The Operator Formalism and The Dual Resonance Model

Abstract

This chapter is on the Operator Formalism for Dual Resonance Models (DRM). This is an important bridge between DRM and string theory. The starting point is the Bardakci-Ruegg form of the dual N-point function, followed by the creation and annihilation operators of a single quantum harmonic oscillator. Coherent states are introduced and their overlap, shift, eigenvalue and evolution are derived. All these are generalized to D-dimensional oscillators with Minkowski signature. After defining Vertex Operators and Propagators, a remarkable formula is displayed that expresses the N-point function as the vacuum expectation values of ordered products of vertex operators and propagators. The oscillators are extended to include the zero modes. The factorizability follows in a straightforward manner. The degeneracy of states is obtained which is seen to rise exponentially. The generators for the Projective Group and their Lie algebra are discussed, along with their realization in terms of oscillators. The classical and quantum Lie algebras of the generalized projective invariance are worked out. The physical states and their construction are worked out. Virasoro’s subsidiary conditions to eliminate all negative norm (Ghost) states are discussed. States like spurious states and zero-norm states are discussed. The issue of eliminating ghosts and the so-called DDF construction is explained. The significance of the critical dimension $D_c=26$ is examined. The chapter ends with a discussion of the Shapiro-Virasoro models.