The European Physical Journal H

, Volume 39, Issue 5, pp 575–589 | Cite as

On the mathematics underlying dispersion relations

Article

Abstract

The history of mathematical methods underlying the study of dispersion relations in physics is discussed. In particular, some misconceptions connected with a theorem known in the physics literature as Titchmarsh’s Theorem are addressed. It is pointed out that the aforementioned theorem is a compilation of two well-known theorems in mathematics, the Paley-Wiener theorem and the Marcel Riesz theorem.

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References

  1. 1.
    Cartwright, M. 1982. Manuscripts of Hardy, Littlewood, Marcel Riesz and Titchmarsh. Bull. Lond. Math. Soc. 14: 472–532.CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Duren, P.L. 1970. Theory of H p -spaces. Academic Press. New York and London.Google Scholar
  3. 3.
    Hilgevoord, J. 1962. Dispersion Relations and Causal Description. North Holland Publ. Amsterdam.Google Scholar
  4. 4.
    Hoffman, K. 1962. Banach spaces of analytic functions. Prentice Hall, Inc. Englewood Cliffs, N.J. Google Scholar
  5. 5.
    Kallmann, H. and H. Mark. 1927. Über die Dispersion und Streuung von Röntgenstrahlen. Z. Phys. 387: 585–604.Google Scholar
  6. 6.
    King, F.W. 2009. Hilbert Transforms, Volumes 1 and 2. Encyclopedia of Mathematics and its Applications 124 and 125. Cambridge University Press, Cambridge.Google Scholar
  7. 7.
    Koosis, P. 1998. Introduction to H p Spaces. Cambridge University Press, Cambridge.Google Scholar
  8. 8.
    Kramers, H.A. 1924. Law of dispersion and Bohr’s theory of spectra. Nature 113: 673–674. ADSCrossRefGoogle Scholar
  9. 9.
    Kramers, H. and W. Heisenberg. 1925. Über die Streuung von Strahlung durch Atome. Z. Phys. 31: 681–708.ADSCrossRefMATHGoogle Scholar
  10. 10.
    Kramers, H.A. 1927. La diffusion de la lumière par les atomes. Atti del congresso internazionale dei fisici II: 545-557. English translation in ter Haar D, 1998 Master of Modern Physics: The Scientific Contributions of H. A. Kramers Princeton University Press, Princeton.Google Scholar
  11. 11.
    Kronig, R. de L. 1926. On the Theory of Dispersion of X-rays. J. Opt. Soc. Am. 12: 547–556.ADSCrossRefGoogle Scholar
  12. 12.
    Ladenburg, X. 1921. Die Quantentheoretische Deutung der Zahl der Dispersionselektronen. Z. Phys. 4:451–468.ADSCrossRefGoogle Scholar
  13. 13.
    Mashregi, J. 2009. Representation Theorems in Hardy Spaces. Cambridge University Press, Cambridge.Google Scholar
  14. 14.
    Nussenzveig, H. 1972. Causality and Dispersion Relations. Academic Press, New York.Google Scholar
  15. 15.
    Paley, R. and N. Wiener. 1934. Fourier Transforms in the Complex Domain. American Mathematical Society Colloquium Publications, Providence, RI.Google Scholar
  16. 16.
    Riesz, M. 1924. Les fonctions conjuguées et les séries de Fourier. C. R. Acad. Sci. Paris, 178: 1464–1467.MATHGoogle Scholar
  17. 17.
    Riesz, M. 1927. Sur les fonctions conjuguées. Math. Z. 27: 218–244.CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Rudin, W. 1987. Real and Complex Analyis, 3rd edn. McGraw-Hill, New York.Google Scholar
  19. 19.
    Sellmeier, E. 1871. Zur Erklärung der abnormen Farbenfolge im Spectrum einiger Substanzen. Ann. Phys. Chem. 219: 272–282. ADSCrossRefGoogle Scholar
  20. 20.
    Stein, E.M. and R. Shakarchi. 2003. Complex Analysis, Vol. 2 of Princeton Lectures in Analysis. Princeton University Press, Princeton.Google Scholar
  21. 21.
    Stein, E.M. and R. Shakarchi. 2005. Real Analysis, Vol. 3 of Princeton Lectures in Analysis. Princeton University Press, Princeton.Google Scholar
  22. 22.
    Stein, E.M. and R. Shakarchi. 2011. Functional Analysis, Vol. 4 of Princeton Lectures in Analysis. Princeton University Press, Princeton.Google Scholar
  23. 23.
    ter Haar, D. 1998. Master of Modern Physics, The Scientific Contributions of H. A. Kramers. Princeton University Press, Princeton.Google Scholar
  24. 24.
    Titchmarsh, E.C. 1926a. Conjugate trigonometrical integrals. Proc. Lond. Math. Soc. 24: 109–130.CrossRefMathSciNetGoogle Scholar
  25. 25.
    Titchmarsh, E.C. 1926b. Reciprocal formulae involving series and integrals. Math. Z. 25: 321–347. Erratum: 26, 496 (1927).CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Titchmarsh, E.C. 1937. Introduction to the Theory of Fourier Integrals, 1st edn. Clarendon Press, Oxford. Google Scholar
  27. 27.
    Toll, J.S. 1956. Causality and the Dispersion Relation: Logical Foundations. Phys. Rev. 104: 1760–1770. ADSCrossRefMathSciNetGoogle Scholar
  28. 28.
    van Kampen, N.G. 1953. S-matrix and Causality Condition: I. Maxwell field. Phys. Rev. 89: 1072–1079. ADSCrossRefMathSciNetMATHGoogle Scholar

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© EDP Sciences and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of MississippiMississippiUSA

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