L. M. Simmons

A new perturbative approach to nonlinear problems

A recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the Lagrangian such as φ4 by (φ2)1+δ and then treating δ as a small parameter. It is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the Lane–Emden, Thomas–Fermi, Blasius, and Duffing equations.

Expansion formulas and addition theorems for Gegenbauer functions

We give a systematic summary of the properties of the Gegenbauer functions Cαλ(x) and Dαλ(x) for general complex degree and order, with emphasis on the functions of the second kind, Dαλ(x), and on results useful in scattering theory. The results presented include Sommerfeld–Watson type expansion formulas and two reciprocal addition formulas for the functions of the second kind.

Limiting Spectra from Confining Potentials.

For confining potentials and large quantum numbers, the bound‐state energies rise more rapidly as a function of n the more rapidly the potential rises with distance. However, the spectrum can rise no faster than n2 in the nonrelativistic case, or n in the relativistic case.

Coherent states for the “isotonic oscillator”☆

Abstract Gaussians are not, as a matter of principle, allowable wave functions for the “isotonic oscillator” system. Appropriate coherent states for nonharmonic potentials are minimum-uncertainty coherent states. Moreover, they provide a better approximation to the classical motion than do gaussians.

δ expansion for a quantum field theory in the nonperturbative regime

The δ expansion, a recently proposed nonperturbative technique in quantum field theory, is used to calculate the dimensionless renormalized coupling constant of a λ(φ2)1+δ quantum field theory in d‐dimensional space‐time at the critical point defined by λ→∞ with the renormalized mass held fixed. The calculation is performed to leading order in δ and compared with previous lattice strong‐coupling calculations. The numerical results are good and provide new evidence that the theory in four dimensions is free for all δ.

The delta expansion in zero dimensions

The recently introduced δ‐expansion (or logarithmic‐expansion) technique for obtaining nonperturbative information about quantum field theories is reviewed in the zero‐dimensional context. There, it is easy to study questions of analytic continuation that arise in the construction of the Feynman rules that generate the δ series. It is found that for six‐ and higher‐point Green’s functions, a cancellation occurs among the most divergent terms, and that divergences that arise from summing over an infinite number of internal lines are illusory. The numerical accuracy is studied in some detail: The δ series converges inside a circle of radius one for positive bare mass squared, and diverges if the bare mass squared is negative, but in all cases, low‐order Pade approximants are extremely accurate. These general features are expected to hold in higher dimensions, such as four.

Triviality of monomial Higgs potentials

Abstract The δ expansion method is used to study all scalar field theories with symmetric monomial self-interactions of the form ( φ 2 ) p in four dimensions. With a momentum cut-off Λ it is shown that these theories are all distinct, admit spontaneous symmetry breaking, and have a nonperturbative structure in the coupling constant λ. In the limit that the momentum cut-off is taken to infinity, it is shown that the δ expansion of all these theories degenerates to the renormalized expansion of a single theory independent of p . That expansion is the renormalized weak-coupling expansion of λφ 4 . It is argued that all theories of this class are the same in the limit Λ → ∞. This universal behavior of all monomial interactions with the discrete symmetry φ → − φ is interpreted as a strong indication that all such theories are free, as λφ 4 is believed to be.

Strong-coupling calculation of the mass ratio in supersymmetric field theory

Abstract The purpose of this paper is to demonstrate the suitability of strong-coupling expansions as a technique for obtaining useful numerical information from a supersymmetric field theory. Strong-coupling methods involve introducing a lattice which explicitly breaks supersymmetry invariance. However, we present compelling evidence that supersymmetry is restored in the continuum limit. Specifically, we calculate the ratio of the fermion mass to the boson mass and show that to seventh order in perturbation theory this ratio approaches one to within one percent.