Proper-Time Formulation of the Dual Resonance Model and Subsidiary Conditions. II

Abstract

We have proposed proper-time formulation of the dual resonance model for a (0) = 1. As in the case of a (0) ~ 1, we discuss the generalized Mobius transformations for an infinite­ dimensional group. (Its algebraic structure has been investigated by Fubini and Veneziano.) Next we elucidate the invariance properties of the proper-time equation for a(O) =1 under this group. Then it is shown that there are infinite conserved currents; they correspond to infinitely many Ward-like identities or subsidiary conditions which were first found by Virasoro. Further we discuss the ghost problem. The dual resonance model has many interesting features such as crossing symmetry and duality. In this model, these properties are connected to the group structure of the theory, or invariance under Mobius transformations. These in­ variance properties are represented in the most compact form by proper-time formulation of the dual resonance model.l) In this paper, we apply proper-time formulation to the dual resonance model of a (0) = 1. The case a (0) = 1 is quite interesting, because it is theoretically invariant under an infinite-dimensional group. Thus, there are infinite Ward-like identities or subsidiary conditions on physical states, as have been found by Virasoro. 2 l Fubini and V eneziano 3 l have discussed the algebraic structure of this infinite-dimensional group. They have also sug­ gested the possibility of eliminating the state with a negative metric. In § 2 we shall discuss the structure of an infinite dimensional group of generalized Mobius transformations and derive the explicit form of the generators. In § 3 we discuss the invariance properties of the proper-time equation under this infinite-dimensional group of a (0) = 1. The group structure is represented in a rather compact form in our formulation. In this formulation we can derive infinitely many Ward-like identities or subsidiary conditions on the physical amplitude, and on the Schrod­ inger functional by using the invariance properties of the proper-time equation. In § 4 we shall discuss the Lagrangian formulation of the dual resonance model of a (0) = 1. We derive the infinite many conserved currents corresponding to our infinite dimensional gauge groups. From this we also derive the infinite many subsidiary conditions. In § 5 we discuss briefly the possibility of eliminating the states with a negative metric.