A Treatment of the Non-polynomial Contributions: Application to Calculate Partition Functions of Strings and Membranes

  • Emilio Elizalde
Part of the Lecture Notes in Physics book series (LNP, volume 855)


In this chapter we consider a very interesting way of dealing with the additional term of non-polynomial type that shows up in the series commutation relevant to the zeta-function regularization theorem, as described in the preceding chapters. The asymptoticity of the series is proven for the important cases which are useful in Physics (e.g., sums over non-complete lattices, mainly coming from Neumann and Robin BC) and cannot be dealt with using the otherwise very powerful formulas (as Jacobi’s theta function identity) obtained from Poisson’s summation in many dimensions. Later, a first physical application to calculate the partition function corresponding to string, membrane and, in general, p-brane theories is investigated in detail. Such theories are commonly termed as fundamental, in any attempt at a rigorous description of QED from first principles.


Partition Function Zeta Function Additional Term Heat Kernel Asymptotic Series 
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.


  1. 10.
    E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994) zbMATHCrossRefGoogle Scholar
  2. 11.
    E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer, Berlin, 1995) zbMATHGoogle Scholar
  3. 14.
    A.A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, S. Zerbini, Analytic Aspects of Quantum Fields (World Scientific, Singapore, 2003) CrossRefGoogle Scholar
  4. 33.
    E. Elizalde, S. Leseduarte, S. Zerbini, Mellin transform techniques for zeta-function resummations, UB-ECM-PF 93/7, arXiv: hep-th/9303126 (1993)
  5. 51.
    E. Elizalde, J. Math. Phys. 31, 170 (1990) MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 56.
    M. Bordag, E. Elizalde, K. Kirsten, J. Math. Phys. 37, 895 (1996) MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 64.
    M. Bordag, E. Elizalde, B. Geyer, K. Kirsten, Commun. Math. Phys. 179, 215 (1996) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 84.
    C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), pp. 261–265 zbMATHGoogle Scholar
  9. 99.
    F.W.J. Olver, Asymptotics and Special Functions (Academic Press, New York, 1974), pp. 71–72 and 80–84 Google Scholar
  10. 109.
    E. Elizalde, J. Phys. A 22, 931 (1989) MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 111.
    J.J. Duistermaat, V.W. Guillemin, Invent. Math. 29, 39 (1975) MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 112.
    N. Bleistein, R.A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1975) zbMATHGoogle Scholar
  13. 113.
    S.A. Molchanov, Russian Math. Surveys 30, 1 (1975) ADSzbMATHCrossRefGoogle Scholar
  14. 114.
    J. Polchinski, Phys. Rev. Lett. 68, 1267 (1992) MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 115.
    J. Polchinski, Phys. Rev. D 46, 3667 (1992) ADSCrossRefGoogle Scholar
  16. 116.
    E. Elizalde, S. Leseduarte, S.D. Odintsov, Phys. Rev. D 48, 1757 (1993) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Space ScienceHigher Council for Scientific ResearchBellaterra (Barcelona)Spain

Personalised recommendations