In Appreciation

Leslie Foldy’s diminutive stature and modest demeanor gave little clue to the powerful intellect responsible for several significant advances in theoretical physics.Two were particularly important. His 1945 theory of the multiple scattering of waves laid out the fundamentals that most modern theories have followed (and sometimes rediscovered), while his work with Siegfried Wouthuysen on the nonrelativistic limit of the Dirac equation opened the way to a wealth of valuable insights. In this article we recall some of the milestones along Foldy’s path through a life in physics. Some of the anecdotes we report here were related to one of the authors (PLT) just before an event in 2000 celebrating Foldy’s 80th birthday, while others were told to us over the course of the nearly forty years during which we were colleagues. Still others were uncovered during the course of WJF’s research for his book, Physics at a Research University: Case Western Reserve 1830–1990 (Cleveland: Case Western Reserve University, 2006). Other details were provided by Foldy’s widow, Roma. Philip L. Taylor is the Perkins Professor of Physics and Professor of Macromolecular Science and Engineering at Case Western Reserve University. William J. Fickinger is Professor Emeritus of Physics at Case Western Reserve University.     

Foldy–Lax approximation on multiple scattering by many small scatterers

The multiple scattering of time harmonic wave emitted by a localized source through a medium with many scatterers can be approximated by an Foldy–Lax self-consistent system when the relative radius of each scatterer is small and the distribution of scatterers is sparse. The scattering amplitude in the Foldy–Lax self-consistent system will be specified in terms of scatterer volume and scattering strength. By neglecting the self-interaction effect, the difference from the exciting field in the Foldy–Lax formula to the analytic wave field given implicitly by the Lippmann–Schwinger integral equation is compared. An upper bound of the difference is obtained in terms of scaled radius and sparsity of the distribution of the scatterers.     

Separation of Dirac's Hamiltonian by Van Vleck transformation

ABSTRACT

The now classic Foldy–Wouthuysen transformation (FWT) was introduced as successive unitary transformations. This fundamental idea has become the standard in later developments such as the Douglas–Kroll transformation (DKT) – but it is not the only possibility. FWT can be seen as a simple special case of the general Van Vleck transformation (VVT) which besides the successive version has another, known as the canonical because of a series of nice mathematical properties discovered gradually over time. The aim of the present paper is to compare the two approaches – which give identical results in the lower orders, but not in the higher. After having recapitalised both, we apply them to Dirac's Hamiltonian for the electron in a constant electromagnetic field, written with so few assumptions about the operators that the mathematical techniques stand out separated from the terminology of relativistic quantum mechanics. FWT for a free particle is dealt with by a recent geometric approach to VVT. The original FWT is continued through the next non-zero orders. DKT is considered with special weight on equivalent formulations of the generalised and the optimised forms introduced by Wolf, Reiher and Hess.     

The Foldy‐Lax approximation of the scattered waves by many small bodies for the Lamé system

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by many small rigid obstacles of arbitrary, Lipschitz regular, shapes in 3D case. We prove that there exist two constants a₀ and c₀, depending only on the Lipschitz character of the obstacles, such that under the conditions and on the number M of the obstacles, their maximum diameter a and the minimum distance between them d, the corresponding Foldy‐Lax approximation of the farfields is valid. In addition, we provide the error of this approximation explicitly in terms of the three parameters and d. These approximations can be used, in particular, in the identification problems (i.e. inverse problems) and in the design problems (i.e. effective medium theory).