Space-Time Approach to the Lattice Dual Resonance Model

We are able to interpret the Veneziano amplitude as the motion of an infinite number of harmonic oscillators or springs between the pairs of quarks and antiquarks. Nambu> and Fubini, Gordon and Veneziano> have succeeded in making the factrization theorem for the Veneziano amplitude by using the above concepts. Susskind> has explained the origin of the Veneziano amplitude in terms of B-S like equation for the harmonic oscillators between the quark-antiquark pairs. Further last year (1970) ~ Fairlie and Nielsen> proposed an analogue model by using the functional integral method, and obtained the integrand of KobaNielsen amplitude.> In order to elucidate the connection of Susskind's or Nielsen's model with the Veneziano amplitude, Hsue, Sakita and Virasoro employed the functional integral method and derived the dual resonance amplitudes (tree, oneloop planar, and one-loop non-planar). However they have not explained the origin of the interaction part which reproduces the· Veneziano amplitude. Dirac> had proposed that the action integral method is valid in quantum mechanics in order to connect the classical description to the quantum description. This method was developed by Feynman,> Schwinger> and many authors. > The action is defined by