A Nonstandard Analysis Approach to the Theory of Quantum Meanfield Systems

Manfred P. H. Wolff

M. O. Farrukh

D. Baillie, P. B. Blakie, Ashton S. Bradley

The phenomenal recent progress with the control of the external degrees of freedom of ultra-cold gases, and the advent of new species with long range interactions presents a range of new and exciting systems for exploring manybody physics. The meanfield description of these systems is challenging and demands the development of new systematic theoretical approaches that lead to tractable and accurate numerical schemes. In this work we present a range of applications of finite temperature meanfield theory to such systems including:

Jeffrey Barrett, Isaac Goldbring

Using the tools of nonstandard analysis, we develop and present an alternative formulation of Bohmian mechanics. This approach allows one to describe a broader assortment of physical systems than the standard formulation of the theory. It also allows one to make predictions in more situations. We motivate the nonstandard formulation with a Bohmian example system that exhibits behavior akin to Earman's (1986) classical space invaders and reverse space invaders. We then use the example to illustrate how the alternative formulation of Bohmian mechanics works.

Tomio Yamakoshi Petrosky, I. Prigogine, S. Tasaki

A. Raab

We use nonstandard analysis to formulate quantum mechanics in hyperfinite-dimensional spaces. Self-adjoint operators on hyperfinite-dimensional spaces have complete eigensets, and bound states and continuum states of a Hamiltonian can thus be treated on an equal footing. We show that the formalism extends the standard formulation of quantum mechanics. To this end we develop the Loeb-function calculus in nonstandard hulls. The idea is to perform calculations in a hyperfinite-dimensional space, but to interpret expectation values in the corresponding nonstandard hull. We further apply the framework to nonrelativistic quantum scattering theory. For time-dependent scattering theory, we identify the starting time and the finishing time of a scattering experiment, and we obtain a natural separation of time scales on which the preparation process, the interaction process, and the detection process take place. For time-independent scattering theory, we derive rigorously explicit formulas for the Mo/ller wave operators and the S-matrix.