RENORMALIZATION-GROUP AND QUANTUM-GRAVITY

We study non-perturbative effects of the quantum fluctuation of gravity by means of the ε{lunate}-expansion around two dimensions and the renormalization group method. By introducing the fractal dimension as an order parameter we show that the space-time possesses a self-similar structure. We also point out that an operator such as the cosmological term, the scalar and fermion mass terms ceases to exist as a local one as the number of matter fields increases. This fact may be taken as a suggestion as to how to answer the naturalness problem, in particular the vanishing of the cosmological term. It is argued that there exists potential pathology in the ε{lunate}-expansion in the 2+ε{lunate} dimensional gravity and we give the solution that it is a double expansion in ε{lunate} and 1 -c where c is the number of matter fields (c=NS+ 1 2NF). © 1990.