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Large-N Expansions

  • Ashok Chatterjee
Part of the Lecture Notes in Chemistry book series (LNC, volume 50)

Abstract

Several quantum mechanical theories and statistical and field theoretic models of physical interest possess continuous group symmetry often in some internal space. Many of these theories admit straightforward extensions in which the number of internal degrees of freedom N may be treated as a free variable parameter. Inflating the given number N to infinity then entails, quite surprisingly, a drastic simplification in the analyses of a diverse class of such theories. The method of large-N expansion hinges on the fact that if the large-N limit of such a problem can be obtained explicitly then the finite-N corrections can be incorporated by introducing a systematic expansion in powers of 1/N and contact with the original theory may be made by substituting for N its given fixed value at the end of the calculation. This inverse-N expansion technique has emerged in recent years as a powerful approximation scheme in fields as disparate as quantum mechanics, nuclear physics, critical phenomena, laser physics and quantum chromodynamics [1,2]. Here we shall however restrict our discussion mainly to quantum mechanics where the large-N expansion was first applied by Ferrei and Scalapino [3] in 1974 and since then interest in this subject has continued unabated.

Keywords

Ground State Energy Critical Phenomenon Helium Atom Angular Momentum Quantum Number Generalize Coherent State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    E. Witten, Phys. Today 33, No. 7 (1980) 38.CrossRefGoogle Scholar
  2. [2]
    L.G. Yaffe, Phys. Today 36, No. 8 (1983) 50.CrossRefGoogle Scholar
  3. [3]
    R.A. Ferrei and D.J. Scalapino, Phys. Rev. A 9 (1974) 846.CrossRefGoogle Scholar
  4. [4]
    R.S. Gangopadhyay, Ph.D. Thesis, University of Calcutta (1985).Google Scholar
  5. [5]
    D.J. Gross, in:Methods in field theory, eds. R. Balian and J. Zinn-Justin (North-Holland, Amsterdam, 1976).Google Scholar
  6. [6]
    S. Ma, Modern theory of critical phenomena, (W.A. Benjamin Inc., 1976).Google Scholar
  7. [7]
    S. Ma, in: Phase transitions and critical phenomena, vol. 6, eds. C. Domb and M.S. Green (Academic Press, New York, 1976) p. 250.Google Scholar
  8. [8]
    T.H. Berlin and M. Kac, Phys. Rev. 86 (1952) 821.CrossRefGoogle Scholar
  9. [9]
    H.E. Stanley, Phys. Rev. 176 (1968) 718.CrossRefGoogle Scholar
  10. [10]
    R. Abe, Prog. Theor. Phys. 48 (1972) 1414.CrossRefGoogle Scholar
  11. [11]
    R. Abe and S. Hikami, Prog. Theor. Phys. 49 (1973) 442.CrossRefGoogle Scholar
  12. [12]
    R. Abe, Prog. Theor. Phys. 49 (1973) 1074.CrossRefGoogle Scholar
  13. [13]
    R.A. Ferrei and D.J. Scalapino, Phys. Rev. Lett. 29 (1972) 413.CrossRefGoogle Scholar
  14. [14]
    J.D. Louck, J. Mol. Spectroscopy 4 (1960) 298.CrossRefGoogle Scholar
  15. [15]
    A. Chatterjee, J. Phys. A: Math. Gen. 18 (1985) 735.CrossRefGoogle Scholar
  16. [16]
    S. Kalara, University of Rochester Report No. UR-812 (1982).Google Scholar
  17. [17]
    J. Ader, Phys. Lett. A 97 (1983) 178.CrossRefGoogle Scholar
  18. [18]
    S. Hikami and E. Brézin, J. Phys. A: Math. Gen. 12 (1979) 759.CrossRefGoogle Scholar
  19. [19]
    L.D. Mlodinow and M. Shatz, J. Math. Phys. 25 (1984) 943.CrossRefGoogle Scholar
  20. [20]
    G. Moreno and A. Zepeda, J. Phys. B: At. Mol. Phys. 17 (1984) 21.CrossRefGoogle Scholar
  21. [21]
    A. Chatterjee, J. Phys. A: Math. Gen. 18 (1985) 1193.CrossRefGoogle Scholar
  22. [22]
    A. Chatterjee, Phys. Rev. A 31 (1986) 2470.CrossRefGoogle Scholar
  23. [23]
    S. Erkoc and R. Sever, Phys. Rev. D 33 (1986) 588.CrossRefGoogle Scholar
  24. [24]
    M. Jameel, J. Phys. A: Math. Gen. 19 (1986) 1967.CrossRefGoogle Scholar
  25. [25]
    C.H. Lai, J. Math. Phys. 28 (1987) 1801.CrossRefGoogle Scholar
  26. [26]
    A. Chatterjee, Phys. Rev. A 35 (1987) 2722.CrossRefGoogle Scholar
  27. [27]
    A. Jevicki and N. Papanicolaou, Nucl. Phys. B 171 (1980) 362.CrossRefGoogle Scholar
  28. [28]
    L.D. Mlodinow and N. Papanicolaou, Ann. Phys. (N.Y.) 128 (1980) 314.CrossRefGoogle Scholar
  29. [29]
    A. Jevicki and B. Sakita, Nucl. Phys. B 185 (1981) 89.CrossRefGoogle Scholar
  30. [30]
    S.R. Wadia, Chicago, Enrico Fermi Institute preprint 80/47 (1980).Google Scholar
  31. [31]
    L.G. Yaffe, Rev. Mod. Phys. 54 (1982) 407.CrossRefGoogle Scholar
  32. [32]
    P.W. Anderson, Phys. Rev. 112 (1958) 1900.CrossRefGoogle Scholar
  33. [33]
    F.A. Berezin, Comm. Math. Phys. 63 (1978) 131.CrossRefGoogle Scholar
  34. [34]
    L.D. Mlodinow and N. Papanicolaou, Ann. Phys. (N.Y.) 131 (1981) 1.CrossRefGoogle Scholar
  35. [35]
    P. du. T. Van der Merwe, J. Chem. Phys. 81 (1984) 5976.CrossRefGoogle Scholar
  36. [36]
    P. du. T. Van der Merwe, J. Chem. Phys. 82 (1985) 5293.CrossRefGoogle Scholar
  37. [37]
    D.J. Doren and D.R. Herschbach, Chem. Phys. Lett. 118 (1985) 115.CrossRefGoogle Scholar
  38. [38]
    D.R. Herschbach, J. Chem. Phys. 84 (1986) 839.CrossRefGoogle Scholar
  39. [39]
    D.J. Doren and D.R. Herschbach, J. Chem. Phys. 87 (1987) 433.CrossRefGoogle Scholar
  40. [40]
    D.Z. Goodson and D.R. Herschbach, Phys. Rev. Lett. 58 (1987) 1628.CrossRefGoogle Scholar
  41. [41]
    J.L. Miramontes and C. Pajares, Nuovo Cimento B 84 (1984) 10.CrossRefGoogle Scholar
  42. [42]
    A. Chatterjee, J. Math. Phys. 27 (1986) 2331.CrossRefGoogle Scholar
  43. [43]
    U. Sukhatme and T. Imbo, Phys. Rev. D 28 (1983) 418.CrossRefGoogle Scholar
  44. [44]
    T. Imbo, A. Pagnamenta and U. Sukhatme, Phys. Rev. D 29 (1984) 1669.CrossRefGoogle Scholar
  45. [45]
    D.J. Doren and D.R. Herschbach, Phys. Rev. A 34 (1986) 2654.CrossRefGoogle Scholar
  46. [46]
    D.J. Doren and D.R. Herschbach, Phys. Rev. A 34 (1986) 2666.Google Scholar
  47. [47]
    S.A. Maluendes, F.M. Fernandez and E.A. Castro, Phys. Lett. A 124 (1987) 215.CrossRefGoogle Scholar
  48. [48]
    T. Imbo and U. Sukhatme, Phys. Rev. Lett. 54. (1985) 2186.CrossRefGoogle Scholar
  49. [49]
    R. Dutt and Y.P. Varshni, Z. Phys. D 2 (1986) 207.CrossRefGoogle Scholar
  50. [50]
    R. Dutt and Y.P. Varshni, J. Phys. B: At. Mol. Phys. 20 (1987) 2437.CrossRefGoogle Scholar
  51. [51]
    Y.P. Varshni, Phys. Rev. A 36 (1987) 3009.CrossRefGoogle Scholar
  52. [52]
    S. Coleman, R. Jackiw and H.D. Politzu, Phys. Rev. D 10 (1974) 2491.CrossRefGoogle Scholar
  53. [53]
    D. Gross and A. Neveu Phys. Rev. D 10 (1974) 3235.CrossRefGoogle Scholar
  54. [54]
    R. Dashen, B. Hasslacher and A. Neveu Phys. Rev. D 13 (1975) 2215.Google Scholar
  55. [55]
    G. ’t Hooft, Nucl. Phys. B 72 (1974) 461.CrossRefGoogle Scholar
  56. [56]
    E. Witten, Nucl. Phys. B 160 (1979) 57.CrossRefGoogle Scholar
  57. [57]
    T.V. Ramakrishnan, in: Valence fluctuations in solids, eds. L.M. Falicov, W. Hanke and M.S. Maple (North-Holland, Amsterdam, 1981) p. 28.Google Scholar
  58. [58]
    N. Read and D.M. Newns, J. Phys. C: Solid State Phys. 16 (1983) 3272.CrossRefGoogle Scholar
  59. [59]
    P. Coleman, Phys. Rev. B 28.(1983) 5255.Google Scholar
  60. [60]
    Y. Goldschmidt, Phys. Rev. B 30 (1984) 1632.CrossRefGoogle Scholar
  61. [61]
    D. Schmeltzer, J. Phys. C: Solid State Phys. 20 (1987) 3131.CrossRefGoogle Scholar
  62. [62]
    C.M. Bender, L.D. Mlodinow and N. Papanicolaou, Phys. Rev. A 25 (1982) 1305.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Ashok Chatterjee
    • 1
  1. 1.Department of Theoretical PhysicsIndian Association for the Cultivation of ScienceJadavpur, CalcuttaIndia

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