Generalised Symmetries and the Ermakov-Lewis Invariant

Abstract

Generalised symmetries and point symmetries coincide for systems of first-order ordinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the system of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has the form of the famous Ermakov-Lewis invariant, but in fact reveals a richer structure.     

The parametric orbits and the form invariance of three-body in one-dimension

In this paper, the differential equations of motion of a three-body interacting pairwise by inverse cubic forces(“centrifugal potential”) in addition to linear forces (“harmonical potential”) are expressed in Ermakov formalism in two-dimension polar coordinates, and the Ermakov invariant is obtained. By rescaling of the time variable and the space coordinates, the parametric orbits of the three bodies are expressed in terms of relative energy H1 and Ermakov invariant. The form invariance of the transformations of two conserved quantities are also studied.     

Lie transformation method on quantum state evolution of a general time-dependent driven and damped parametric oscillator

A variety of dynamics in nature and society can be approximately treated as a driven and damped parametric oscillator. An intensive investigation of this time-dependent model from an algebraic point of view provides a consistent method to resolve the classical dynamics and the quantum evolution in order to understand the time-dependent phenomena that occur not only in the macroscopic classical scale for the synchronized behaviors but also in the microscopic quantum scale for a coherent state evolution. By using a Floquet U-transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamic behaviors of a driven and damped parametric oscillator to obtain the optimal solutions by means of invariant parameters of K$K$s to combine with Lewis–Riesenfeld invariant method. This approach can discriminate the external dynamics from the internal evolution of a wave packet by producing independent parametric equations that dramatically facilitate the parametric control on the quantum state evolution in a dissipative system. In order to show the advantages of this method, several time-dependent models proposed in the quantum control field are analyzed in detail.     

YM2:Continuum expectations,lattice convergence,and lassos

The two dimensional Yang-Mills theory (YM₂) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM₂-measure space.     

Invariant Solutions of Inhomogeneous Universe with Electromagnetic Field in Lyra Geometry

The present study deals with the cylindrically symmetric inhomogeneous cosmological models for perfect fluid distribution with electro-magnetic field in Lyra geometry. Lie group analysis has been used to identify the generators (symmetries) that leave the given system of partial differential equations (field equations) invariant. With the help of canonical variables associated with these generators, the assigned system of partial differential equations is reduced to an ordinary differential equations whose simple solutions provide nontrivial solutions of the original system. They obtained a new class of invariant (similarity) solutions by considering the potentials of metric and displacement field are functions of coordinates t and x. The physical behavior of the derived models are also discussed.     

Hyperbolic Low-Dimensional Invariant Tori¶and Summations of Divergent Series

We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coefficients that grow at most as a power of a factorial and a remainder that to any order N is bounded by the (N+1)-st power of the argument times a power of N!. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory. Received: 9 July 2001 / Accepted: 26 October 2001     

Correlations of RMT characteristic polynomials and integrability: Hermitean matrices

Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of τ functions, we (i) identify a zoo of hierarchical relations satisfied by τ functions in an abstract infinite-dimensional space and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasis is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.     

Classification of Fully Nonlinear Integrable Evolution Equations of Third Order

Abstract

A fully nonlinear family of evolution equations is classified. Nine new integrable equations are found, and all of them admit a differential substitution into the Korteweg-de Vries or Krichever-Novikov equations. One of the equations contains hyperelliptic functions, but it is transformable into the Krichever-Novikov equation by a differential substitution that only involves elliptic functions.     

GROUP ANALYSIS OF SYSTEM EQUATIONS,SYMMETRY REACTION-DIFFUSION BRUSSELATOR

General formulas of group classification of the system of nonlinear reaction-diffusion equations(SNRDE) with two unknown functions and four independent variables are given. Group classifying equations of the traveling wave system of-equations corresponding to SNRDE are also obtained.When these results are applied to analyzing Brusselator, we dis-covered that the system of differential equations describing it has only invariant group of spacetime translation.     

Operator products in 2-dimensional critical theories

A noteworthy feature of certain conformally invariant 2-dimensional theories, such as the Ising and 3-state Potts models at the critical point, is the existence of “degenerate primary fields” associated with nullvectors of the Virasoro algebra. Such fields are endowed with a remarkably simple multiplication table under the operator product expansion, known as the fusion rules. In addition, correlation functions made up of these fields satisfy a system of linear homogeneous partial differential equations. We show here that these two properties are intimately related: for any n-point function, the number of conformally invariant solutions to the system of equations equals the number of times that the identity operator appears in the fusion of all n fields in the correlator. This theorem permits the calculation of some apparently intractable correlation functions. Finally, we generalize these ideas to the Neveu-Schwarz sector of superconformal theories.     

Gauge theory of gravitation: (4+N)-dimensional theory

(4+N)-dimensional theory is studied using the method of differential geometry. The invariant line element is uniquely determined by the connection one-form which is invariant under the local gauge transformations. Generalized Lorentz equations are derived as the geodesic equations. One of these equations is that for a spinning point particle in gravitation which violates the strong equivalence principle.     

Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants. Received: 26 May 1999 / Accepted: 9 November 1999     

Symmetry Reductions of the Lax Pair of the Four-Dimensional Euclidean Self-Dual Yang-Mills Equations

Abstract

The reduction by symmetry of the linear system of the self-dual Yang-Mills equations in four-dimensions under representatives of the conjugacy classes of subgroups of the connected part to the identity of the corresponding Euclidean group under itself is carried out. Only subgroups leading to systems of differential equations nonequivalent to conditions of zero curvature without parameter, or to systems of uncoupled first order linear O.D.E.’s are considered. Lax pairs for a modified form of the Nahm’s equations as well as for systems of partial differential equations in two and three dimensions are written out.     

Nonlinear diffusion under a time dependent external force: q-maximum entropy solutions

Nonlinear diffusion equations provide useful models for a number of interesting phenomena, such as diffusion processes in porous media. We study here a family of nonlinear Fokker-Planck equations endowed both with a power-law nonlinear diffusion term and a drift term with a time dependent force linear in the spatial variable. We show that these partial differential equations exhibit exact time dependent particular solutions of the Tsallis maximum entropy (q-MaxEnt) form. These results constitute generalizations of previous ones recently discussed in the literature [C. Tsallis, D.J. Bukman, Phys. Rev. E 54, R2197 (1996)], concerning q-MaxEnt solutions to nonlinear Fokker-Planck equations with linear, time independent drift forces. We also show that the present formalism can be used to generate approximate q-MaxEnt solutions for nonlinear Fokker-Planck equations with time independent drift forces characterized by a general spatial dependence. Received 25 April 2001 and Received in final form 6 June 2001     

Group and Renormgroup Symmetry of Quasi-Chaplygin Media

Abstract

Results of renormgroup analysis of a quasi-Chaplygin system of equations are presented. Lie-Bäcklund symmetries and corresponding invariant solutions for different “Chaplygin” functions are obtained. The algorithm of construction of a group on a solution (renormgroup) using two different approaches is discussed.     

Near mass shell singularities in quantum electrodynamics

We discuss the near mass shell infrared behavior of QED by performing an explicit sum over all Feynman diagrams in the eikonal approximation. We review the infrared singularities of exclusive amplitudes in particular limits ((a) small photon mass or dimension ≠ 4, (b) equal off shell p_i², (c) large momentum transfers) as special cases of a general parametric formula. In the parametric representation the infrared singularities always exponentiate. This allows us to derive simple differential equations for Laplace transforms of the scattering amplitudes. Similar differential equations have been conjectured to hold in QCD and we summarize the present evidence regarding this assumption.     

Discrete Symmetries and Generalized Fields of Dyons

We have studied the different symmetric properties of the generalized Maxwell’s–Dirac equation along with their quantum properties. Applying the parity (℘), time reversal ( T\mathcal{T} ), charge conjugation (C\mathcal{C}) and their combined effect like parity time reversal (PT\mathcal{PT}), charge conjugation and parity (CP\mathcal{CP}) and CPT\mathcal{CP}T transformations to various equations of generalized fields of dyons, it is shown that the corresponding dynamical quantities and equations of dyons are invariant under these discrete symmetries.     

General Solutions of Relativistic Wave Equations II: Arbitrary Spin Chains

A construction of relativistic wave equations on the homogeneous spaces of the Poincaré group is given for arbitrary spin chains. Parametrizations of the field functions and harmonic analysis on the homogeneous spaces are studied. It is shown that a direct product of Minkowski space time and two-dimensional complex sphere is the most suitable homogeneous space for the physical applications. The Lagrangian formalism and field equations on the Poincaré and Lorentz groups are considered. A boundary value problem for the relativistically invariant system is defined. General solutions of this problem are expressed via an expansion in hyperspherical functions defined on the complex two-sphere. PACS numbers: 02.30.Gp, 02.60.Lj, 03.65.Pm, 12.20.-m