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Prog. Theor. Phys. Vol. 52 No. 3 (1974) pp. 1031-1041

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Functional Approach to the Critical Dimension in Veneziano's Model

Masatsugu Minami

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606

(Received April 8, 1974)

Abstract:

We give an explicit derivation of the critical dimension of space-time associated with the Veneziano model starting from the path-integral expression based on a Dirichlet-type boundary-value problem. It is shown that our expression for four-leg amplitudes happens to reduce to the beta-function formula of Veneziano only when the effective number of transverse dimensions is 24. Up to the goal, the transformation properties are not self-evident since our representation has a batch of the theta functions in the integrand. Our next concern is therefore with a discussion how the Möbius invariance or the reciprocal invariance is connected with the critical dimensionality.


URL : http://ptp.ipap.jp/link?PTP/52/1031/
DOI : 10.1143/PTP.52.1031

[ Full Text PDF : FREE ACCESS (824K) ] Citation:

References:

  1. M. Minami, Prog. Theor. Phys. 50 (1973), 2027[PTP].
  2. L. R. Ford, Automorphic Functions, 2nd ed. (Chelsea, N. Y., 1951).
  3. J. Lehner, Discontinuous Groups and Automorphic Functions (Amer. Math. Soc. R. I. 1964).
  4. S. Mandelstam, Nucl. Phys. B 64 (1973), 205[CrossRef].
  5. M. Minami, Prog. Theor. Phys. 46 (1971), 614[PTP]; ibid. 48 (1972), 633[PTP].
  6. D. J. Gross, A. Neveu, J. Scherk and J. H. Schwarz, Phys. Rev. D 2 (1970), 697[APS].
  7. V. I. Smirnov, A Course of Higher Mathematics, transld by D. E. Brown (Pergamon, Oxford, 1964), vol. III, part 2.
  8. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge, 1962).
  9. L. Brink and H. B. Nielsen, Phys. Lett. B 43 (1973), 319[CrossRef].
  10. M. Minami, Prog. Theor. Phys. 45 (1971), 927[PTP].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 52 No. 6 (1974) pp. 1890-1901 :
    A Green's Function Method for Cremmer and Gervais' Vertex
    Masatsugu Minami
  2. Progress of Theoretical Physics Vol. 53 No. 1 (1975) pp. 237-246 :
    Modular Group and Non-Cyclic Symmetry of the Veneziano Formula
    Masatsugu Minami
  3. Progress of Theoretical Physics Vol. 57 No. 6 (1977) pp. 2150-2151 :
    Note on Nahm's Partition Function of the Dual Spectrum. II
    Masatsugu Minami
  4. Progress of Theoretical Physics Vol. 58 No. 5 (1977) pp. 1603-1621 :
    Functional Method for the Off-Shell Dual Amplitudes
    Masatsugu Minami
  5. Progress of Theoretical Physics Vol. 59 No. 2 (1978) pp. 590-606 :
    Towards a Satisfactory Invariant Measure of Dual-Resonance Amplitudes
    Masatsugu Minami
  6. Progress of Theoretical Physics Vol. 59 No. 4 (1978) pp. 1361-1375 :
    Dual Amplitudes with Two Off-Shell Lines
    Masatsugu Minami

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