The concept of time in 2D gravity

Abstract

We show that the "time" ts defined via spin clusters in the Ising model coupled to 2d gravity leads to a fractal dimension dh(s) = 6 of space-time at the critical point, as advocated by Ishibashi and Kawai. In the unmagnetized phase, however, this definition of Hausdorff dimension breaks down. Numerical measurements are consistent with these results. The same definition leads to dh(s) = 16 at the critical point when applied to flat space. The fractal dimension dh(s) is in disagreement with both analytical prediction and numerical determination of the fractal dimension dh(g), which is based on the use of the geodesic distance tg as "proper time". There seems to be no simple relation of the kind ts = t9dh(g)/dh(s) as expected by dimensional reasons.