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Il Nuovo Cimento A (1971-1996)

, Volume 67, Issue 1, pp 29–47 | Cite as

Duality in operator formalism

  • S. Fubini
  • G. Veneziano
Article

Summary

We propose a form for a dual resonant amplitude in which both duality and factorization are explicitly exhibited. This result is achieved by considering in detail the transformation properties of the operators appearing in the model under the fundamental projective transformation of Koba and Nielsen.

Дуализм в операторном формализме

Резюме

Мы предлагаем форму для дуальной резонансной амплитуды, в которой явно проявляются и дуализм и факторизация. Этот результат достигается путем подробного рассмотрения свойств преобразований операторов, которые возникают в этой модели при основном проективном преобразовании Коба и Нильсена.

Riassunto

Si propone una forma per l'ampiezza duale risonante in cui, sia la dualità, sia la fattorizzazione sono esplicitamente esibite. Questo risultato è ottenuto studiando in dettaglio le proprietà degli operatori rispetto alla trasformazione proiettiva di Koba e Nielsen.

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References

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    The correspondence between theb operators here and thea operators ofI is\({{b_n^{( + )} \to a^{(n)} } \mathord{\left/ {\vphantom {{b_n^{( + )} \to a^{(n)} } {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }},{{b_n^{( - )} \to a^{ + (n)} } \mathord{\left/ {\vphantom {{b_n^{( - )} \to a^{ + (n)} } {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }}\). There is also of course a four-vector index μ implicitly understood ina andb.Google Scholar
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    Because of the discussion of Sect.3 one must be careful for the zero frequency operatorsp 02. To be rigorous one should keep up to the end the operatorsx p 02 acting on <0‖ and eliminate them only after establishing eq. (4.13).Google Scholar
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    At this point it is amusing to speculate a little on the meaning of the projective variablez (ϱ in this Section). If we define a variable τ byz=exp⌊−iτ⌋ the natural internation of τ is that of some time variable. Indeed the η function of equation (A.6) of the Appendix becomes η(τy−ηr), and so do the θ functions in the volume element. This is not unnatural if τ is a time. Furthermore the commutator withL 3, which is related to the energy, gives a dilation inz. i.e. a shift in τ. Finally we can rewriteQ(z) asQ(z)=q 0+ip0logz=q 0+p0τ, which again resembles the position of a particle of momentump at the time τ. The identification of τ with a proper time variable was indeed put forward byY. Miyamoto (Tokyo University preprint (1969)). A variable ξ=ilogz was also considered byY. Nambu in ref. (1) Enrico Fermi Institute preprint COO264-507 (1969).Google Scholar
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    We should note however that it is not possible to treat the nonzero mass case without somewhat altering the formalism. The ground state ‖0>, for instance, no longer corresponds to a physical one-particle state as it is in the zero-mass case. It may very well be that one needs nontrivial modifications of the formalism of Sect.3 in order to treat the case of nonzero masses in a fully consistent group-theoretical way.Google Scholar

Copyright information

© Società Italiana di Fisica 1970

Authors and Affiliations

  • S. Fubini
    • 1
  • G. Veneziano
    • 1
  1. 1.Laboratory for Nuclear Science and Physics DepartmentMassachusetts Institute of TechnologyCambridge

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