Correlation Functions in Two-Dimensional Dilaton Gravity

The Liouville approach is applied to the quantum treatment of the dilaton gravity in two dimensions. The physical states are obtained from the BRST cohomology and correlation functions are computed up to three-point functions. For the N = 0 case (i.e., without matter), the cosmological term operator is found to have the discrete momentum that plays a special role in the c = 1 Liouville gravity. The correlation functions for arbitrary numbers of operators are found in the N = 0 case, and are nonvanishing only for specific “chirality” configurations. The two-dimensional gravity interacting with a dilaton field and matter fields has attracted much attention recently ever since the work on the black hole evaporation [1]. Many efforts are devoted to study the Hawking radiation and the black hole evaporation in the model [1]–[4]. Most of the works have eventually employed the semi-classical approximation, which is often blamed to be the possible origin of diseases in this problem. Therefore it is very desirable to have a full quantum treatment of the dilaton gravity even for a restricted class of models. In two dimensions, we can perhaps hope to understand quantum theory of the dilaton gravity without using the semi-classical approximations. The continuum approach of the Liouville theory is most suitable in discussing the dilaton gravity, since the theory is nonlinear and it is difficult to invent the discretized version of the model such as matrix models. The purpose of our paper is to determine the physical states and the gauge invariant operators by a BRST analysis and compute their correlation functions in the dilaton gravity coupled to N massless free scalar fields applying the methods used in the Liouville theory. We shall take the recently proposed models of dilaton gravity [4] that are conformally invariant. We obtain the BRST cohomology for the case N ≥ 24 that we cannot find in the existing literature. Apart from the usual string states with momentum and oscillator excitations, we find that there are only a few physical states with nontrivial ghost numbers. This situation is very similar to the case of 0 < N < 24 analyzed in ref. [5], but is in sharp contrast to the case of N = 0 which is essentially the same system as the c = 1 two-dimensional gravity, except that one of the field is of negative metric. Similarly to ref. [4], we determine the cosmological term by demanding that it should be a gauge (BRST) invariant operator which reduces to the classical cosmological term in the weak coupling limit. The cosmological term in the case of N = 0 is particularly interesting. We find that the momentum of the cosmological term is precisely at the smallest of the discrete momenta where the characteristic symmetry structure of the c = 1 two-dimensional gravity has been observed as the ground ring [6]. By applying the methods used in the Liouville theory [7], [8] we compute the correlation functions of gauge (BRST) invariant operators with momentum. The correlation functions up to three gauge invariant operators can be obtained for the general N > 0 case, and exhibit singularity structures expected from the physical state spectrum. For the N = 0 case (without matter fields), we obtain correlation