Exploring geometric properties

4.13 Exploring geometric properties

Since the effective geometry in both phases is rather singular, different ways of measuring length may lead to inequivalent definitions of “dimension”. A common (local) notion is derived from the volume of a geodesic ball with radius $r$ . Usually the radius is measured in terms of the geodesic distance $d 1$ (the minimal number of links), or dual geodesic distance $d 4$ (the minimal number of links of the dual graph). Alternatively, one may consider the number $n(r)$ of 4-simplices in spherical shells of thickness 1 at distance $r$ , and define a fractal dimension by $⟨n(r)⟩ ∼ rdF−1$ [90

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To extract a global length scale, one may use the averages $⟨d1⟩$ , $⟨d4⟩$ , or consider the average “radius of the universe” $⟨r(T )⟩$ [11 ] to obtain a cosmological Hausdorff dimension $d CH$ , or the “average intrinsic linear extent” $−2∑ L = ⟨V i,j d4(i,j,T )⟩$ (see, for example, [75 , 76 ]). Away from the phase transition, the fractal dimensions associated with these geometric construction are more or less equivalent and give $dF = ∞$ in the crumpled phase and $dF ≈ 2$ in the elongated phase. It is difficult to measure the dimension close to the transition point.

It will not be straightforward to interpret the behaviour of observables (defined in analogy with the continuum theory), since in most of the phase space the geometry of the simplicial complex is far from approximating a metric 4-manifold. In search of a semiclassical interpretation for geometric observables, an alternative notion of local curvature for a simplicial manifold was suggested by de Bakker and Smit [90 ], based on a continuum expansion of the volume of a geodesic ball. Assuming furthermore that independent of $n$ , $n$ -volumes of balls with radius $r$ behave like regions on $n n+1 S ⊂ ℝ$ , they extracted scaling relations for various geometric quantities for an intermediate range for $r$ . This line of thought was pursued further in [185 ].

Close to the phase transition, one may investigate the behaviour of test particles (ignoring back-reactions on the geometry). Comparing the mass extracted from the one-particle propagator with the energy of the combined system obtained from the two-particle propagator [88 , 92 , 93 ], one does indeed find evidence for gravitational binding.