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Prog. Theor. Phys. Vol. 46 No. 5 (1971) pp. 1528-1548

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Detailed Wave Equation and Dual Amplitude*

— Relativistic Quantum Dynamics of String Model of Hadrons —

Takehiko Takabayasi

Department of Physics, Nagoya University, Nagoya

(Received May 21, 1971)

Abstract:

Relativistic quantum dynamics of a one-dimensional elastic continuum (string) is represented by a “detailed wave equation” which is defined at every point of the string and has local coupling to an external field at the end of string. The actual form of this wave equation is severely determined from its self-consistency, which is guaranteed by the existence of a closed algebra for the “invariant hamiltonian density operator”. The algabra in turn directly represents the “gauge invariance” of the theory. The invariant hamiltonian density operator consists of real and imaginary parts, of which the latter represents nonlocal interaction inside the string and exhibits a remarkable Hilbert transform relation. The detailed wave equation is brought to the set of the global wave equation corresponding to the wave equation in the usual sense and an infinite number of subsidiary conditions which work to suppress unphysical states. The unconventional definition of normal model operators {Cµr} enables a uniform treatment of external and internal motions and the factorization of all wave operators into convolutions. The theory leads to dual amplitude for a particular value of intercept. A completely equivalent formulation of theory with the introduction of a proper-time like parameter is also given, which supplies further insights. Finally speculations are made for generalizing the model to more realistic cases.


URL : http://ptp.ipap.jp/link?PTP/46/1528/
DOI : 10.1143/PTP.46.1528


* For convenience of publication this paper was separated into two parts. This is the first part which contains the first five sections out of a total of eight sections. The latter half is published in Vol. 46 No. 6. (See Ref. 1.)

[ Full Text PDF : FREE ACCESS (1123K) ] Citation:

References:

  1. The second part of this paper; T. Takabayasi, Prog. Theor. Phys. 46 (1971), 1924[PTP].
    cf. T. Takabayasi, Prog. Theor. Phys. Suppl. No. 41 (1968), 130[PTP]; Proceedings of Nobel Symposium on Elementary Particle Theory (1968), p. 157.
  2. T. Takabayasi, Nuovo Cim. 33 (1964), 668; Prog. Theor. Phys. 34 (1965), 124[PTP]; Phys. Rev. 139 (1965), B1381[APS].
  3. Y. Katayama, E. Yamada and H. Yukawa, Prog. Theor. Phys. 33 (1965), 541[PTP].
    H. Yukawa, Proceedings of International Conference on Elementary Particles, Kyoto (1965), p. 139.
    K. Fujimura, T. Kobayashi and M. Namiki, Prog. Theor. Phys. 43 (1970), 73[PTP]; ibid. 44 (1970), 193[PTP].
    D. Ito, Soryusiron Kenkyu (mimeographed circular in Japanese) 41 (1970), C5.
    F. Gürsey and M. Koca, Nuovo Cim. A 1 (1971), 429.
  4. Chan Hong-Mo, Phys. Lett. B 28 (1969), 425[CrossRef].
    K. Bardakci and H. Ruegg, Phys. Rev. 181 (1969), 1884[APS].
    C. Goebel and B. Sakita, Phys. Rev. Lett. 22 (1969), 259[APS].
    Z. Koba and H. B. Nielsen, Nucl. Phys. B 10 (1969), 633[CrossRef]; ibid. 12 (1969), 517[CrossRef].
  5. S. Fubini and G. Veneziano, Nuovo Cim. A 64 (1969), 811.
    K. Bardakci and S. Mandelstam, Phys. Rev. 184 (1969), 1640[APS].
  6. T. Takabayasi, Prog. Theor. Phys. 43 (1970), 1117[PTP]; Soryusiron Kenkyu (mimeographed circular in Japanese) 41 (1970), C11.
  7. Y. Nambu, Proceedings of International Conference on Symmetries and Quark Models (1969), p. 269.
  8. L. Susskind, Phys. Rev. D 1 (1970), 1182[APS]; Nuovo Cim. A 69 (1970), 457.
  9. S. Fubini, D. Gordon and G. Veneziano, Phys. Lett. B 29 (1969), 679[CrossRef].
  10. T. Takabayasi, Prog. Theor. Phys. 44 (1970), 1429[PTP]; Soryusiron Kenkyu (mimeographed circular in Japanese) 42 (1970), 153.
  11. T. Takabayasi, preprint (October 1970); Soryusiron Kenkyu 43 (1971), 168 (Talk presented at Res. Inst. Fund. Phys., December 1970).
  12. M. A. Virasoro, Phys. Rev. D 1 (1970), 2933[APS].
  13. S. Fubini and G. Veneziano, preprint CTP, No. 128 (1970).
  14. Analysis of the problem is being made also by several authors.
    cf. E. del Giudice and P. di Vecchia, preprint (1970);
    R. C. Brower and C. B. Thorn, preprint (1971).
  15. Y. Miyamoto, Prog. Theor. Phys. 43 (1970), 556[PTP]; ibid. 43 (1970), 1620[PTP].
  16. The gauge invariance of the dual amplitude was treated from a different standpoint by some authors, who obtained earlier the first generator Λ1-Λ0.
    See F. Gliozzi, Lett. Nuovo Cim. 2 (1969), 846;
    C. B. Chiu, S. Matsuda and C. Rebbi, Phys. Rev. Lett. 23 (1969), 1526[APS].
  17. Proper-time-like parameter is employed in Refs. 7), 8) and 15).
    See also T. Shirafuji, Prog. Theor. Phys. 45 (1971), 501[PTP].

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 46 No. 6 (1971) pp. 1924-1936 :
    Detailed Wave Equation and Dual Amplitude*
    Takehiko Takabayasi
  2. Progress of Theoretical Physics Vol. 47 No. 3 (1972) pp. 1026-1046 :
    Detailed Wave Equation for Extended Model Underlying the Dual Amplitude
    Takehiko Takabayasi
  3. Progress of Theoretical Physics Vol. 47 No. 6 (1972) pp. 2090-2106 :
    On the Interaction of Extended Particles
    Tetsuo Gotō
  4. Progress of Theoretical Physics Vol. 48 No. 3 (1972) pp. 1024-1036 :
    Difference Equation and Dual Resonance Model
    Sigefumi Naka
  5. Progress of Theoretical Physics Vol. 48 No. 4 (1972) pp. 1308-1323 :
    Plateau's Problem and the Virasoro Conditions in the Theory of Duality
    Masatsugu Minami
  6. Progress of Theoretical Physics Vol. 48 No. 4 (1972) pp. 1324-1337 :
    Field Theory of Dual-Resonance Model
    Yuji Nakawaki and Takesi Saito
  7. Progress of Theoretical Physics Vol. 48 No. 5 (1972) pp. 1718-1741 :
    Bilocal Wave Equation for Baryons Lying on Linear Trajectories and Its Extension to a Linear Chain Model
    Takehiko Takabayasi
  8. Progress of Theoretical Physics Vol. 48 No. 6 (1972) pp. 2008-2023 :
    Derivation of the Virasoro Conditions in the String Model
    Gaku Konisi
  9. Progress of Theoretical Physics Vol. 49 No. 5 (1973) pp. 1724-1745 :
    Classical and Quantum Mechanics of a One-Dimensional Relativistic Elastic Continuum
    Takehiko Takabayasi
  10. Progress of Theoretical Physics Vol. 51 No. 1 (1974) pp. 262-283 :
    Theory of Relativistic String and Super-Wave Equation. I
    Takehiko Takabayasi
  11. Progress of Theoretical Physics Vol. 51 No. 1 (1974) pp. 284-298 :
    Generalized Conformal Gauge Conditions in Dual Models
    Gaku Konisi and Takesi Saito
  12. Progress of Theoretical Physics Vol. 51 No. 2 (1974) pp. 571-591 :
    Theory of Relativistic String and Super-Wave Equation. II
    Takehiko Takabayasi
  13. Progress of Theoretical Physics Vol. 52 No. 3 (1974) pp. 1051-1058 :
    Difference Equation and an Extended Particle Model of Hadrons
    Sigefumi Naka
  14. Progress of Theoretical Physics Vol. 52 No. 6 (1974) pp. 1910-1928 :
    General-Covariant Approach to Relativistic String Theory
    Takehiko Takabayasi
  15. Progress of Theoretical Physics Vol. 53 No. 3 (1975) pp. 849-870 :
    Dynamical Theory of Hadron Interactions Based upon Extended Particle Picture. I
    Osamu Hara
  16. Progress of Theoretical Physics Vol. 53 No. 4 (1975) pp. 1162-1177 :
    Strong Interaction Property of Double Dirac String
    Masasi Fujigaki
  17. Progress of Theoretical Physics Vol. 53 No. 5 (1975) pp. 1499-1504 :
    Dilatation Transformation in the String Model
    Yūichi Chikashige, Masataka Hosoda and Satoru Saito
  18. Progress of Theoretical Physics Vol. 57 No. 2 (1977) pp. 632-642 :
    Field Theory in 2 + ε Dimensions and Critical Dimension in the String Model. I
    Yūichi Chikashige and Yasunori Fujii
  19. Progress of Theoretical Physics Supplement No.67 (1979) pp. 1-68 :
    Relativistic Mechanics of Confined Particles as Extended Model of Hadrons
    Takehiko Takabayasi
  20. Progress of Theoretical Physics Supplement No.67 (1979) pp. 115-208 :
    Dynamical Theory of Hadrons Based upon Extended Particle Picture
    Osamu Hara
  21. Progress of Theoretical Physics Supplement No.86 (1986) pp. 81-92 :
    Theory of Relativistic String and Multilocal Model
    Takehiko Takabayasi

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