A generalized Schrödinger formalism as a Hilbert space representation of a generalized Liouville equation

A generalized Schrödinger formalism has been presented which is obtained as a Hilbert space representation of a Liouville equation generalized to include the action as a dynamical variable, in addition to the positions and the momenta. This formalism applied to a classical mechanical system had been shown to yield a similar set of Schrödinger like equations for the classical dynamical system of charged particles in a magnetic field. The novel quantum-like predictions for this classical mechanical system have been experimentally demonstrated and the results are presented.     

A simple formalism for the BPS monopole

A simple formalism for the BPS monopole is obtained by generalizing the ADHM construction of multi-instantons to a Hilbert space. Both the potential itself and the Green's functions for different isospin can be obtained with very little effort from the instanton formulae.     

Integrable systems as nonlinear realizations of infinite-dimensional symmetries: The Liouville equation example

The Liouville equation is shown to have a natural interpretation in terms of the nonlinear realization of an infinite parameter conformal group in 1+1-dimensions. The relevant zero-curvature representation and Bäcklund transformations get a simple treatment in this approach. The proposed construction can hopefully be generalized to other integrable systems.     

Quasiclassical formalism and the Andreev equation

In this article I shall make explicit the connection between the quasiclassical Green’s function and the Andreev quasiparticle energies and wavefunctions. The physical meaning of the components of the Green’s function is elucidated.     

Comment on the Liouville equation and multisolitons

It has been asserted in the literature that a particular solution of the Liouville equation, ?_xtσ = e^σ, can be interpreted as an N-soliton. It is shown here that this solution id in fact familiar one-soliton written in a misleading form.     

A simple application of the Newman-Penrose spin coefficient formalism. II

This is the second paper giving a simple application of the Newman-Penrose formalism to be used by those learning this theory. Here we solve the spherically symmetric Einstein-Maxwell problem for the Reissner-Nordstrøm solution using the Newman-Penrose formalism. As in our previous example the calculation is carried out using a Minkowski null tetrad.     

A simple application of the Newman-Penrose spin coefficient formalism. I

As a simple application of the Newman-Penrose spin coefficient formalism, useful for beginners, we find the vacuum spherical symmetry (Schwarzschild) solution. The calculations also show that all spherically symmetric metrics are Petrov typeD.     

Solving the Kadanoff-Baym equations for inhomogeneous systems: application to atoms and molecules

We implement time propagation of the nonequilibrium Green function for atoms and molecules by solving the Kadanoff-Baym equations within a conserving self-energy approximation. We here demonstrate the usefulness of time propagation for calculating spectral functions and for describing the correlated electron dynamics in a nonperturbative electric field. We also demonstrate the use of time propagation as a method for calculating charge-neutral excitation energies, equivalent to highly advanced solutions of the Bethe-Salpeter equation.     

Liouville Integrability of Conservative Peakons for a Modified CH equation

The modified Camassa-Holm (also called FORQ) equation is one of numerous cousins of the Camassa-Holm equation possessing non-smoth solitons (peakons) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev H¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the H¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].     

Irreversibility: The Liouville equation in the Gell-Mann-Goldberger limit

Irreversible asymptotic behaviour of the diagonal part of density operator ?_d both for t → ∞ and for t₀ → -∞ limits can be in very simple way by applying Gell-Mann-Goldberger limit directly to the Liouville equation.     

Quantum equations of motion and the Liouville equation

Equations of motion for explicitly time-dependent operators in the Heisenberg and Schrödinger pictures, respectively, are reviewed. A simple transformation reduces the related equation in the Heisenberg picture to closed form. An algorithm is introduced for the classical limit which, in either picture, returns the classical equation of motion for dynamical functions. Applications of this algorithm to the equation of motion for the density matrix reduces it to the classical Liouville equation. A property related to this algorithm is established for the commutator of any two analytic functions of finite polynomials of conjugate variables. Thus, if \(\hat F\) and ? are two such functions and \(\hat F^\prime \) and ?′ contain an arbitrary permutation of variables, then $$[\hat F,\hat G] = [\hat F^\prime ,\hat G^\prime ] + \hbar ^2 \hat{A}$$ where  is a remainder commutator. In the classical limit [ \(\hat F\) , ?] is invariant to such permutation.     

Quantum mechanics in the K-field formalism: Field equation

The paper considers a field equation for representing a microparticle as an extended object constructed from that field (K field). It is shown that in contrast to the equation of an S field (i.e., Schrödinger equation), the boundary-value problem for the K-field equation cannot be used for a unique determination of the parameters of a microparticle in steady states. An auxiliary equation, therefore, is necessary to ensure that the solution of the K -field equation is unique. This is in complete agreement with the physical meaning of K field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 30–33, August, 1995.     

Solving the Bethe-Salpeter equation for positronium

We propose a Coulomb-like kernel for the relativistic two-fermion Bethe-Salpeter equation, to be used as the lowest-order approximation in systematic perturbative calculation of bound-state energy levels in QED. The kernel is symmetric in the two fermions and for the exchange of in and out momenta. The resulting equation is exactly soluble, unlike previously considered unperturbed kernels. We give explicitly the Green function and eigen-functions. We also discuss the problem of the behaviour of the wave functions at zero relative coordinate in connection with the contribution to energy levels from the one-photon annihilation channel in QED.     

Hamiltonian formalism of the Landau-Lifschitz equation for a spin chain with full anisotropy

Abstract

The Hamiltonian formalism of the Landau-Lifschitz equation for a spin chain with full anisotropy is formulated completely, which constructs a stable base for further investigations.     

Variational method for lattice systems: General formalism and application to the two-dimensional Ising model in an external field

A simple variational approach to the eigenvalue problem of the transfer operator is proposed. After reducing the transfer operator according to the symmetries of the Hamiltonian, the leading eigenvalues of the irreducible blocks can be evaluated by elementary variational principles. Hence the thermodynamics and a large class of correlation functions of lattice systems can be calculated. Following a natural truncation scheme the results can be improved in a systematic way. The high accuracy and the convergence of the method is demonstrated by two-dimensional Ising model. As a first application, the thermodynamics of the two-dimensional Ising ferro-and antiferromagnet in an external field is studied. We show how the same method can be used to obtain zero-temperature properties of interacting quantum lattice systems.     

Selfadjointness of the Liouville operator for infinite classical systems

We study some properties of the time evolution of an infinite one dimensional hard cores system with singular two body interaction. We show that the Liouville operator is essentially antiselfadjoint on the algebra of local observables. Some consequences of this result are also discussed.Research partially supported by the Consiglio Nazionale delle Ricerche, G.N.F.M.     

Renormalization and envelope function formalism for incommensurate systems

Summary We study the localization-delocalization transition in one-dimensional incommensurate crystals both numerically and analytically. From the numerical point of view we provide an implementation of the renormalization method, which allows to process with high accuracy millions of sites (whenever necessary). From the analytic point of view we extend the envelope function concepts to incommensurate potentials, smoothly varying on lattice constant scale. The control of the transition is made by numerical calculation of the Lyapunov exponent: it presents surprising aspects of universality and simplicity, with plateaux, linear regions and, at times, much more complicated behaviours. The envelope function method, together with semi-analytic considerations, allows to understand, in a number of situations, the presence of mobility edges, pseudo-mobility edges, and gaps in the energy spectrum. In honour of Prof. Fausto Fumi on the occasion of his retirement from teaching.