C-THEOREM AND SPECTRAL REPRESENTATION

Zamolodchikov's c-theorem is reformulated by using the spectral representation for the two-point function of the stress tensor. This approach makes explicit the unitarity constraints on the field theory and implements a nice physical picture of the renormalization group flow. An attempt is made to generalize the theorem above two space-time dimensions. There are two candidate c-functions, the spectral densities for spin-zero and spin-two intermediate states. The latter one is ruled out by means of examples. The spin-zero density can satisfy a generalized c-theorem, if the corresponding "central charge" is well defined at the fixed points. A meaningful charge is obtained by defining the theory on curved hyperbolic space. However, its limit to flat space needs some assumptions which seem to hold for free theories only. As a by-product, the trace anomaly in four dimensions is related to the spectral densities.