How to Reduce Practically Any Problem to One Dimension

  • Daniel C. Mattis
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 23)


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.


Soliton Stein Kelly Summing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Footnotes and References

  1. 1.
    C. Lanczos, J. Res. N.B.S. 45, 255 (1950). See alsoMathSciNetGoogle Scholar
  2. 1a.
    G. Strang, “Linear Algebra and its Applications”, Academic, New York, 1975, pp. 276 et seq.Google Scholar
  3. 2.
    R. Haydock, J. Phys. A7, 2120 (1974)ADSMathSciNetGoogle Scholar
  4. 2a.
    R. Haydock, J. Phys. ibid A10, 461 (1977)ADSGoogle Scholar
  5. 2b.
    R. Haydock, V. Heine and M.J. Kelly, J. Phys. C5, 2845 (1972) andADSGoogle Scholar
  6. 2c.
    R. Haydock, V. Heine and M.J. Kelly, J. Phys. C8, 2591 (1975).ADSGoogle Scholar
  7. 3.
    J. Stein and U. Krey, J. Magn. and Magn. Mat. 6, 196 (1977)CrossRefADSGoogle Scholar
  8. 3a.
    J. Stein and U. Krey, Solid State Comm. 27, 797 (1978)CrossRefADSGoogle Scholar
  9. 3b.
    J. Stein and U. Krey, Solid State Comm. 27, ibid., 1405 (1978)CrossRefADSGoogle Scholar
  10. 3c.
    J. Stein and U. Krey, Z. Phys. B34, 287 (1979); alsoADSGoogle Scholar
  11. 3d.
    J. Stein and U. Krey, R. Haydock, Phil. Mag. B37, 97 (1978).Google Scholar
  12. 4.
    D.C. Mattis and R. Raghavan, Phys. Lett. 75A, 313 (1980).ADSGoogle Scholar
  13. 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
  14. 6.
    K.S. Dy, Shi-Yu Wu and T. Spratlin, Phys. Rev. B20, 4237 (1979).ADSGoogle Scholar
  15. 7.
    See various chapters in E.H. Lieb and D.C. Mattis, “Mathematical Physics in One Dimension”, Academic, New York, 1965; orGoogle Scholar
  16. 7a.
    P.J. Dean, Rev. Mod. Phys. 44, 127 (1972).CrossRefADSGoogle Scholar
  17. 8.
    M.L. Mehta, “Random Matrices”, Academic, New York, 1967.MATHGoogle Scholar
  18. 9.
    P.A. Wolff, Phys. Rev. 124, 1030 (1961).CrossRefADSGoogle Scholar
  19. 10.
    D.C. Mattis, Ann. of Phys. (N.Y.) 89, 45 (1975).CrossRefADSGoogle Scholar
  20. 12.
    D.C. Mattis, Phys. Rev. Lett. 19, 1478 (1967).CrossRefADSGoogle Scholar
  21. 13.
    J. Moser, Am. Scientist 67, 1689 (1979).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • Daniel C. Mattis
    • 1
  1. 1.Physics DepartmentUniversity of UtahSalt Lake CityUSA

Personalised recommendations