Scaling in quantum gravity

Abstract

The 2-point function is the natural object in quantum gravity for extracting critical behavior: The exponential fall off of the 2-point function with geodesic distance determines the fractal dimension $d_H$ of space-time. The integral of the 2-point function determines the entropy exponent $γ$, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2-point function connects $γ$ and $d_H$ by a quantum gravity version of Fisher's scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation.

Figures (3)

• Figure 1: The inequality (\ref{['6']}). Two triangulations with marked links separated by distances $r_1$ and $r_2$ can be glued together to a triangulation where the marked links has a distance $r_1+r_2$ but the same number of triangles by cutting open a marked link in each of the triangulations to a 2-loop boundary and glue together the two boundaries.
• Figure 2: The graphical representation of the 2-point function for branched polymers. The dashed line represents the unique shortest path between the two marked vertices. The "blobs" represent the contribution from all rooted polymers branching out from a vertex.
• Figure 3: The 2-point function represented as a summation over 2-loop functions times 1-loop functions.