How to Reduce Practically Any Problem to One Dimension
A large number of interesting problems of theoretical physics are exactly soluble in one spatial dimension (1D). These include studies in statistical mechanics of interacting or of random systems (solved by diagonalization of a transfer matrix), many-body problems (solved by a Bethe ansatz), nonlinear dynamics (leading to solitons), magnetism (reducible by the Jordan-Wigner transformation to a many-fermion problem), etc. Some of these studies apply to physically interesting systems such as polymers, magnetic molecules, and energy transport in long chain molecules, yet others might appear to be of academic interest only. Surprisingly, even some far-fetched work in 1D may ultimately find important physical applications in three spatial dimensions. In this paper I shall endeavor to demonstrate what sorts of problems, arising in arbitrary dimensions, can be successfully reduced to 1D where they can be analyzed and ultimately solved.
KeywordsSoliton Stein Kelly Summing
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Footnotes and References
- 1a.G. Strang, “Linear Algebra and its Applications”, Academic, New York, 1975, pp. 276 et seq.Google Scholar
- 3d.J. Stein and U. Krey, R. Haydock, Phil. Mag. B37, 97 (1978).Google Scholar
- 5.R.F. Hausman Jr. and C.F. Bender, in “Methods of Electronic Structure Theory”, H. F. Schaefer III, Ed., Plenum, New York 1977, p. 319.Google Scholar
- 7.See various chapters in E.H. Lieb and D.C. Mattis, “Mathematical Physics in One Dimension”, Academic, New York, 1965; orGoogle Scholar
- 13.J. Moser, Am. Scientist 67, 1689 (1979).Google Scholar