Skip to main content
Springer Nature Link
Log in

The Long Way to the Statistical Bootstrap Model

  • Chapter
Hot Hadronic Matter

Part of the book series: NATO ASI Series ((NSSB,volume 346))

Abstract

The statistical bootstrap model (SBM) is a statistical model of strong interactions based on the observation that hadrons not only form bound and resonance states but also decay statistically into such states if they are heavy enough. This leads to the concept of a possibly unlimited sequence of heavier and heavier bound and resonance states, each being a possible constituent of a still heavier resonance, while at the same time being itself composed of lighter ones. We call these states clusters (in the older literature heavier clusters are called fireballs; the pion is the lightest “one-particle-cluster”) and label them by their masses. Let ρ(m)dm be the number of such states in the mass interval {m, dm}; we call ρ(m) the “SBM mass spectrum”. Bound and resonance states are due to strong interactions; if introduced as new, independent particles in a statistical model, they also simulate the strong interactions to which they owe their existence. To simulate all attractive strong interactions we need all of them (including the not yet discovered ones), that is: we need the complete mass spectrum ρ(m). To simulate repulsive forces we may use proper cluster volumes à la Van der Waals. In order to obtain the full mass spectrum, we require that the above picture, namely that a cluster is composed of clusters, be self-consistent. This leads to the “bootstrap condition and/or bootstrap equation” for the mass spectrum ρ(m). The bootstrap equation (BE) is an integral equation embracing all hadrons of all masses. It can be solved analytically with the result that the mass spectrum ρ(m) has to grow exponentially. Consequently any thermodynamics employing this mass spectrum has a singular temperature T 0 generated by the asymptotic mass spectrum: ρ(m) ∼ exp(m/T 0). Today this singular temperature is interpreted as the temperature where (for baryon chemical potential zero) the phase transition (hadron gas) ⇔ (quark-gluon plasma) occurs.

“It is the nature of a hypothesis when once a man has conceived it, that it assimilates everything to itself, as proper nourishment; and, from the first moment of your begetting it, it generally grows the stronger by everything you see, hear, read or understand. This is of great use.”1

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Laurence Sterne, “The life and Opinions of Tristram Shandy Gentleman” (London, 1760), Book II, Ch. 19.

    Google Scholar 

  2. R. Hagedorn, in “Quark Matter ’84”, Helsinki Symposium, ed. K. Kajantie, published in “Lecture Notes in Physics” 221 (Springer, Heidelberg, 1985).

    Google Scholar 

  3. E.L. Feinberg, Physics Reports 5 (1972) 237.

    Article  ADS  Google Scholar 

  4. H. Yukawa, Proc.Phys.Math.Soc.Japan 17 (1935) 48.

    Google Scholar 

  5. W. Heitler, Proc.Cambr.Phil.Soc. 37 (1941) 291.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. W. Heitler and H.W. Peng, Proc.Cambr.Phil.Soc. 38 (1942) 296.

    Article  ADS  Google Scholar 

  7. L. Janossy, Phys.Rev. 64 (1943) 345.

    Article  ADS  Google Scholar 

  8. W. Heisenberg, Z.Phys. 101 (1936) 533.

    Article  ADS  Google Scholar 

  9. N.M. Dulles and W.D. Walker, Phys.Rev. 93 (1954) 215.

    Article  ADS  Google Scholar 

  10. J. Nishimura, Soryushiron Kenkyu 12 (1956) 24.

    Google Scholar 

  11. G. Cocconi, Phys.Rev. 111 (1958) 1699.

    Article  ADS  Google Scholar 

  12. F.W. Büsser et al. (CERN-Columbia-Rockefeller Coll.), Phys.Lett. 46B (1973) 471, and a wealth of further papers in conference reports and review articles.

    Google Scholar 

  13. S. Takagi, Progr.Theor.Phys. 7 (1952) 123.

    Article  ADS  Google Scholar 

  14. P. Ciok, T. Coghen, J. Gierula, R. Holyński, A. Jurak, M. Miesowicz, T. Saniewska, O. Stanisz and J. Pernegr, Nuovo Cimento 8 (1958) 166 and 10 (1958) 741 [the second without O. Stanisz].

    Google Scholar 

  15. J. Gierula, Fortschr.d.Physik 11 (1963) 109.

    Article  ADS  Google Scholar 

  16. N. Bohr, Nature 137 (1936) 344.

    Article  ADS  MATH  Google Scholar 

  17. V.F. Weisskopf, Phys.Rev. 52 (1937) 295.

    Article  ADS  Google Scholar 

  18. I. Frenkel, Soviet Phys. 9 (1936) 533.

    Google Scholar 

  19. H. Koppe, Phys.Rev. 76 (1949) 688.

    Article  ADS  MATH  Google Scholar 

  20. H. Koppe, Zs.f.Naturforschung 3a (1948) 251.

    ADS  Google Scholar 

  21. E. Fermi, Progr.Theor.Phys. 5 (1950) 570.

    Article  MathSciNet  ADS  Google Scholar 

  22. E. Fermi, Phys.Rev. 81 (1951) 683.

    Article  ADS  MATH  Google Scholar 

  23. W. Heisenberg, Naturwissenschaften 39 (1952) 69.

    Article  ADS  Google Scholar 

  24. E. Beth and C.E. Uhlenbeck, Physica 4 (1937) 915.

    Article  ADS  MATH  Google Scholar 

  25. L. Landau and E. Lifshitz, “Statistical Physics” (MIR, Moscow, 1967).

    Google Scholar 

  26. R. Hagedorn, in Cargèse Lectures, Vol. 6 (1973), p. 643, ed. E. Schatzmann (Gordon and Breach, New York, 1973).

    Google Scholar 

  27. K. Huang, “Statistical Mechanics” (John Wiley and Sons, New York, 1963).

    Google Scholar 

  28. J. Bernstein, R. Dashen and S. Ma, Phys.Rev. 187 (1969) 1.

    Article  Google Scholar 

  29. S.Z. Belenkij, Nucl.Phys. 2 (1956) 259.

    Article  Google Scholar 

  30. L. Sertorio, Rivista del Nuovo Cimento 2 (1979) No 2.

    Google Scholar 

  31. E Cerulus and R. Hagedorn, Suppl.Nuovo Cimento 9 (1958) 646, 659.

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Cerulus, Nuovo Cimento 19 (1961) 528.

    Article  MathSciNet  MATH  Google Scholar 

  33. F. Cerulus, Nuovo Cimento 22 (1961) 958.

    Article  MathSciNet  Google Scholar 

  34. R. Hagedorn, Nuovo Cimento 15 (1960) 434.

    Article  Google Scholar 

  35. S.Z. Belenkij, V.M. Maksimenko, A.I. Nikisov and I.L. Rozental, Usp.Fiz.Nauk 62 (1957) 1, in Russian.

    Google Scholar 

  36. S.Z. Belenkij, V.M. Maksimenko, A.I. Nikisov and I.L. Rozental Fortschr.d.Physik 6 (1958) 524, same in German.

    Article  ADS  Google Scholar 

  37. G. Cocconi, Nuovo Cimento 33 (1964) 643.

    Article  Google Scholar 

  38. J. Orear, Phys.Lett. 13 (1964) 190.

    ADS  Google Scholar 

  39. G. Fast, R. Hagedorn and L.W. Jones, Nuovo Cimento 27 (1963) 856.

    Article  Google Scholar 

  40. G. Fast and R. Hagedorn, Nuovo Cimento 27 (1963) 208.

    Article  Google Scholar 

  41. T.E.O. Ericson, Phys.Lett. 4 (1963) 258.

    Article  ADS  Google Scholar 

  42. A. Bialas and V.F. Weisskopf, Nuovo Cimento 35 (1965) 1211.

    Article  Google Scholar 

  43. G. Auberson and B. Escoubès, Nuovo Cimento 36 (1965) 628.

    Article  Google Scholar 

  44. H. Satz, Nuovo Cimento 37 (1965) 1407.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Vandermeulen, Bull.Soc.Roy.Sci.Liège (Belgium) 34 (1965) Nos 1–2.

    Google Scholar 

  46. I. Pomeranchuk, Dokl.Akad.Nauk SSSR 78 (1951) 889.

    Google Scholar 

  47. R. Hagedorn, Suppl. Nuovo Cimento 3 (1965) 147.

    Google Scholar 

  48. R. Carreras, “Picked up for you this week”, No 26 (CERN, Geneva, 1993).

    Google Scholar 

  49. Particle Data Group, Phys.Rev. D50 (1994) No 3, Part I.

    Google Scholar 

  50. W. Nahm, Nucl.Phys. B45 (1972) 525.

    Article  MathSciNet  ADS  Google Scholar 

  51. S. Frautschi, Phys.Rev. D3 (1971) 2821.

    ADS  Google Scholar 

  52. R. Hagedorn and J. Ranft, Suppl. Nuovo Cimento 6 (1968) 169.

    Google Scholar 

  53. H. Grote, R. Hagedorn and J. Ranft, “Particle Spectra” (CERN, Geneva, 1970).

    Google Scholar 

  54. J. Yellin, Nucl.Phys. B52 (1973) 583.

    Article  ADS  Google Scholar 

  55. E. Schröder, Z.Math.Phys. 15 (1870) 361.

    MATH  Google Scholar 

  56. G.J.H. Burgers, C. Fuglesang, R. Hagedorn and V. Kuvshinov, ZPhys. C46 (1990) 465.

    Google Scholar 

  57. H. Satz, Phys.Rev. D20 (1979) 582.

    ADS  Google Scholar 

  58. K. Redlich and L. Turko, Z.Phys. C5 (1980) 201.

    MathSciNet  ADS  Google Scholar 

  59. R. Hagedorn, I. Montvay and J. Rafelski, Proc. Workshop on Hadronic Matter at Extreme Energy Density, Erice, 1978, p. 49 eds. N. Cabibbo and L. Sertorio (Springer Science+Business Media New York, 1980).

    Google Scholar 

  60. R. Hagedorn and J. Rafelski, Phys.Lett. 97B (1980) 136.

    ADS  Google Scholar 

  61. B. Touschek, Nuovo Cimento B58 (1968) 295.

    ADS  Google Scholar 

  62. A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys.Rev. D19 (1974) 3471.

    MathSciNet  ADS  Google Scholar 

  63. J.I. Kapusta, Nucl.Phys. B196 (1982) 1.

    Article  ADS  Google Scholar 

  64. K. Redlich, Z.Phys. C21 (1983) 69.

    ADS  Google Scholar 

  65. N. Cabibbo and G. Parisi, Phys.Lett. 59B (1975) 67.

    ADS  Google Scholar 

  66. R. Hagedorn and J. Rafelski, Bielefeld Symposium on “Statistical Mechanics of Quarks and Hadrons”, p. 237, ed. H. Satz (North Holland, Amsterdam, 1981).

    Google Scholar 

  67. J. Rafelski and R. Hagedorn, as in Ref. (63), p. 253.

    Google Scholar 

  68. M.I. Gorenstein, Yad.Fiz. 31 (1980) 1630.

    Google Scholar 

  69. M.I. Gorenstein, V.K. Petrov and G.M. Zinovjev, Phys.Lett. 106B (1981) 327.

    ADS  Google Scholar 

  70. M.I. Gorenstein, G.M. Zinovjev, V.K. Petrov and V.P. Shelest, Teor.Mat.Fiz. 52 (1982) 346.

    Google Scholar 

  71. R. Hagedorn, Z.Phys. C17 (1983) 265.

    ADS  Google Scholar 

  72. M.I. Gorenstein, S.I. Lipskikh and G.M. Zinovjev, Z.Phys. C22 (1984) 189.

    ADS  Google Scholar 

  73. E.A. Guggenheim, J.Chem.Phys. 7 (1939) 103.

    Article  ADS  Google Scholar 

  74. J. Baacke, Acta Phys.Pol. B8 (1977) 625.

    Google Scholar 

  75. J. Letessier and A. Tounsi, Nuovo Cimento 99A (1988) 521.

    ADS  Google Scholar 

  76. J. Letessier and A. Tounsi, Phys.Rev. D40 (1989) 2914.

    ADS  Google Scholar 

  77. J. Letessier and A. Tounsi, Preprint PAR/LPTHE 90-17, Univ. Paris VII (1990) presented at IInd Internat. Bodrum Physics School, 1989, University of Istanbul, Turkey.

    Google Scholar 

  78. K. Fabricius and U. Wambach, Nucl.Phys. B62 (1973) 212.

    Article  MathSciNet  ADS  Google Scholar 

  79. M. Chaichian, R. Hagedorn and M. Hayashi, Nucl.Phys. B92 (1975) 445.

    Article  ADS  Google Scholar 

  80. J. Engels, K. Fabricius and K. Schilling, Phys.Lett. 59B (1975) 477.

    ADS  Google Scholar 

  81. J. Kripfganz, Nucl.Phys. B100 (1975) 302.

    Article  ADS  Google Scholar 

  82. M. Chaichian and M. Hayashi, Phys.Rev. D15 (1977) 402.

    ADS  Google Scholar 

  83. M. Levinson, Kgl.Danske Vid.Selsk.Mat.Fys.Med. 25 (1949) No 9.

    Google Scholar 

  84. R. Hagedorn and I. Montvay, Nucl.Phys. B59 (1973) 45.

    Article  ADS  Google Scholar 

  85. GJ. Hamer and S. Frautschi, Phys.Rev. D4 (1971) 2125.

    ADS  Google Scholar 

  86. F. Karsch, in “QCD 20 Years Later”, Aachen Symposium, 1992, Vol. 2, p. 717, eds. P.M. Zerwas and H.A. Kastrup, (World Scientific, Singapore, 1993).

    Google Scholar 

  87. J. Fingberg, U. Heller and F. Karsch, Nucl.Phys. B392 (1993) 493.

    Article  ADS  Google Scholar 

  88. R. Hagedorn and J. Ranft, Nucl.Phys. B48 (1972) 157.

    Article  ADS  Google Scholar 

  89. J. Letessier and A. Tounsi, Nuovo Cimento 13A (1973) 557.

    ADS  Google Scholar 

  90. R. Hagedorn, Rivista del Nuovo Cimento 6 (1983) No 10.

    Google Scholar 

  91. A.T. Laasanen, C. Ezell, L.J. Gutay, W.N. Schreiner, P. Schublin, L. von Lindern and F. Tlirkot, Phys.Rev.Lett. 38 (1977) 1.

    Article  ADS  Google Scholar 

  92. A. Bussière et al., Nucl.Phys. B174 (1980) 1.

    Article  ADS  Google Scholar 

  93. J. Letessier and A. Tounsi, Nuovo Cimento 99A (1988) 521.

    ADS  Google Scholar 

  94. R. Hagedorn and J. Rafelski, Phys.Lett. 97B (1980) 136.

    ADS  Google Scholar 

  95. L.D. Landau, “Collected Papers”, ed. D. Ter Haar (Gordon and Breach, New York, 1965).

    Google Scholar 

  96. Yu.B. Rumer, Soviet Phys. (JETP) 11 (1960) 1365 [Translation from original: JETP 38 (1960) 1899].

    MathSciNet  Google Scholar 

  97. G.F. Chew, Science 161 (1968) 762.

    Article  ADS  Google Scholar 

  98. G.F. Chew, Physics Today 23 (1970) 23.

    Article  Google Scholar 

  99. G.F. Chew and S. Mandelstam, Nuovo Cimento 19 (1961) 752.

    Article  MathSciNet  MATH  Google Scholar 

  100. F. Zachariasen, Phys.Rev.Lett. 7 (1961) 112.

    Article  ADS  Google Scholar 

  101. Laurence Sterne, “The Life and Opinions of Tristram Shandy Gentleman” (London, 1761), Book IV, Ch. 27.

    Google Scholar 

  102. E.L. Feinberg, Sov. Phys. Usp 18 (1976) 624 [Translation from original: Usp. Fiz. Nauk 116 (1975) 709].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Physics Division, CERN, CH-1211, Geneva 23, Switzerland

    Rolf Hagedorn

Authors
  1. Rolf Hagedorn
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LPTHE, Université Paris 7, Paris, France

    Jean Letessier

  2. Gesellschaft für Schwerionenforschung, Darmstadt, Germany

    Hans H. Gutbrod

  3. CERN, Genèva, Switzerland

    Hans H. Gutbrod

  4. University of Arizona, Tucson, Arizona, USA

    Johann Rafelski

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media New York

About this chapter

Cite this chapter

Hagedorn, R. (1995). The Long Way to the Statistical Bootstrap Model. In: Letessier, J., Gutbrod, H.H., Rafelski, J. (eds) Hot Hadronic Matter. NATO ASI Series, vol 346. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1945-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-1945-4_2

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5798-8

  • Online ISBN: 978-1-4615-1945-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics