Nonlinear Realization Research Articles

Group-theoretic approach to perfect fluid equations with conformal symmetry

The method of nonlinear realizations is a convenient tool for building dynamical realizations of a Lie group, which relies solely upon structure relations of the corresponding Lie algebra. The goal of this work is to discuss advantages and limitations of the method, which is here applied to construct perfect fluid equations with conformal symmetry. Four cases are studied in detail, which include the Schrodinger group, the l-conformal Galilei group, the Lifshitz group, and the relativistic conformal group.

Higher dualisations of linearised gravity and the {A}_1^{+++} algebra

The non-linear realisation based on {A}_1^{+++} is known to describe gravity in terms of both the graviton and the dual graviton. We extend this analysis at the linearised level to find the equations of motion for the first higher dual description of gravity that it contains. We also give a systematic method for finding the additional fields beyond those in the non-linear realisation that are required to construct actions for all of the possible dual descriptions of gravity in the non-linear realisation. We show that these additional fields are closely correlated with the second fundamental representation of {A}_1^{+++} .

Kepler problem and chiral effective dynamics.

It is shown that by an appropriate canonical transformation, Kepler dynamics can be put in the form which allows one to exhibit the structure of the symmetry transformations related to the superintegrability. They appear to fit nicely into a general scheme of nonlinear realizations. In new coordinates, the Kepler dynamics results from dimensional reduction of that describing low-energy mesons with spontaneously broken chiral symmetry.

Dynamical realizations of the Lifshitz group

Dynamical realizations of the Lifshitz group are studied within the group-theoretic framework. A generalization of the 1d conformal mechanics is constructed, which involves an arbitrary dynamical exponent z. A similar generalization of the Ermakov-Milne-Pinney equation is proposed. Invariant derivative and field combinations are introduced, which enable one to construct a plethora of dynamical systems enjoying the Lifshitz symmetry. A metric of the Lorentzian signature in (d+2)-dimensional spacetime and the energy-momentum tensor are constructed, which lead to the generalized Ermakov-Milne-Pinney equation upon imposing the Einstein equations. The method of nonlinear realizations is used for building Lorentzian metrics with the Lifshitz isometry group. In particular, a (2d+2)-dimensional metric is constructed, which enjoys an extra invariance under the Galilei boosts.

Nonlinear realizations of Lie superalgebras

For any decomposition of a Lie superalgebra into a direct sum of a subalgebra and a subspace without any further resctrictions on and we construct a nonlinear realization of on The result generalizes a theorem by Kantor from Lie algebras to Lie superalgebras. When is a differential graded Lie algebra, we show that it gives a construction of an associated -algebra.

Beyond the standard model effective field theory with b→cτ−ν¯

Electroweak interactions assign a central role to the gauge group $SU(2{)}_{L}\ifmmode\times\else\texttimes\fi{}U(1{)}_{Y}$, which is either realized linearly (SMEFT) or nonlinearly (e.g., HEFT) in the effective theory obtained when new physics above the electroweak scale is integrated out. Although the discovery of the Higgs boson has made SMEFT the default assumption, nonlinear realization remains possible. The two can be distinguished through their predictions for the size of certain low-energy dimension-6 four-fermion operators: for these, HEFT predicts $O(1)$ couplings, while in SMEFT they are suppressed by a factor ${v}^{2}/{\mathrm{\ensuremath{\Lambda}}}_{\mathrm{NP}}^{2}$, where $v$ is the Higgs vev. One such operator, ${O}_{V}^{LR}\ensuremath{\equiv}(\overline{\ensuremath{\tau}}{\ensuremath{\gamma}}^{\ensuremath{\mu}}{P}_{L}\ensuremath{\nu})(\overline{c}{\ensuremath{\gamma}}_{\ensuremath{\mu}}{P}_{R}b)$, contributes to $b\ensuremath{\rightarrow}c{\ensuremath{\tau}}^{\ensuremath{-}}\overline{\ensuremath{\nu}}$. We show that present constraints permit its non-SMEFT coefficient to have a HEFTy size. We also note that the angular distribution in $\overline{B}\ensuremath{\rightarrow}{D}^{*}(\ensuremath{\rightarrow}D{\ensuremath{\pi}}^{\ensuremath{'}}){\ensuremath{\tau}}^{\ensuremath{-}}(\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\nu}}_{\ensuremath{\tau}}){\overline{\ensuremath{\nu}}}_{\ensuremath{\tau}}$ contains enough information to extract the coefficients of all new-physics operators. Future measurements of this angular distribution can therefore tell us if non-SMEFT new physics is really necessary.

SO(5) Landau model and 4D quantum Hall effect in the SO(4) monopole background

We investigate the $SO(5)$ Landau problem in the $SO(4)$ monopole gauge field background by applying the techniques of the non-linear realization of quantum field theory. The $SO(4)$ monopole carries two topological invariants, the second Chern number and a generalized Euler number, specified by the $SU(2)$ monopole and anti-monopole indices, $I_+$ and $I_-$. The energy levels of the $SO(5)$ Landau problem are grouped into $\text{Min}(I_+, I_-) +1$ sectors, each of which holds Landau levels. In the $n$-sector, $N$th Landau level eigenstates constitute the $SO(5)$ irreducible representation with $(p,q)_5=(N+I_+ + I_--n, N+n)_5$ whose function form is obtained from the $SO(5)$ non-linear realization matrix. In the $n=0$ sector, the emergent quantum geometry of the lowest Landau level is identified as the fuzzy four-sphere with radius being proportional to the difference between $I_+$ and $I_-$. The Laughlin-like wavefunction is constructed by imposing the $SO(5)$ lowest Landau level projection to the many-body wavefunction made of the Slater determinant. We also analyze the relativistic version of the $SO(5)$ Landau model to demonstrate the Atiyah-Singer index theorem in the $SO(4)$ gauge field configuration.

(Super)Schwarzian mechanics

In this paper we revisit the construction of supersymmetric Schwarzians using nonlinear realizations. We show that mathcal{N} = 0, 1, 2, 3, 4 supersymmetric Schwarzians can be systematically obtained as certain projections of Maurer-Cartan forms of superconformal groups after imposing simple conditions on them. We also present the supersymmetric Schwarzian actions defined as the integrals of products of Cartan forms. In contrast with the previous attempts to obtain the super-Schwarzians within nonlinear realizations technique, our set of constraints does not impose any restriction on the super-Schwarzians.

Constrained superfields in dynamical background

We study the nonlinear realization of supersymmetry in a dynamical/cosmological background in which derivative terms like kinetic terms are finite. Starting from linearly realized theories, we integrate out heavy modes without neglecting derivative terms to obtain algebraic constraints on superfields. Thanks to the supersymmetry breaking contribution by the kinetic energy, the validity of constrained superfields can be extended to cosmological regimes and phenomena such as reheating after inflation, kinetic-energy domination, and the kinetic and standard misalignment of axion.

Gravity Models with Nonlinear Symmetry Realization

Validity of three gravity models with non-linear realization of conformal symmetry previously discussed in literature is addressed. Two models are found to be equivalent up to a change of coset coordinates. It was found that models contain ghost degrees of freedom that may be excluded by an introduction of an additional symmetry to the target space. One model found to be safe in early universe. The others found to lack spin-2 degrees of freedom and to have peculiar coupling to matter degrees of freedom.

Dynamical symmetries, coherent states and nonlinear realizations: Towards noncommutative gravity

Based on the gravity model as a gauge theory previously by the authors, the introduction of noncommutative structures in the very geometry is discussed and developed in a new different way. Taking into account a linear Lagrangian in curvature and considering a pair of coherent vectors responsible, among other things, for the symmetry breaking, the introduction of noncommutativity in the very structure of the geometry is achieved. The fact that the vectors are coherent states ensure not only the natural and total quantization of the model, but also the formulation of the noncommutative structure, in particular by the introduction of a star product from the convolutory properties thereof. This new star product, in a sharp contrast with other proposals, is independent of the Bargmann index or other parameters depending on the dimension of the representation of the structure group of the noncommutativity. We also explain in the same way that a matrix model, through the properties of these coherent vectors, can be easily formulated. The pros and cons of the implementation of star products versus a theory of matrices based on coset coherent states are briefly discussed.

Change in Hamiltonian General Relativity with Spinors

In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence (lack of a time-like Killing vector field), one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time philosophy tends to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a 3+1 version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate 3+1-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory (unlike, say, massive photons), it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors.

What is the most suitable Lyapunov function?

• Algorithm for Hamilton energy function in dynamical system is clarified. • Hamilton energy function is the most suitable Lyapunov function for dynamical control. • Control of energy flow is the most effective way to synchronization control of chaotic systems. Lyapunov function provides feasible estimation and prediction of nonlinear system stability, and useful guidance for adaptive control in chaos and synchronization approach. In case of synchronization and control of chaotic systems, the involvement of adjustable gains in the Lyapunov function can be effective to optimize the convergence of orbits to stability and controllers within finite transient period. As a result, shorter transient period and lower power consumption can be approached by detecting the most suitable gains in the controllers and parameter observers. In this paper, we claim that the most suitable Lyapunov function can be the Hamilton energy for chaotic systems and more nonlinear dynamical systems, and so the parameter region for stability and controllability can be detected exactly, in addition, the reliability of controllers can be confirmed in practical way. Furthermore, the Lorenz and improved Chua oscillators in chaotic states are presented to confirm the dependence of Hamilton energy and stability on the intrinsic parameters and variables. It indicates that control of energy flow can be an effective scheme to control chaos in nonlinear systems and synchronization realization between chaotic systems, neurons and networks.

Electroweak effective theory and beyond Standard Model resonances

We consider a non-linear realization of the electroweak symmetry-breaking pattern S U(2)L × S U(2)R/S U(2)L+R to construct a low-energy effective theory, later extended by the inclusion of heavy new-physics resonances. After assuming appropriate high-energy constraints given by Weinberg sum-rules and the asymptotic behaviour of form-factors, we obtain relations between resonance masses and some low-energy effective couplings. These predictions are compared with current experimental data and some resonance mass bounds are inferred.

Remarks on a D(2,1;a) super-Schwarzian derivative

It was recently demonstrated that N=1,2,3,4 super-Schwarzian derivatives can be constructed by applying the method of nonlinear realisations to finite-dimensional superconformal groups OSp(1|2), SU(1,1|1), OSp(3|2), SU(1,1|2), respectively, thus avoiding the use of superconformal field theory techniques. In this work, a similar construction is applied to the exceptional supergroup D(2,1;a), which describes the most general N=4 supersymmetric extension of SL(2,R), with the aim to study possible candidates for a D(2,1;a) super-Schwarzian derivative.

Estimating extreme characteristics of stochastic non-linear systems

The nature of the ocean is random and, as a result, marine structures face extreme, non-linear load effects. Estimating extreme responses of said structures often reduces to purely probabilistic approaches involve a considerable amount of conjecture. A method is developed in this paper to conserve original system information without the need for high fidelity Monte Carlo Simulations. Here, an iterative linearization of a non-linear system is developed to estimate extremes for the non-linear system. The Matched Upcrossing Equivalent Linear System (MUELS) method finds linear systems with zero-upcrossing periods equivalent to that of the non-linear system under a specified forcing spectrum. These linear systems are used as input into the Design Loads Generator, where ensembles of extreme linear realizations at the exposure period of interest, and the input time series that lead to those extremes, are generated. These input time series are valid inputs into the non-linear system and create a set of non-linear realizations that approximate the set of non-linear extremes at the exposure period of interest. As an example, this paper introduces and applies this method to the Duffing Oscillator at various levels of increasing cubic stiffness, forced by an ITTC wave spectrum, for exposure periods as long as 108 hours representing O(1010) zero-crossing maxima. Quantitative comparisons are made with Monte Carlo simulations, GEVD estimates, and expected values of time series near extreme local maxima.

Non-relativistic and Carrollian limits of Jackiw-Teitelboim gravity

We construct the non-relativistic and Carrollian versions of Jackiw-Teitelboim gravity. In the second order formulation, there are no divergences in the non-relativistic and Carrollian limits. Instead, in the first order formalism, some divergences can be avoided by starting from a relativistic BF theory with (A)dS2 × ℝ gauge algebra. We show how to define the boundary duals of the gravity actions using the method of non-linear realisations and suitable Inverse Higgs constraints. In particular, the non-relativistic version of the Schwarzian action is constructed in this way. We derive the asymptotic symmetries of the theory, as well as the corresponding conserved charges and Newton-Cartan geometric structure. Finally, we show how the same construction applies to the Carrollian case.

Tangent space symmetries in general relativity and teleparallelism

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonormality. We borrow techniques from the methodology of non-linear realizations to analyze this group-subgroup structure. “Parallel maps” are used to relate tangent space at different points. “Parallelisms” across a finite region of the manifold may be built up from them. These are used to define Weitzenböck connections and Levi-Civita connections. This provides a new formulation of teleparallel gravity, in which the tetrad field is viewed as a field-valued group element relating the coordinate basis to the frame basis used in defining a parallelism. This formulation separates the metric degrees of freedom from those associated with the choice of parallelism. The group element can be combined by matrix multiplication with Lorentz transformations of frame or with other Jacobian matrices. We show how this facilitates a new understanding of inertial forces and local Lorentz transformations. The analysis is also applied to translations of the coordinates. If they are constant across spacetime, this has no effect on the tangent space bases. If the translation parameters become fields, they induce general linear transformations of the coordinate basis; however, the tetrad components can only be expressed in terms of translations on a flat spacetime.

Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems With Unknown Dynamics

Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This letter describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm is constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.

Chiral waves on the Fermi-Dirac sea: Quantum superfluidity and the axial anomaly

We show that as a result of the axial anomaly, massless fermions at zero temperature define a relativistic quantum superfluid. The anomaly pole implies the existence of a gapless Chiral Density Wave (CDW), i.e. an axion-like acoustic mode of an irrotational and dissipationless Hamiltonian perfect fluid, that is a correlated fermion/anti-fermion pair excitation of the Fermi-Dirac sea. In D=2 dimensions the chiral superfluid effective action coincides with that of the Schwinger model as e→0, and the CDW acoustic mode is precisely the Schwinger boson. Since this identity holds also at zero chiral chemical potential, the Dirac vacuum itself may be viewed as a quantum superfluid state. The CDW collective boson is a U(1) chiral phase field, which is gapless as a result of a novel, non-linear realization of Goldstone's theorem, extended to this case of symmetry breaking by an anomaly. A new local form of the axial anomaly bosonic effective action in any D even spacetime is given, consistent with superfluidity, and its quantization is shown to be required by the anomalous Schwinger terms in fermion current commutators. In QED4 this collective Goldstone mode appears as a massless pole in the axial anomaly triangle diagram, and is responsible for the macroscopic non-dissipative currents of the Chiral Magnetic and Chiral Separation Effects, as well as the Anomalous Hall Effect. In a constant uniform magnetic field an exact dimensional reduction from D=4 to D=2 occurs and the collective e+e− CDW chiral pair excitation propagating along the magnetic field direction is a Chiral Magnetic Wave, which acquires a mass gap M2=e3B/2π2. Possible realizations and tests of the theory of collective bosonic excitations due to the anomaly in Dirac/Weyl materials are briefly discussed.