Nonperturbative Approximation Research Articles

Vacuum structures in Hamiltonian light-front dynamics

Hamiltonian light-front dynamics of quantum fields may provide a useful approach to systematic nonperturbative approximations to quantum field theories. We investigate inequivalent Hilbert-space representations of the light-front field algebra in which the stability group of the light front is implemented by unitary transformations. The Hilbert space representation of states is generated by the operator algebra from the vacuum state. There is a large class of vacuum states besides the Fock vacuum which meet all the invariance requirements. The light-front Hamiltonian must annihilate the vacuum and have a positive spectrum. We exhibit relations of the Hamiltonian to the nontrivial vacuum structure.

Effective action for SU(N) at a finite temperature.

Techniques are developed in order to study static magnetic screening and other nonperturbative aspects of QCD at high temperatures. In particular, a covariant derivative expansion of the one-loop effective action is presented and then modified by an infinite resummation so as to provide agreement with the exactly calculable one-loop effective potential. Essential to this technique is a self-consistently defined infrared cutoff which determines the prefactor in semiclassical calculations. Using this prefactor, densities of monopole and dyon plasmas are calculated, and it is found that if such plasmas do exist at finite temperature, then the solitons involved must be overlapping one another. It is also shown that no consistent perturbative or nonperturbative approximation can give rise to a linear term in the SU(2) effective potential, since such a term would not be gauge invariant. Finally, contour plots of the SU(3) ${\mathit{A}}_{0}$ effective potential are presented.

Glueballs as bound states of massive transverse gluons

It is shown how the glueball spectrum can be obtained from the Bethe-Salpeter equation using nonperturbative approximations. The masses are estimated to be (3.5+or-0.4) Lambda QCD for the 0+ and 2+ glueballs and (2.8+or-0.3) Lambda QCD for the 1- glueball.

Closed-form intermediate representations of many-body propagators and resolvent matrices.

A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}$〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$\ensuremath{\Vert}C${\mathrm{^}}_{\mathit{J}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉, where C${\mathrm{^}}_{\mathit{J}}$ denotes a physical excitation operator and \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.

Microscopic optical potentials for time-dependent quantum systems

The numerical solution of the time-dependent Schrödinger equation often relies on the expansion of the wavefunction in terms of a truncated set of orbitals spanning a finite model space. The projection of the Schrödinger equation onto a finite subspace of the Hilbert space necessarily leads to an effective interaction described by a complex operator. The imaginary part of this optical potential accounts globally for the coupling between the model space and its explicitly neglected complement. We discuss a nonperturbative approximation to this optical potential and give some details for the numerical evaluation of the matrix elements involved.

LEADING INFRARED LOGARITHMS AND VACUUM STRUCTURE OF QCD3

QCD 3 is a superrenormalizable massless theory; therefore off-mass-shell infrared divergences appear in the loop expansion. We show how certain infrared divergences can be subtracted by changing the boundary conditions in the functional integral, letting the vector potentials approach non-zero constant values at infinity. Infrared divergences, in the Green's functions, come together with powers of logarithms of the external momenta, and among the infrared divergences we deal with, there are those that give rise to the leading and first subleading logarithms. We show how for two-point functions it is possible to sum the leading and first subleading logarithms to all orders. This procedure defines a nonperturbative approximation for QCD 3. We find that in the ultraviolet region these summations are well defined, while in the infrared region, some additional prescription is needed to make sense out of them. From the construction developed to cancel infrared divergences, we derive the presence of arbitrarily large field strengths in the vacuum. We respect Lorentz invariance, because in this construction, one averages all directions and strengths of these fields. Possible applications to the infrared problem of QCD 4 are indicated.

Nonperturbative square-well approximation to a quantum theory

The possibility of expressing the solution to a φ2P quantum field theory as a series in powers of 1/P is proposed. Such a series would be nonperturbative in its dependence on the fundamental parameters of the theory such as the mass and the coupling constant. The first term in such a series describes a field in an infinite-dimensional square-well potential. In this paper, the quantum-mechanical Hamiltonian H=p2+q2P is studied as a model calculation and the expansion of the energy levels as series in powers of 1/P is examined. The method of matched asymptotic expansions to determine the first five terms in the series for all energy levels is used. The results are compared with extensive numerical calculations of the ground-state energy and it is found that the series is extremely accurate: When P=2, the five-term series has a relative error of 6%, when P=10 the relative error is 0.009%, and when P=200 the relative error is 3.4×10−9%.

Electroweak physics

The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilson’s RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity.

Path-summation approach to the dynamics of radiative phenomena

We investigate a new form of the path-summation method, which is related to the Feynman-Vernon influence-functional approach, within a fairly standard model describing the two-level atom interacting with the (quantum) electromagnetic field. Our method, being essentially a nonperturbative one, is developed with perspective of application to dynamical problems, where the parameters of the (weak) atom-field interaction are combined with the parameters describing the (strong) pulse of radiation in the initial condition and/or in situations where one desires to calculate the time evolution of the field quantities. However, in this paper, after displaying the path-summation method in its general form, we test its possibilities just in the simplest case of populational-difference time evolution. The populational difference is formally expressed as a series in the level-splitting parameter. In the terms of this series, field averaging is explicitly performed and the physical analysis of the path contributions leads to a hierarchical class of nonperturbative partial-summation approximation based on the damping of the field correlations. We propose a trial form of the “correlation-damping” function, which enables the explicit discussion of the resulting time evolution. Finally, we compare the dynamics so obtained with the usual approach using the perturbative (in the atom-field interaction) expansion of the memory operator.

Nonperturbative approximations in the Walecka model beyond the mean field approximation

On the basis of formal solutions for the nucleon field in Walecka's ellective meson theory we derive nonperturbative approximations neglecting free mesonic degrees of freedom and anti-nucleons. In lowest order the method yields the Hartree-Fock equations. Higher approximations describe the propagation of particles and holes correlated to particle-hole excitations.

Quantum systems with external electromagnetic fields: The large mass asymptotics

The large mass asymptotics of the quantum evolution problem for a system of charged particles that mutually interact through scalar fields and couple to an arbitrary time-varying external electromagnetic field is rigorously described. If K(x,t; y,s;m) denotes the coordinate space propagator (time evolution kernel) of this system, the singular perturbation behavior of K as mass m→∞ is expressed in terms of a gauge invariant asymptotic expansion. In terms of the external fields and interparticle interactions, this expansion provides a nonperturbative approximation for the propagator K that is valid for all particle coordinates x, y and for finite time displacements t−s. For the class of analytic scalar and vector fields that are defined as Fourier transforms of time-dependent measures, the existence of this asymptotic series for K in powers of (m)−1 is established for both real and complex masses. Explicit bounds for the error term are obtained and a manifestly gauge invariant transport recurrence relation is derived that uniquely determines all the coefficient functions of the asymptotic series. The small time asymptotic expansion of K is shown to be embedded within the large mass expansion.

The O− nonet and solution of the U(1) problem in a nonperturbative approximation of quantum chromodynamics

The O− nonet and solution of the U(1) problem in a nonperturbative approximation of quantum chromodynamics

Approximate theory for multiphoton ionization of an atom by an intense field.

We formulate a nonperturbative approximation for treating the multiphoton ionization of an atom. Our approximate ionization amplitude is the time average of the matrix element for the electron, having absorbed enough photons to reach the continuum threshold, to jump into the continuum state represented by the Kroll-Watson scattering wave function.

Strongly correlated electron ground-state energy approximations for Anderson-like models

We report preliminary results of convergence properties for nonperturbative resolvent approximations to Anderson-like models of magnetic ions in metals. Our study is initially focused on the spin- 1/2 Anderson model for magnetic impurities, but the methods studied can include multiplet and crystal-field effects which are needed for more accurate descriptions of real systems. We will compare the nonperturbative Lanczos method (tridiagonalization) and similar truncation schemes to exact ground-state energies for the impurity model and assess the efficacy of these nonperturbative approaches to understanding the Anderson lattice, heavy fermions, and other strongly interacting electronic systems.

Finite-dimensional approximation of the scattering matrix

A finite-dimensional nonperturbative approximation scheme of the time-evolution operator and the S matrix for relativistic field theories is discussed. It is amenable to computer calculations. Parallels with lattice-field theory are drawn. The method is outlined for the ϕ4 theory. Equivalence to standard perturbation theory in the weak-coupling regime is obtained in the limit of the approximation parameters. The method is tested numerically for nonrelativistic proton–proton s-wave scattering and the the ϕ4 model in the weak-coupling regime in 1 + 1 dimensions. In both examples, convergence to the reference solution is found.

Finite-dimensional approximation of the S-matrix

We suggest a time-dependent, finite-dimensional, nonperturbative approximation scheme for the S-matrix of relativistic quantum field theories. The method is presented for the φ 4-theory. In the weak coupling regime our method tends to the results of perturbation theory in all orders. We present results of a numerical test calculation in 1+1 dimensions for a small coupling constant which gives satisfactory agreement with the result of perturbation theory.

Convergence of High-Intensity Expansions for Atomic Ionization

We show that a frequently used nonperturbative approximation for atomic ionization rates is canceled out when corrections are taken into account. This explains the strong gauge dependence of previous results. A convergent and gauge-invariant expansion is obtained. Numerical results show that its first term, which may be calculated analytically in many cases, describes very well the time-dependent behavior of the ionization probability for very strong fields.

Hopping transport on a randomly substituted lattice in the presence of dilute deep traps

Our recent theory of hopping transport of excitations or charge carriers among particles randomly distributed on the sites of a lattice is extended here to include the presence of a low concentration trap species. We present an exact diagrammatic analysis of the configuration-averaged Green function of the pauli master equation for this problem. We obtain a non-perturbative approximation to the Green function that is expected to be accurate at both high and low donor concentration for low trap concentration. The approximation can be carried out for any lattice type in which any two sites can be transformed into one another by a symmetry operation, and a transfer rate of any distance dependence. We present calculations of the observable in a steady-state excitation transfer and trapping experiment that illustrate the effect of dimensionality and of the distance dependence of the transfer rate on this observable.

Hopping transport on a randomly substituted lattice in the presence of dilute deep traps

Our recent theory of hopping transport of excitations or charge carriers among particles randomly distributed on the sites of a lattice is extended here to include the presence of a low concentration trap species. We present an exact diagrammatic analysis of the configuration-averaged Green function of the pauli master equation for this problem. We obtain a non-perturbative approximation to the Green function that is expected to be accurate at both high and low donor concentration for low trap concentration. The approximation can be carried out for any lattice type in which any two sites can be transformed into one another by a symmetry operation, and a transfer rate of any distance dependence. We present calculations of the observable in a steady-state excitation transfer and trapping experiment that illustrate the effect of dimensionality and of the distance dependence of the transfer rate on this observable.

Quenched master field equations for unitary matrix models

We derive the quenched master field equations for constrained systems such as the O( N) non-linear sigma model, U( N)×U( N) chiral models, and U( N) lattice gauge theory. The master equations are algebraic, and involve quenching in both the space and “fifth” (Langevin) time directions. The quenched master field for the O(N) nonlinear sigma model is found exactly. The 0-dimensional unitary matrix model is solved perturbatively, and we recover the Gross-Witten result. The master equation for the U( N)×U( N) chiral model is set up for non-perturbative approximation methods, and some qualitative results are obtained.