Expansion In Coupling Constant Research Articles

On the phenomenon of dynamical friction for particles interacting by the inverse square law

The dynamics of the homogeneous positive background of a one-component plasma (OCP) is taken into account — compensated system — in order to modify the first coefficient of the coupling constant expansion of the average velocity of a test electron derived in a previous letter for an uncompensated OCP.

Logarithmic perturbation expansions in nonrelativistic quantum mechanics

We discuss an alternative method of performing coupling constant perturbation expansions in nonrelativistic quantum mechanics. This method, called logarithmic perturbation theory, yields new expressions for any-order corrections En to an unperturbed bound-state energy which do not involve cumbersome sums over intermediate unperturbed states. In one dimension, these corrections En can be evaluated using a simple explicit form containing a small number of integrals. In more than one dimension the approach is systematic but computations require the solution of well-defined partial differential equations. For order n=2 this equation is identical to that appearing in the method of Dalgarno and Lewis. Numerous illustrative examples are presented.

On the usefulness of the 't Hooft and Adler transformations of the running coupling constant in perturbative quantum chromodynamics

It is argued that the 't Hooft transformation of the running coupling constant, in which the two-loop renormalization group (RG) function becomes exact, will be useful in the framework of perturbative quantum chromodynamics at least to three-loop order. On the other hand, the coupling constant expansion obtained by the Adler transformation, in which the RG equation takes its one-loop form, may suffer from large corrections in a finite order.

Virtual stationary points

Using ideas from the Morse theory, a type of limiting field configurations is introduced which is expected to yield non-perturbative contributions to coupling constant expansions of topologically non-trivial quantum field theories. Examples are given for two-dimensional non-linear σ models and four-dimensional gauge fields. For the O(3) σ model the complete set is determined. The results are compared with the exact S matrices in two dimensions.

Variational matrix Padé approximations in potential scattering and low-energy Lagrangian field theory

In the first part, we develop in potential scattering theory the method of operator Padé approximations (OPA), starting from the variational principle for phase shifts. We show that the lowest approximation, constructed from the first two terms of the perturbative expansion of the Green function, is, for the Schrödinger equation, the exact solution of the problem. However the practical calculation of the OPA requires the inversion of an operator. This inversion is achieved by discretizing the Hilbert space: the OPA is replaced by an MPA (matrix Padé approximation). The “best choice” of the discrete basis of the Hilbert space is given, at each energy, angular momentum, parity, and internal quantum numbers, by the variational principle for phase shifts. The quality of the results is such, that, for potentials giving rise to phase-shifts varying over hundreds of degrees, it is possible, using only the first two terms of the Green function perturbative expansion, to reproduce the exact phase shifts within one part in one thousand, at any energy; this result is achieved with a discrete basis involving no more than two to four vectors. New theorems are derived which ensure the reliability of this method. In the second part, we show how this formalism extends naturally to Lagrangian field theory in the low-energy region. As a first application we give preliminary results for nucleon-nucleon scattering. The information coming from the Green function is essential to obtain the existence of the deuteron and dineutron from the Yukawa Lagrangian, without any adjustable parameter or hard core. All the partial waves L ⩾ 2 are correctly reproduced. The zeroes of the S and P wave phase shifts are still missing: this is very likely due to the fact that, in this preliminary calculation, only the spin part of the amplitude is taken off-shell. This effect happens in potential scattering when the momenta are kept on-shell. The complete analysis involving the introduction of the off-shell momenta is in progress. In the conclusion, we show, by giving a simple example, that the variational matrix Padé approximation is no longer a rational fraction in the coupling constant (expansion parameter), but may have the full analyticity structure of the exact solution. It then defines what could be called a hyper-Padé approximant.

Strong coupling polaron theory and translational invariance

We develop a systematic, manifestly translation invariant, strong coupling theory for nonrelativistic Hamiltonians of the polaron type. As in earlier strong coupling theories, the position of the polarization well is a collective coordinate. The field is expanded in a set of basis functions centered about the well with three amplitudes deleted. A particle coordinate relative to the polarization center is introduced. The new coordinates are introduced using a point canonical transform leading to a Hermitian Hamiltonian, with properly normalized wavefunctions, and with a Jacobian that is evaluated in closed form. All subsequent approximations to the states are manifestly translation invariant. For the ground state the energy of the recoil terms to leading order depend on the coupling constant g as g −4. The intrinsic part of the Hamiltonian determines the energy terms of order g 4 and g 0. An adiabatic canonical transformation is used to calculate all terms through order g −4. The coefficients depend on the Green's function for the electron in a static potential well. We determine the first three terms in the inverse coupling constant expansion of the effective mass.

VII. The quantum theory of solitons and other non-linear classical waves

It has been suspected for some time that certain solutions of non-linear classical field equations indicate that the corresponding quantum theory possesses interesting, even unsuspected properties [1]. A group of us in Cambridge [2] (U.K. and U.S.A.) has developed an approach to the quantum theory which consists of a systematic coupling constant expansion for various quantum mechanical objects: transition amplitudes, S-matrix elements. The first term in this expansion frequently leads to a differential equation which is identical with an equation of the classical theory. Unlike the method of the Princeton group [31,ours is not a semi-classical calculation; we do not look to the classical equations for a description of the dynamics which is then quantized by Bohr— Sommerfeld or WKB approximations. Rather we encounter “classical” equations in the full quantum theory; they determine lowest order approximations to quantum mechanical objects. Also we have a procedure for calculating higher order corrections. We shall consider theories involving a single boson field in one space dimension ~p(t,x) whose dynamics is governed by a Lagrangian density C. [The restriction to an unphysical dimensionality of space is for simplicity. The phenomena that we are describing exist also in the physical situation of three spatial dimensions. However such models necessarily contain fields with spin; this leads to computational tedium.] Of special interest will be static [time independent] solutions of the classical Euler—Lagrange equations. The full time dependent equation is

Non-local structure of field theories with non-renormalizable interaction

Quantized field theories of the second kind, the so-called non-renormalizable theories, are investigated. It has recently been shown that a number of those theories actually can be renormalized. We show that field theories of the second kind, pertaining to derivative couplings, possess non-local structure. The full recoil neutral PS-PV theory in its non-linear form is discussed. The non-localizability of both the unrenormalized and the renormalized theories is due to the fact that the infinite set of topologically independent direct interactions, which characterizes PS-PV theory, is equivalent to a new type of indirect interaction. The non-localizability manifests itself in an indeterminacy in space-time of the light-cone. This indeterminacy can be interpreted in terms of a statistical spread in space-time of the positions of two interacting point-nucleons over a space-time volume the size of which is determined by the coupling constant |g| of the linear version of PS-PV theory. The length |g| loses its meaning as a coupling in the non-linear formalism in favour of the nucleon mass and plays now the role of a fundamental structure constant. In particular, |g| determines the region in which the equivalence theorem and the 1/r3-potential become invalid and repulsive-core potentials begin to act. In virtue of the oscillatory behaviour of the non-tempered momentum space operators in the high energy region, positive-definiteness conditions can be established only for regions |kv2|≳|1/g2| and |xv2|≳|1/g2|. The non-localizability is concentrated in these domains and |g| seems to play the role of the meson Compton wave length. A new interpretation of the non-linear PS-PV theory is suggested. It is conjectured that also β interactions give rise to non-local structures. Multiple propagators are introduced to render the formal coupling constant expansions in theories of the second kind mathematically meaningful without producing non-renormalizable infinities. Arguments are advanced against the mathematical consistency of the concept of creation (annihilation) and localizability of more than one point-particle at a single space-time point.

Some Notes on Nonrenormalizable Field Theory

Some nonrecoil, derivative coupling theories, which are exactly soluble, are analyzed in order to discover what it is that produces the nonrenormalizable behavior typical of derivative coupling. This behavior is found to arise as the result of an essential singularity which the operators of the derivative theories possess at the origin in coordinate space and the resulting branch point the Fourier transforms of these operators possess at the origin in the complex coupling constant domain. This branch point causes the breakdown of the expansion of the Fourier transform of the operators in powers of the coupling constant and introduces the nonrenormalizable infinities. It is shown that a coordinate space coupling constant expansion is possible, and that a Fourier transform of the operators of the derivative theories may be defined by analytic continuations and made finite by renormalization.