Djordje Minic Research Supported in Part by the Robert A. Welch Foundation and Nsf Grant Phy 9009850

The c = 1 string in the Liouville field theory approach is shown to possess a nontrivial tree-level S-matrix which satisfies factorization property implied by unitary, if all the extra massive physical states are included. Research supported in part by the Robert A. Welch Foundation and NSF Grant PHY 9009850 Adress after September 1, 1991: Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 String amplitudes and their factorization properties in the Liouville field theory [1, 2] are not yet well understood at present. In this note we show that the spherical c = 1 string amplitudes computed from the Liouville theory according to conventional vertex operator normalization are nontrivial and satisfy correct factorization. The renormalized amplitudes computed by Gross and Klebanov [3] imply that S = 1. This result agrees with the so-called “bulk” S-matrix in the collective field theory of c = 1 quantum gravity. We, on the other hand, follow Polyakov [4] who considered unrenormalized amplitudes and find a unitary S-matrix not equal to 1. The S-matrix we are considering is also different from the ones computed in matrix models [5] (see also [6,7]), which are “wall” amplitudes. As a necessary byproduct of our analysis, a host of extra states appears in the spectrum of the theory. These states are not present in the naive light-cone analysis of the spectrum (according to this there is only one degree of freedom, the center of mass of the string, so-called “tachyon”), but appear naturally both in factorization of amplitudes as intermediate states and in the pole structure of the corresponding world-sheet operator product expansion (OPE). We hope that our results will shed some light on difficult issues such as factorization and nature of spectrum in the Liouville theory. Due to the apparent difference of our results and conclusions on [3], a subtle relation is to be expected between the continuous collective field theory à la Das-Jevicki and the related Liouville theory of c = 1 quantum gravity. The organizaton of this note is as follows. First, we review some well-known facts about c = 1 string theory. Then we discuss factorization of amplitudes at tree level. Lastly, we concentrate on the OPE’s of the underlying world-sheet theory in attempt to understand the pole structure of the factorization formulae. In the path integral approach to string theory, the S-matrix is computed from sum over surfaces with insertion of local vertex operators of correct dimensions. One checks