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Title: On Conditional Wiener Integrals and a Novel Approach to the Fermion Sign Problem
Authors: Dumas, Warwick Michael
Supervisors: Tretyakov, Michael
Award date: 7-Jul-2010
Presented at: University of Leicester
Abstract: The path-integral formulation of nonrelativistic quantum mechanics was introduced by Feynman in 1948. The use of Path Integral Monte Carlo can be put on a rigorous footing using conditional Wiener integrals. This thesis addresses the topics both of numerical error and of Monte Carlo error. A piecewise constant numerical method which is of second order of accuracy for computing conditional Wiener integrals for a rather general class of sufficiently smooth functional is proposed. The method is based on simulation of Brownian bridges via the corresponding stochastic differential equations (SDEs) and on ideas of the weak-sense numerical integration of SDEs. A convergence theorem is proved. Special attention is paid to integral-type functionals. Results of some numerical experiments are presented. In a further part of the research, the goal is to develop Monte Carlo methods for fermion simulations that are resistant to the explosion of variance which happens due to the fermion sign problem. A novel approach is developed which represents a radical departure from the current approaches. This is based on the principle of using a geometrical interpretation of the problem in order to find ways to maximize the negative covariance between the countersigned functional contributions. The fundamental connection between quantum exchange and the fermion sign problem is exploited. It is shown that this leads to a mathematical proof of the well-known exact solution to the sign problem for 1-dimensional fermion systems, and also to a novel exact solution in the case of a pair of 2-dimensional fermions.
Type: Thesis
Level: Doctoral
Qualification: PhD
Appears in Collections:Theses, Dept. of Mathematics
Leicester Theses

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2010dumaswmphd.pdfThesis2.24 MBAdobe PDFView/Open
lasfirst.cSource code for an experiment in Ch. 718.93 kBascii textView/Open
lasfirst.hSource code for an experiment in Ch. 7 847 Bascii textView/Open
auxitime.cAuxiliary source file needed to compile code653 Bascii textView/Open
0.fbaInitial Fibonacci data for setting up the random number generator22.5 kBascii textView/Open

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