Logarithmic approximations to polynomial Lagrangians

Abstract

A new perturbative computational scheme for solving quantum field theory is proposed. The interaction term in the Lagrangean is expanded about a free-theory form, the expansion involving powers of logarithms of the fields. The resulting perturbation series appears to have a finite radius of convergence and numerical results for simple models are good.PACS numbers: ll.lO.Ef Perturbation methods have played a central role in the quest for approximate solutions to quantum-field-theory models. One can distinguish between two different kinds of perturbation series: a natural expansion, which is a series in powers of a physical parameter that appears in the functional-integral representation of the theory, and an artificial expansion, which is a series in powers of a new parameter 5, which has been introduced temporarily as an expansion parameter for computational purposes. Weak-coupling expansions in powers of the coupling constant X, strong-coupling expansions in powers of lA, and semiclassical (loop) expansions in powers of h are all natural perturbation expansions. Unfortunately these natural expansions suffer a number of disadvantages. Weak-coupling series are divergent and may not even be asymptotic to the solution of the theory. Semiclassical approximations also give divergent series, are very difficult to obtain beyond leading orders, and therefore may give very poor numerical results. ^ The computation of strong-coupling series requires the introduction of a lattice and the subsequent taking of a continuum limit; such series are often very slowly converging with many terms being required to give a reasonable approximation. The principal diflficulty with natural perturbation expansions is that the analytic dependence of the solution to the theory on the physical parameters is lost; by the physical constants being forced to play the role of expansion parameters they are no longer available to display adequately the true functional dependence of the physical theory on them.'^ The advantage of artificial perturbation expansions is that, if a parameter 5 is inserted in a clever way, the resulting series in powers of 8 may be easy to compute and rapidly convergent. Moreover, the terms in this expansion may exhibit a very nontrivial dependence on the physical parameters of the theory. One such perturbation scheme is the large-TV expansion, where A^ is the number of components of a scalar field. In nonrelativistic quantum mechanics large-TV expansions are surprisingly successful.^ For a (O^)^ theory the very first term in the large-TV expansion defines a nontrivial and renormalizable quantum field theory."* Also quantum chromodynamics at large TV displays interesting theoretical and phenomenological features.^ In this Letter we propose the possibility of introducing an artificial perturbation parameter 5 in the exponent of the interaction term^; that is, we consider a Lagrangean of the formwhere d is the space-time dimensionality, X is dimensionless, and Af is a mass parameter that sets the dimensions of the interacti...

Convergence of the linearδexpansion for the shift inTcfor Bose-Einstein condensation

“…The linear δ expansion is much easier to apply than the nonlinear δ expansion in which the lagrangian that interpolates between L 0 and L involves an operator that is a nonlinear function of δ [23]. The lagrangian L 0 for the exactly solvable theory typically involves an arbitrary parameter m. At δ = 1, the lagrangian (22) is independent of m. If a perturbation series in δ converges at δ = 1, it must converge to a value that is independent of m. However at any finite order in the δ expansion, results depend on m. As m varies over its physical range, the prediction for the observable often extends out to ±∞.…”

Nonperturbative calculation of symmetry breaking in quantum field theory

“…For a scalar quantum field theory, the conventional approach [2] has been to introduce the parameter δ into the Euclidean Lagrangian density…”

A comment on the relationship between differential and dimensional renormalization

“…* Another way to regulate this massless λφ n theory is to keep d at its integer value but take n to be non-integer, n = N + δ with N an integer: this is the "δ-expansion" of Bender and collaborators [12].…”