Relativistic Quantum Mechanics of One - Dimensional Mechanical Continuum and Subsidiary Condition of Dual Resonance Model

After the dual resonance model was formulated in terms of infinitely many oscillators by Nambu, Veneziano and Fubini,l) it has been frequently suggested that the underlying string model of hadrons furnishes the multiparticle dual am,. plitudes.) One of the most crucial problems in the dual resonance model, however, is the existence of ghosts which are unphysical states having negative norms or space-like momenta. For eliminating ghosts, Fubini and Veneziano have found a Ward-like identity which has been generalized by Virasoro.) However, the Ward-like identity in their form is abstract and its relation to the so-called string model of hadrons is obscure. On the other hand, Takabayasi has proposed new relativistic quantum mechanical equations of one-dimensional string whic};l are defined at each material point on the string.) Following Takabayasi's, formalism, subsidiary conditions proposed by Virasoro are contained in his new quantum mechanical equations. It is, however, not· clear whether his new formulation is equivalent to ordinary quantum mechanics or not. Recently, Hara) has pointed out that Virasoro's condition is derived from the invariance under a general coordinate transformation of the Lagrange coordinates whi.ch specify each material point on the string. He has also shown that Virasoro's algebra is derived from the algebra of the general coordinates transformation. In this note, we would like to show that relativistic quantum mechanics of a one-dimensional object with uniform mass density is equivalent to the so-called "string" model of hadrons with Virasoro's subsidiary conditions. Our argument is as follows: Starting from a Lagrangian which is invariant under a general coordinate transformation of the Lagrange parameters and local time transforma-