Clustering and Clusters

  • André Krzywicki
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 31)


When one is lecturing in a centre for interdisciplinary research, it is, I imagine, appropriate to start by emphasizing the interdisciplinary character of the topic one is going to discuss. In my case this does not present any difficulty. Clustering is a pictorial term for positive correlations in an ensemble of randomly distributed bodies or events, which can be represented in some space by a system of random points. Probabilistic techniques which enable one to study such systems constitute the theory of stochastic point processes (the word “process” is used because in applications one often considers time series of correlated events). The theory of stochastic point processes found applications in radio physics, in epidemiology, in studying certain aspects of road traffic, in quantum optics etc.


Transverse Momentum Cluster Model Random Point Multiplicity Distribution Inclusive Probability 
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  1. 1.
    See e. g. R.L. Stratonovich, Topics in the Theory of Random Noise, Vol. 1, Gordon and Breach, New York 1973; also O. Macchi, Processus Ponctuels et Coincidences, Thesis, Université de Paris-Sud, 1972.Google Scholar
  2. 2.
    See Z. Koba, Proceedings CERN-JINR School of Physics Ebeltohoft 1973 and references therein; following Koba we employ the operator techniques first used in this context by K.J. Biebl and J. Wolf, Nucl. Phys. B44, 301 (1972)CrossRefGoogle Scholar
  3. 3.
    M. Le Bellac, Phys. Lett, B37, 413 (1971)ADSGoogle Scholar
  4. 4.
    R.C. Arnold and G.H. Thomas, Phys. Lett. B47, 371 (1973); G.H. Thomas, Phys. Rev. D8, 3042 (1973).ADSGoogle Scholar
  5. 5.
    A. Krzywicki, C. Quigg and G.H. Thomas, Phys. Lett. B57, 369 (1975).Google Scholar
  6. 6.
    An exhausive bibliography on cluster models would take much more space than we have at our disposal. The list of early papers, where the model has been applied to the phenomenology of multiparticle production, includes: P. Pirila and S.Pokorski, Phys. Lett. B43, 502 (1973); C. Quigg and G.H. Thomas, Phys. Rev. D7, 2752 (1973); A. Bialas, K. Fialkowski and K. Zalewski, Phys. Lett. B45, 337 (1973); J. Ranft and G. Ranft, Phys. Lett. B45, 43 (1973); E.L. Berger and G.C. Fox, Phys. Lett. B471 162 (1973) S. Pokorski and L. Van Hove, Acta Phys. Polon. B5, 229 (1974); F. Hayot and A. Morel, Nucl. Phys. B68, 323 (1974); E.L. Berger, Nucl. Phys. B85, 61 (1975) Further references can be found in the following review papers: J. Ranft, Proceedings of the 5th International Symposium on Many Particle Hadrodynamics, Leipzig 1974; K. Zalewski, Proceedings of the 17th International Conference on High Energy Physics, London 1974; P. Darriulat, Proceedings of the 6th International Colloquium on Multiparticle Reactions, Oxford 1975; L. Foa, Physics Reports C22, 1(1975); See also refs. 8, 20, 21, 23.Google Scholar
  7. 7.
    W. Feller, An Introduction to Probability Theory and its Applications, vols. I and II, Willey & Sons Inc. New York 1971.Google Scholar
  8. 8.
    C. Quigg, P. Pirila and G.H. Thomas, Phys. Rev. Lett. 34, 290 (1975). It is tacitly assumed in this paper that po =O.ADSCrossRefGoogle Scholar
  9. 9.
    A.H. Mueller, Phys. Rev. D2, 2963 (1970)ADSGoogle Scholar
  10. 10.
    See e. g. the following reviews: W.R. Frazer et al., Rev. Mod. Phys. 44, 284 (1972); Chan Hong-Mo, Proceedings of the CERN School of Physics, Grado 1972; P. Salin, Ecole d’fite de Physique des Particules, Gif 1973 and references therein.ADSCrossRefGoogle Scholar
  11. 11.
    The multiperipheral model of Amati, Bertocchi, Fubini, Stanghellini and Tonin is the prototype of SRO ideas: see the review by S. Fubini, Proceedings of the Scottich Universities Summer School 1963 and references therein; K.G. Wilson, Proceedings of the ScottishGoogle Scholar
  12. 12.
    Universities Suiruner School 1973 and references therein; R.P. Feynman, Photon Hadron Interactions, Benjamin Inc., Reading Mass. 1973; the most recent review is, to my knowledge, the one by M. LeBellac, CERN Academic Training Programme, CERN 76 - 14 (1976).Google Scholar
  13. 13.
    See the paper by K.G. Wilson quoted in our ref. 11.Google Scholar
  14. 14.
    Argonne/Fermilab/ Stony Brook Collaboration and Michigan/Rochester Collaboration as quoted in our ref. 20.Google Scholar
  15. 15.
    See the long list of references given by P. Darriulat in his review quoted in our ref. 6.Google Scholar
  16. 16.
    J. Derre et al., French-Soviet Collaboration, Saclay preprint M-12 (1974); C. Bromberg et al., Phys. Rev. D12, 1224 (1975); D. Fong et al, Phys. Lett. B61, 99 (1976); J. Lamsa et al., Phys. Rev. Let. 37, 73 (1976).Google Scholar
  17. 17.
    A. Krzywicki and D. Weingarten, Phys. Lett. B50, 265 (1074);Google Scholar
  18. 18.
    A. Krzywicki, Nucl. Phys. B86, 296 (1974) and Proceedings of the 10th Rencontre de Moriond (1975)Google Scholar
  19. 19.
    D. Weingarten, Phys. Rev. D11, 1924 (1975) and Phys. Rev. D13 1494 (1976).Google Scholar
  20. 20.
    See the review by A. Krzywicki, Proceedings of the 6th International Colloquium on Multiparticle Reactions, Oxford 197 5 and references therein.Google Scholar
  21. 21.
    P. Darriulat, already quoted in our ref. 6; T. Ferbel, E. Majorana School of Subnuclear Physics, Erice 1976 (Rochester preprint UR-587, to be published in the Proceedings of the School).Google Scholar
  22. 22.
    See e. g. D.R. Snider, Phys. Rev. D11, 140 (1975).ADSCrossRefGoogle Scholar
  23. 23.
    P. Pirila, C. Quigg and G.H. Thomas, Phys. Rev. D12 92 (1975)ADSCrossRefGoogle Scholar
  24. 24.
    See e. g. F. Hayot, Lett, al Nuovo Cim., V2, 676 (1975).CrossRefGoogle Scholar
  25. 25.
    E.L. Berger, G.C. Fox and A. Krzywicki, Phys. Lett. B43, 132 (1973).Google Scholar
  26. 26.
    A. Bialas, M. Jacob and S. Pokorski, Nucl. Phys. B75 259 (1974).ADSCrossRefGoogle Scholar
  27. 27.
    A. Arneodo and G. Plaut, Nucl. Phys. B107, 275 (1976) and Nice preprint NTH-76/3 (1976), to be published.Google Scholar
  28. 28.
    See e. g. the reviews quoted in our ref. 6.Google Scholar
  29. 29.
    S.R. Amendolia et al., Nuovo Cim., 3J 17 (1976)ADSCrossRefGoogle Scholar
  30. 30.
    L. Stodolsky, Phys. Rev. Lett. 28, 60 (1972); the importance of this argument for cluster models has been emphasized in the paper by Pokorski and Van Hove, quoted in our ref. 6.ADSCrossRefGoogle Scholar
  31. 31.
    See the review by J. Whitmore, Physics Reports, C27, 187 (1976) and references therein.ADSCrossRefGoogle Scholar
  32. 32.
    Cf. R. Jengo, A. Krzywicki and B. Petersson, Nucl. B65 319 (1973) and references therein.Google Scholar
  33. 33.
    J.R. Klauder and E.C.G. Sudarshan, Fundamentals of Quantum Optics, Benjamin Inc., 1968.Google Scholar

Copyright information

© Plenum Press, New York 1978

Authors and Affiliations

  • André Krzywicki
    • 1
  1. 1.Laboratoire de Physique Théorique et Particules ElémentairesOrsayFrance

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