The concept of time in 2D quantum gravity

Abstract

We show that the ``time'' t_s defined via spin clusters in the Ising model coupled to 2d gravity leads to a fractal dimension d_h(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 d_h(s)=16 at the critical point when applied to flat space. The fractal dimension d_h(s) is in disagreement with both analytical prediction and numerical determination of the fractal dimension d_h(g), which is based on the use of the geodesic distance t_g as ``proper time''. There seems to be no simple relation of the kind t_s = t_g^{d_h(g)/d_h(s)}, as expected by dimensional reasons.