Exact Solutions of Non-Autonomous Quantum Systems with Quadratic Hamiltonian

Based on algebraic dynamics, we present an algorithm to obtain exact solutions of the Schrodinger equation of non-autonomous quantum systems with Hamiltonian expressed in quadratic function of creation and annihilation operators of bosons. The Hamiltonian is treated as a linear function of generators of a symplectic group. Similar to the canonical transformation of classical dynamics, we employ a set of gauge transformations to gradually transform the Hamiltonian to a linear function of Cartan operators. The exact solutions are obtained by inverse gauge transformations. When the system is autonomous, this algorithm can obtain the normal mode of the Hamiltonian, as well as the eigenstates and eigenvalues.     

Weyl group, <Emphasis Type="Italic">CP</Emphasis> and the kink-like field configurations in the effective <Emphasis Type="Italic">SU</Emphasis>(3) gauge theory

Effective Lagrangian for Yang-Mills gauge fields invariant under the standard space-time and local gauge SU(3) transformations is considered. It is demonstrated that a set of twelve degenerated minima exists as soon as a nonzero gluon condensate is postulated. The minima are connected to each other by the parity transformations and Weyl group transformations associated with the color su(3) algebra. The presence of degenerated discrete minima in the effective potential leads to the solutions of the effective Euclidean equations of motion in the form of the kink-like gauge field configurations interpolating between different minima. Spectrum of charged scalar field in the kink background is discussed.     

Gauge invariance and gauge independence of the S-matrix in nonrelativistic quantum mechanics and relativistic quantum field theories

The gauge independence of transition rates as opposed to the gauge invariance of the equations of motion and gauge dependence of operators and state vectors is critically examined and explicitly demonstrated, both in nonrelativistic quantum mechanics and quantum field theory. Time independent as well as time dependent gauge transformations are explicitly analyzed using several techniques in order to clarify the physical content and significance of gauge independence and the conditions for its applicability.     

Algebro-Geometric Solutions with Characteristics of a Nonlinear Partial Differential Equation with Three-Potential Functions

With the help of a simple Lie algebra, an isospectral Lax pair, whose feature presents decomposition of element (1, 2) into a linear combination in the temporal Lax matrix, is introduced for which a new integrable hierarchy of evolution equations is obtained, whose Hamiltonian structure is also derived from the trace identity in which contains a constant γ to be determined. In the paper, we obtain a general formula for computing the constant γ. The reduced equations of the obtained hierarchy are the generalized nonlinear heat equation containing three-potential functions, the mKdV equation and a generalized linear KdV equation. The algebro-geometric solutions (also called finite band solutions) of the generalized nonlinear heat equation are obtained by the use of theory on algebraic curves. Finally, two kinds of gauge transformations of the spatial isospectral problem are produced.     

Slow motion approximations in general relativity II. Gauge transformations

When the field equations of general relativity are expanded in powers of a small parameter, the general covariance of the exact theory implies a corresponding gauge invariance of the equations obtained in the expansion. In a slow motion expansion, the derivation of this gauge transformation is complicated by the fact that the time coordinate is singled out for special treatment. In a previous paper, a new (3 + 1)-dimensional decomposition of the field equations was obtained which is particularly suitable as a starting point for slow motion approximations. The present paper gives a systematic method, again using covariant techniques throughout, for obtaining the corresponding gauge transformations to arbitrarily high accuracy. The calculations are explicitly carried out as far as is required in the 2 1/2-post-Newtonian approximation.     

A local approach to dimensional reduction: II. Conformal invariance in Minkowski space

We consider the problem of obtaining conformally invariant differential operators in Minkowski space. We show that the conformal electrodynamics equations and the gauge transformations for them can be obtained in the frame of the method of dimensional reduction developed in the first part of the paper. We describe a method for obtaining a large set of conformally invariant differential operators in Minkowski space.     

Light-cone gauge Schwinger model

Nakawaki's Coulomb gauge solution to the Schwinger model is transformed to light-cone gauge. Various options for maintaining the gauge invariance necessary to satisfy the equations of motion are discussed. Satisfactory light-cone gauge solutions are found and are used to study light-cone quantization, the calculation of the dynamical operators and properties of the vacua in the light-cone representation. The solutions found here can be used to justify previous light-cone Tamm-Dancoff calculations performed by others.     

Seiberg--Witten Monopoles in Three Dimensions

Dimensional reduction of the Seiberg--Witten equations leads to the equations of motion of a U(1) Chern--Simons theory coupled to a massless spinorial field. A topological quantum field theory is constructed for the moduli space of gauge equivalence classes of solutions of these equations. The Euler characteristic of the moduli space is obtained as the partition function which yields an analogue of Casson's invariant.A mathematically rigorous definition of the invariant isdeveloped for homology spheres using the theory of spectral flow ofself-adjoint Fredholm operators.     

Bäcklund-calogero group and general form of integrable equations for the two-dimensional Gelfand-Dikij-Zakharov-Shabat problem bilocal approach

The general form of the integrable equations and their Bäcklund transformations connected with the general two-dimensional Gelfand-Dikij-Zakharov-Shabat spectral problem is found within the framework of the generalized AKNS method. The bilocal tensor product of the solutions of the spectral problem is used successively, which essentially simplifies the calculations of recursion operators. The transformation properties of the integrable equations and Bäcklund transformations under the gauge group are discussed.     

General theory of renormalization of gauge invariant operators

We study the question of renormalization of gauge invariant operators in the gauge theories. Our discussion applies to gauge invariant operators of arbitrary dimensions and tensor structure. We show that the gauge noninvariant (and ghost) operators that mix with a given set of gauge invariant operators form a complete set of local solutions of a functional differential equation. We show that this set of gauge noninvariant operators together with the gauge invariant operators close under renormalization to all orders. We obtain a complete set of local solutions of the differential equation. The form of these solutions has recently been conjectured by Kluberg Stern and Zuber. With the help of our solutions, we show that there exists a basis of operators in which the gauge noninvariant operators “decouple” from the gauge invariant operators to all orders in the sense that eigenvalues corresponding to the eigenstates containing gauge invariant operators can be computed without having to compute the full renormalization metrix. We further discuss the substructure of the renormalization matrix.     

Lie algebroids,Lie groupoids and TFT

We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.     

Bäcklund transformations for finite-dimensional integrable systems: a geometric approach

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parameterized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel–Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by ad hoc methods, and we find Bäcklund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.     

Bäcklund transformations and the construction of the Atiyah-Ward ansätze for self-dual SU(2) gauge fields

Simple expressions for the ansätze of Atiyah and Ward are given in Yang's R gauge. Some non-linear transformations connecting solutions of the field equations are exhibited. These transformations are analogous to the Bäcklund transformations of the Sine-Gordon equation, etc., and provide an essentially algebraic inductive proof of the ansätze. Neither the sources for the anti-self-dual Maxwell fields used in the ansätze nor the reality conditions are discussed.     

BRS gauge and ghost field supersymmetry in gravity and supergravity

Becchi-Rouet-Stora transformations are obtained for the following systems: (i) Pure Einstein gravity in first order form with vierbein and spin connection as independent fields. (ii) First order Einstein gravity coupled to Yang-Mills fields. (iii) Pure supergravity. For the first two systems the results are as in Yang-Mills theory. But for conventional supergravity the BRS transformations leave the effective action invariant only if the classical equations of motion are satisfied. New transformations of the gauge fields of supergravity have been proposed under which the supersymmetry algebra closes. The corresponding BRS transformations do leave the effective action invariant without the need to use the classical equation of motion; moreover, as in Yang-Mills theories, they are nilpotent and have unit Jacobian.     

Degenerate Sklyanin algebras

New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.     

Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation

By considering a new discrete isospectral eigenvalue problem, two hierarchies of integrable positive and negative lattice models are derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And, each equation in the resulting hierarchies is integrable in Liouville sense. Moreover, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.     

Boson expansions and Hartree-Bogoliubov theory

If the boson operators in boson expansions of the Hamiltonian are suitably replaced by c-numbers regarded as variational parameters, solutions of the Hartree-Bogoliubov or Hartree-Fock-Bogoliubov equations are obtained. The c-number replacement in the boson equations of motion yields solutions of the corresponding time-dependent self-consistent field equations.     

Solving Quantum—Nonautonomous System with Non—Hermitian Hanmiltonians by Algebraic Method

A convenient method to exactly solve the quantum-nonautonomous systems with non-Hermitian Hamiltonians is proposed.It is shown that a nonadiabatic complete biorthonormal set can be easily obtained by the gauge transformation method in which the algebraic structure of systems has been used.The nonuitary evolution operator is also found by choosing a special gauge function.All auxiliry parameters introduced in the present approach are only determined by some algebraic equations.The dynamics of two quantum-nonautonomous systems ruled by non-Hermitian Hamiltonians,including a two-photon ionization process involving two-state only and a mesoscopic RLC circuit with a source,are treated as the demonstration of our general approach.     

Improved Speed and Shape of Ion-Acoustic Waves in a Warm Plasma

The basic set of fluid equations can be reduced to the nonlinear Kortewege-de Vries (KdV) and nonlinear Schrödinger (NLS) equations. The rational solutions for the two equations has been obtained. The exact amplitude of the nonlinear ion-acoustic solitary wave can be obtained directly without resorting to any successive approximation techniques by a direct analysis of the given field equations. The Sagdeev's potential is obtained in terms of ion acoustic velocity by simply solving an algebraic equation. The soliton and double layer solutions are obtained as a small amplitude approximation. A comparison between the exact soliton solution and that obtained from the reductive perturbation theory are also discussed.