Regularization of Quantum Gravity in the Matrix Model Approach

We study divergence problem of the partition function in the matrix model approach for two· dimensional quantum gravity. We propose a new model V(¢)=1/2Tr¢2+g/NTr¢4+dN 4 (Tr¢4)2 and show that in the sphere case it has no divergence problem and the critical exponent is of pure gravity. 23 Recently quantization of two dimensional surfaces has been studied extensively by many people. It is generally believed that in doing so, we can treat a theory of two dimensional quantum gravity and string theory in a non-perturbative manner. To study two dimensional surfaces the idea of N x N hermitean matrix model was brought about in the large N limit, but some difficulties exist in this approach; the most serious problem appears when calculating the partition function of pure gravity. In fact for the simplest ¢4 matrix model, the partition function diverges. To avoid this difficulty we may change the model into the ¢4+¢6 model/) but the latter also has the same divergence. What is the action of the model that does not have this difficulty? We find that ¢4 + (¢4? model has good property in the sphere case by means of saddle point approximation. We consider a matrix model which has the following action,