Time-dependent variational method for sine-Gordon quantum field theory

The sine-Gordon model in 1+1 dimensions is studied within the Schrödinger framework for field theory. In particular we evaluate the effective potential and examine the finiteness ofm(t), the soliton mass, for allt.     

The source galerkin method for scalar field theory

This is part I of a two-part series on the Source Galerkin method. This approach is based on the differential formulation of quantum field theory. On a finite lattice, the functional differential equations for a theory in the presence of an external source becomes a set of coupled differential equations for the generating functional Z. Systematic approximations to these equations are found using the Galerkin method. Calculations are straightforward to perform, and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. In addition, bosons and fermions are treated in a symmetric manner. In this paper, we consider power series solutions for scalar field theory in D = 2, 3,4. Propagators and mass gaps are calculated for a number of systems. The calculations in this paper were made on a work station of modest power using a fourth order polynomial expansion for lattices of size 8², 4³, 2⁴ in 2D 3D, and 4D. In part II we consider the fermionic formulation.     

A variational phase field method for curve smoothing

A variational phase field model is proposed for curve smoothing, in which a weight function is associated with the similarity measure term in the model so that important geometric features could be well preserved. Finite element approximation of the proposed model is given for its numerical implementation. Since the model has a linear weak variational form, the discretized system could be solved efficiently by many existing solution techniques. An effective algorithm is also developed, for the purpose of feature preservation, to automatically determine the weight from the given data. Various numerical examples are presented to demonstrate effectiveness and robustness of the proposed method.     

A variational principle for bound states in quantum field theory

We develop a Rayleigh-Ritz variational method for estimating relativistic, multi-particle bound state energies in any (weak-coupling) quantum field theory. A comparison is made with bound state energies derived from the Bethe-Salpeter equation in the Wick-Cutkosky model. Possible applications to QCD are discussed.     

A variational approach to second-order multisymplectic field theory

This paper presents a geometric-variational approach to continuous and discrete second-order field theories following the methodology of [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395]. Staying entirely in the Lagrangian framework and letting Y denote the configuration fiber bundle, we show that both the multisymplectic structure on J³Y as well as the Noether theorem arise from the first variation of the action function. We generalize the multisymplectic form formula derived for first-order field theories in [Marsden, Patrick, Shkoller, Comm. Math. Phys. 199 (1998) 351–395], to the case of second-order field theories, and we apply our theory to the Camassa–Holm (CH) equation in both the continuous and discrete settings. Our discretization produces a multisymplectic-momentum integrator, a generalization of the Moser–Veselov rigid body algorithm to the setting of nonlinear PDEs with second-order Lagrangians.     

A variational approach to bound states in quantum field theory

We consider here in a toy model an approach to bound state problem in a nonperturbative manner using equal time algebra for the interacting field operators. The potential is replaced by offshell bosonic quanta inside the bound state of nonrelativistic particles. The bosonic dressing is determined through energy minimisation, and mass renormalisation is carried out in a nonperturbative manner. Since the interaction is through a scalar field, it does not include spin effects. The model however nicely incorporates an intuitive picture of hadronic bound states in which the gluon fields dress the quarks providing the binding between them and also simulate the gluonic content of hadrons in deep inelastic collisions.     

A variational principle for Newton-Cartan theory

In the framework of a space-time theory of gravitation a variational principle is set up for the gravitational field equations and the equations of motion of matter. The general framework leads to Newton's equations of motion with an unspecified force term and, for irrotational motion, to a restriction on the propagation of the shear tensor along the streamlines of matter. The field equations obtained from the variation are weaker than the standard field equations of Newton-Cartan theory. An application to fluids with shear and bulk viscosity is given.     

The coherent state variational algorithm: A numerical method for solving large-N gauge theories

This paper presents a new approach for studying large-N gauge theories which directly exploits the classical nature of the N → ∞ limit. This method supplies a practical algorithm for computing and minimizing the classical hamiltonian (or effective action) which governs N = ∞ dynamics, and allows one to calculate physical quantities such as the mass spectrum or scattering amplitudes of glueballs or mesons. Two different implementations of the basic ideas are discussed; one variant provides an algorithm for constructing N = ∞ master field matrices, while the other works directly with a list of expectation values of physical operators. Algorithms are developed for both the hamiltonian and euclidean formulations of lattice gauge theories. The inclusion of fermions in the hamiltonian version is also described. Detailed tests of the method in the context of the exactly solvable one-plaquette model are presented.     

A variational method for hydrogen chemisorption

Starting from the Hartree-Fock approximation a general variational ansatz for the ground state of the metal-hydrogen system is presented which properly describes the strong correlation effects occuring at larger distances from the surface. As a simple example the Anderson model is studied. The calculated ground state energies are in excellent agreement with exact model calculations.     

A numerical comparison between two unconstrained variational formulations

In an effort to relieve the often cumbersome burden of meeting the requirements of the end conditions and to unify the solution formulation for boundary- and initial-value problems, unconstrained variational statements have been introduced in conjunction with some approximate methods. In the case of a boundary value problem, it is shown in this paper that two different variational statements can be established: one is arrived at by the use of the Lagrange multipliers, the other by energy considerations. The numerical convergence of the solutions associated with finite element schemes involving use of one of these two different variational statements is compared with that of the other. In the case of an initial value problem, both formulations can again be established when the adjoint field variable and the adjoint variational statement are introduced. The numerical data presented here indicate that while both methods generate excellent convergent results for the boundary problem, the method of stiff springs yields results which show much better convergence for the initial value problem than those achieved by Lagrange multipliers.     

A phase field model of deformation twinning: Nonlinear theory and numerical simulations

A continuum phase field theory and corresponding numerical solution methods are developed to describe deformation twinning in crystalline solids. An order parameter is associated with the magnitude of twinning shear, i.e., the lattice transformation associated with twinning. The general theory addresses the following physics: large deformations, nonlinear anisotropic elastic behavior, and anisotropic phase boundary energy. The theory is applied towards prediction of equilibrium phenomena in the athermal and non-dissipative limit, whereby equilibrium configurations of an externally stressed crystal are obtained via incremental minimization of a free energy functional. Outcomes of such calculations are elastic fields (e.g., displacement, strain, stress, and strain energy density) and the order parameter field that describes the size and shape of energetically stable twin(s). Numerical simulations of homogeneous twin nucleation in magnesium single crystals demonstrate fair agreement between phase field solutions and available analytical elasticity solutions. Results suggest that critical far-field displacement gradients associated with nucleation of a twin embryo of minimum realistic size are 4.5%–5.0%, with particular values of applied shear strain and equilibrium shapes of the twin somewhat sensitive to far-field boundary conditions and anisotropy of twin boundary surface energy.     

Massive gauge field in source theory. II

In an earlier source theory investigation it was shown that the vacuum polarization of a massive gauge field is finite, provided that the conservation of current is imposed everywhere, including the interior of the sources. It is shown in the present paper that radiative corrections to a vertex function (two-particle production by an external source) are also finite if the same requirement about current conservation is imposed. In other words one-loop corrections turn out to be finite in both calculations. The cancellations leading to convergence may be understood in terms of Ward identities. The three form factors that appear are not only shown to be finite but are also explicity found.     

Five-dimensional unified field theory: The source function

In Kaluza's five-dimensional unified field theory the restriction for the 55 component of the metric tensor ₅₅=1 demands that the 15 equations for the unified field be weakened. Equations which have been proposed have identically vanishing trace. The equations then admit only a radiation field as source of the gravitational field. By relaxing the condition, this limitation is avoided, while retaining the striking successes of the five-dimensional approach. A scalar function, determined by the 15th field equation apart from integration constants, provides source terms for both the gravitational and electromagnetic fields, in the latter case of polarization type.     

Massive gauge field in source theory. I

The vacuum polarization is calculated for a massive gauge field according to the methods of source theory. The spectral integral turns out to be convergent and the numerical coefficient, which is (1/3)(e²/4π²) in spinor electrodynamics, is here (3f/16)(g²/4π²) where f depends on the group and f = 1 for SU(2).     

LocalH-theorem for a kinetic variational theory

A localH-theorem is derived for a recently proposed extension of Enskog kinetic theory to a dense model fluid composed of particles with interactions extending beyond a hard core.On leave from: Katedra Fizyki, Uniwersytetu Szczecinskiego, 70-451 Szczecin, Poland.     

A Schwinger variational method for the Bloch equation

A Schwinger variational principle has been derived for use in quantum, manybody systems at finite temperatures. The variational principle is a stationary expression for the density matrix which may be iterated to improve an approximate density matrix. It also can be used to find stationary expressions for observables. If an approximate, parametrized density matrix is used, the parameters are varied to find the regions where the variational principle is stationary. The variational density matrix obtained with the optimal parameters can be regarded as optimal for that observable. The method has been applied to two model problems, a particle in a box and two hard spheres at finite temperatures. The advantages and shortcomings of the method are discussed.     

A direct variational method for non-conservative system stability

Hamilton's principle is applied to analyze the problem of the stability of equilibrium of a discrete, holonomic, and scleronomic mechanical system under conservative and non-conservative (position and/or velocity dependent) forces. At the stability limit, the vanishing of the second order terms (in the deviations from equilibrium) of the total action change functional leads to the condition that the matrix of a certain quadratic form be singular; this yields the eigenvalue (frequency-load) curve. The flutter loads follow by setting the frequency derivative of the determinant of this matrix equal to zero; the energetic interpretation of this latter is also given. When the non-conservative forces go to zero it is shown that one recovers the well-known discrete conservative system stability criterion. An application follows, and finally in an Appendix various relevant time-integral equalities are summarized and interpreted.     

A variational approach to distribution function theory

We present a novel formalism for the generation of integral equations for the distribution functions of fluids. It is based on a cumulant expansion for the free energy. Truncation of the expansion at theKth term and minimization of the resulting approximation leads to equations for the distribution functions up toKth order.The formalism is not limited to systems with two-body interactions and does not require the addition of closure relations to yield a complete set of equations. In fact, it automatically generates superposition approximations, such as the Kirkwood three-body superposition approximation or the Fisher-Kopeliovich four-body one.The conceptual approach is adapted from the cluster variation method of lattice theory.