Duality in operator formalism

S. Fubini; G. Veneziano

doi:10.1007/bf02728411

journal article May 01, 1970

Duality in operator formalism

S. Fubini G. Veneziano

Il Nuovo Cimento A Vol. 67 No. 1 pp. 29-47 · Springer Science and Business Media LLC

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References

15

[1]

S. Fubini, D. Gordon andG. Veneziano:Phys. Lett.,29 B, 679 (1969), hereafter referred to as I;Y. Nambu: Enrico Fermi Institute preprint COO264-507 (1969). 10.1016/0370-2693(69)90109-9

[2]

For a review of the subject see II.M. Chan:The generalized Veneziano model, CERN preprint No. TH-1057 (1969).

[3]

S. Fubini andG. Veneziano:Nuovo Cimento,64 A, 811 (1969) (hereafter referred to as II), MIT preprint No. 81 (1969);K. Bardakçi andS. Mandelstam:Phys. Rev.,184, 1640 (1969). 10.1007/bf02758835

[4]

D. Amati, C. Bouchiat andJ. L. Gervais:Lett. Nuovo Cimento,2, 399 (1969). 10.1007/bf02755620

[5]

S. Sciuto:Lett. Nuovo Cimento. 2, 411 (1969), see also ref. (6)L. Caneschi, A. Schwimmer andG. Veneziano:Phys. Lett.,30 B, 351 (1969). 10.1007/bf02755622

[6]

L. Caneschi, A. Schwimmer andG. Veneziano:Phys. Lett.,30 B, 351 (1969);L. Caneschi andA. Schwimmer: SLAC preprint (1969). 10.1016/0370-2693(69)90503-6

[7]

F. Gliozzi:Lett. Nuovo Cimento,2, 846 (1969);C. B. Chiu, S. Matsuda andC. Rebbi:Phys. Rev. Lett.,23, 1526 (1969);C. B. Thorn: Berkeley preprint (1969);D. Amati, M. LeBellac andD. Olive:Nuoro Cimento,66 A, 815 (1969). The potentially important result that more Ward identities appear for special values of the trajectory intercept has been found byM. A. Virasoro: University of Wisconsin preprint No. 600-267 (1969). 10.1007/bf02755080

[8]

Z. Koba andH. B. Nielsen:Nucl. Phys., B10, 633 (1969). 10.1016/0550-3213(69)90331-9

[9]

E. Donini andS. Sciuto: University of Torino preprint (1969) (to be published inAnn. of Phys.).

[10]

Equations (2.12), (2.13) and (2.14) have been independently obtained byC. B. Chiu, S. Matsuda andC. Rebbi: Cal Tech preprint (1969).

[11]

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.

[12]

Because of the discussion of Sect.3 one must be careful for the zero frequency operatorsp 0 2 . To be rigorous one should keep up to the end the operatorsx p 0 2 acting on <0‖ and eliminate them only after establishing eq. (4.13).

[13]

These appropriate combinations can be almost guessed by looking at the master equation given byY. Nambu in ref. (1) Enrico Fermi Institute preprint COO264-507 (1969). with the interaction term included.

[14]

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).

[15]

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.

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