the Dzyaloshinskii–Moriya interaction. We consider configurations of the magnetization that are perturbations of two well-separated domain wall and study their evolution under the Landau–Lifshitz–Gilbert flow associated to |$E$|⁠. Our main result is that, if the two walls have opposite speed, these configurations are asymptotically stable, up to gauges intrinsic to the invariances of the energy |$E$|⁠. Our analysis builds on the framework developed in [ 4 ], taking advantage that it is amenable to space localisation. [ABSTRACT FROM AUTHOR]

Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations, converting the defined Lax pairs to some simpler forms, are found. The direct method and its possible applications to other types of evolution equations are discussed. [ABSTRACT FROM AUTHOR]

We study the perturbations up to the 2nd-order for a power-law inflation driven by a scalar field in synchronous coordinates. We present the 1st-order solutions, and analytically solve the 2nd-order perturbed Einstein equation and scalar field equation, give the 2nd-order solutions for all the scalar, vector, and tensor metric perturbations, as well as the perturbed scalar field. During inflation, the 1st-order tensor perturbation is a wave and is decoupled from other perturbations, the scalar metric perturbation and the perturbed scalar field are coupled waves, propagating at the speed of light, differing from those in the dust and relativistic fluid models. The 1st-order vector perturbation is not wave and just decreases during inflation. The 2nd-order perturbed Einstein equation is similar in structure to the 1st-order one, but various products of the 1st-order perturbations occur as the effective source, among which the scalar–scalar coupling is considered in this paper. The solutions of all the 2nd-order perturbations consist of a homogeneous part similar to the 1st-order solutions, and an inhomogeneous part in a form of integrations of the effective source. The 2nd-order vector perturbation is also a wave since the effective source is composed of the 1st-order waves. We perform the residual gauge transformations between synchronous coordinates up to the 2nd-order, and identify the 1st-order and 2nd-order gauge modes. [ABSTRACT FROM AUTHOR]

We consider Yang–Mills theory with a compact structure group G on a Lorentzian 4-manifold M = ℝ × Σ such that gauge transformations become identity on a submanifold S of Σ (framing over S ⊂ Σ). The space S is not necessarily a boundary of Σ and can have dimension k ≤ 3. Framing of gauge bundles over S ⊂ Σ demands introduction of a G -valued function ϕ S with support on S and modification of Yang–Mills equations along ℝ × S ⊂ M. The fields ϕ S parametrize non-equivalent flat connections mapped into each other by a dynamical group S changing gauge frames over S. It is shown that the charged condensate ϕ S is the Stueckelberg field generating an effective mass of gluons in the domain S of space Σ and keeping them massless outside S. We argue that the local Stueckelberg field ϕ S can be responsible for color confinement. We also briefly discuss local breaking of symmetries in gravity. It is shown that framing of the tangent bundle over a subspace of spacetime makes gravitons massive in this subspace. [ABSTRACT FROM AUTHOR]

In this paper, we formulate, for the first time, in a systematic manner, Galilean relativistic Born–Infeld action in detail. Exploiting maps connecting Lorentz relativistic and Galilean relativistic vectors, we construct the two limits (electric and magnetic) of Galilean relativistic Born–Infeld action from usual relativistic Born–Infeld theory. An action formalism is thereby derived. From this action, equations of motion are obtained either in the potential or field formulation. Galilean version of duality transformations involving the electric and magnetic fields are defined. They map the electric limit relations to the magnetic ones and vice-versa, exactly as happens for Galilean relativistic Maxwell theory. We also explicitly show the Galilean boost and gauge invariances of the theory in both limits. [ABSTRACT FROM AUTHOR]

The conformal version of the antisymmetric second-order tensor field in four spacetime dimensions does not have gauge invariance extensively discussed in the literature for more than half a century. Our first observation is that, when coupled to fermions, only the conformal version provides renormalizability of the theory at the one-loop level. General considerations are supported by the derivation of one-loop divergences in the fermionic sector, indicating good chances for asymptotic freedom. The arguments concerning one-loop renormalizability remain valid in the presence of self-interactions and the masses for both fermion and antisymmetric tensor fields. In the flat spacetime limit, even regardless the conformal symmetry has gone, there is an expectation to meet renormalizability in all loop orders. [ABSTRACT FROM AUTHOR]

The universality of classical thermodynamics rests on the central limit theorem, due to which, measurements of thermal fluctuations are unable to reveal detailed information regarding the microscopic structure of a macroscopic body. When small systems are considered and fluctuations become important, thermodynamic quantities can be understood in the context of classical stochastic mechanics. A fundamental assumption behind thermodynamics is therefore that of coarse graining, which stems from a substantial lack of control over all degrees of freedom. However, when quantum systems are concerned, one claims a high level of control. As a consequence, information theory plays a major role in the identification of thermodynamic functions. Here, drawing from the concept of gauge symmetry—essential in all modern physical theories—we put forward a new possible intermediate route. Working within the realm of quantum thermodynamics, we explicitly construct physically motivated gauge transformations which encode a gentle variant of coarse graining behind thermodynamics. As a first application of this new framework, we reinterpret quantum work and heat, as well as the role of quantum coherence. [ABSTRACT FROM AUTHOR]