Stochastic quantization, associated supersymmetry and Nicolai map

  • Jnanadeva Maharana
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 208)


In these lectures we present an introduction to the method of stochastic quantization due to Parisi and Wu and demonstrate the equivalence between stochastic and canonical methods of quantization in perturbation theory using the techniques due to Floratos and Iliopoulos. The lower critical dimension of spin systems with and without Gaussian random fields is discussed to illustrate the underlying supersymmetry of the system associated with stochastic differential equations. The Nicolai map is constructed for a simple model and its interpretation as a stochastic evolution equation is envisaged in a supersymmetric quantum mechanical model. The constraints on supersymmetry breaking, the Witten index and conjugation operations are discussed with some simple examples. The appendix contains relevant definitions and formulas of stochastic processes.


Spin System Supersymmetry Breaking Langevin Equation Supersymmetric Theory Witten Index 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L.D. Faddeev and V.N. Popov, Phys. Lett. 25B, 29 (1967); R.P. Feynmann, Acta Phys. Polonica 26, 697 (1963); For a review see E. Abers and B.W. Lee, Phys. Rep. 9C, 1 (1973).Google Scholar
  2. 2.
    V.N. Gribov, Nucl. Phys. BT39, 1 (1978).Google Scholar
  3. 3.
    G. Parisi and Y.-S. Wu, Sc. Sin. 24, 483 (1981).Google Scholar
  4. 4.
    P. Langevin, Comptes rendus 146, 530 (1908); Excellent exposition to stochastic processes relevant for physicist can be found inGoogle Scholar
  5. 4a.
    S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943); M.C. Wang and G.E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945).Google Scholar
  6. 5.
    G. Parisi, Nucl. Phys. B180, 378 (1981); B205, 337 (1982); F. Fucito and E. Marinari, Nucl. Phys. B190, (1981).Google Scholar
  7. 6.
    J. Alfaro and B. Sakita, Phys. Lett. 121B, 339 (1983).Google Scholar
  8. 7.
    D. Zwanziger, Nucl. Phys. B192, 259 (1981); L. Baulieu and D. Zwanziger, Nucl. Phys. B193, 161, (1981).Google Scholar
  9. 8.
    E. Floratos and J. Iliopoulos, Nucl. Phys. B214, 392 (1983).Google Scholar
  10. 9.
    J. Breit, S. Gupta and A. Zaks, IAS Princeton preprint. Also see, Chaturdevi, A.K. Kapoor and G. Srinivasan, Hyderabad Univ.Preprints.Google Scholar
  11. 10.
    G. Parisi and N. Sourlas, Phys. Rev. Lett. 43, 744 (1979).Google Scholar
  12. 11.
    Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 (1975).Google Scholar
  13. 12.
    H. Nicolai, Phys. Lett. 89B, 341 (1980); 101B, 396 (1981); Nucl. Phys. B176, 419 (1980).Google Scholar
  14. 13.
    S. Cecôtti and L. Girardello, Nucl. Phys. {uB226}, 417 (1983)Google Scholar
  15. 13a.
    N. Sakai and I. Sakamoto, Nucl. Phys. B229, 173 (1983).Google Scholar
  16. 14.
    E. Witten, Nucl. Phys. B202, 253 (1982).Google Scholar
  17. 15.
    For reviews see R. Graham, Springer Tracts in Modern Physics 66, 1 (1973); P.C. Hohenberg and B. Halperin, Rev. Mod. Phys. 49, 435 (1977); R.F. Fox, Phys. Rep. 48C, 179 (1978).Google Scholar
  18. 16.
    Discussion on the analogy between statistical mechanical system and quantum field theory can be found in A. Strominger, Ann. Phys. 146, 419 (1983), β ↔ h Action ↔ Energy. Also see J. DeAlfaro, S. Fubini and G. Furlan, CERN preprint 3426 (1982).Google Scholar
  19. 17.
    K. Symanzik in Lectures in theoretical physics III, Ed. E. Brittin et al. Int. Science New York 1961.Google Scholar
  20. 18.
    For an introduction to supersymmetry see S. Joglekar. These proceedings. Note that the supersymmetry of egs.(70)–(73) is generated by an anticommuting four-vector. Thus it is different from the usual supersymmetry in field theory which is generated by an anticommuting spinor.Google Scholar
  21. 19.
    E. Gozzi preprint CCNY-HEP-83/4.Google Scholar
  22. 20.
    E. Witten,Trieste Lectures (1981); P. Salmanson and J.W. Van Holten,Nucl. Phys. B196, 509 (1982); F. Cooper and B. Freedman, Ann. Phys. 146, 262-T1T383).Google Scholar
  23. 21.
    P.T. Matthews and A. Salam, Nuovo Cimento 2, 120 (1955).Google Scholar
  24. 22.
    F.A. Berezin, The Method of Second Quantization. Academic Press, New York, 1966.Google Scholar
  25. 23.
    H. Nicolai in Group theoretic methods in physics (Springer, Heidelberg 1978).Google Scholar
  26. 24.
    S. Cecotti and L. Girardello, Ann. Phys. 145, 81 (1983).Google Scholar
  27. 25.
    G. Parisi and N. Sourlas, Nucl. Phys. B206, 321 (1983). Also see J.L. Cardy, Phys. Letters 125B, 470 (1983); A. Klein and J.F.Perez, Phys. Letters 125B, 473 (1987T; A. Niemi, Phys. Rev. Lett. 49, 1808 (1982).Google Scholar
  28. 26.
    S. Cecotti and L. Girardello, Phys. Lett. 110B, 39 (1982).Google Scholar
  29. 27.
    L. Alvarez-Gaumes, Commun. Math. Phys. 90, 61 (1983) and Harvard preprint HUTP-83/A035, L. Girardello, C Imbimbo and S. Mukhi, Phys. Lett. 132B, 69 (1983).Google Scholar
  30. 28.
    J.L. Doob, Ann. Math. 43, 351 (1942). *** DIRECT SUPPORT *** A3418165 00004Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Jnanadeva Maharana
    • 1
  1. 1.Institute of PhysicsBhubaneswarIndia

Personalised recommendations