3.4 The phase structure3 Quantum Regge Calculus3.2 Higher-derivative terms

3.3 First simulations

The first numerical studies of the Regge action were undertaken by Berg [38, 39Jump To The Next Citation Point In The Article] and Hamber and Williams [118Jump To The Next Citation Point In The Article, 119]. Berg performed a Monte-Carlo simulation of the pure-curvature action for hypercubic tex2html_wrap_inline2477 - and tex2html_wrap_inline2479 -lattices with simplicial subdivision (see [40Jump To The Next Citation Point In The Article] for a description of the method). He used the scale-invariant measure tex2html_wrap_inline2481 . To avoid the divergence that results from rescaling the link lengths, he kept the total volume constant by performing an overall length rescaling of all links after each move. This amounts to fixing a typical length scale tex2html_wrap_inline2483, where tex2html_wrap_inline2485 is the expectation value of the 4-simplex volume.

For k =0, he found a negative average curvature tex2html_wrap_inline2489, and some evidence for a canonical scaling behaviour of lengths, areas and volumes. For tex2html_wrap_inline2491, he obtained a negative (positive) average deficit angle tex2html_wrap_inline2493 and a positive (negative) tex2html_wrap_inline2489 . A more detailed analysis for tex2html_wrap_inline2497 led Berg [39] to conclude that there exists a critical value tex2html_wrap_inline2499 (presumably a first-order transition [40]), below which tex2html_wrap_inline2489 is convergent, whereas above it diverges. (Myers [169] has conjectured that it may be possible to perform a similar analysis for Monte-Carlo data for the Lorentzian action.)

By contrast, Hamber and Williams [118] simulated the higher-derivative action

  equation385

on tex2html_wrap_inline2477 - and tex2html_wrap_inline2283 -lattices, using a time-discretized form of the Langevin evolution equation (see also [106]). For technical reasons, one uses barycentric instead of Voronoi volumes. They employed the scale-invariant measure tex2html_wrap_inline2507, where tex2html_wrap_inline2509 enforces an ultra-violet cutoff tex2html_wrap_inline2511 . They investigated the average curvature tex2html_wrap_inline2513 and squared curvature tex2html_wrap_inline2515 (scaled by powers of tex2html_wrap_inline2517 to make them dimensionless), as well as tex2html_wrap_inline2519 and tex2html_wrap_inline2521 . For tex2html_wrap_inline2523, one finds a negative tex2html_wrap_inline2525 and a large tex2html_wrap_inline2515, indicating a rough geometry. For small a, one observes a sudden decrease in tex2html_wrap_inline2515, as well as a jump from large positive to small negative values of tex2html_wrap_inline2525 as tex2html_wrap_inline2329 is increased. For large a, tex2html_wrap_inline2525 is small and negative, and the geometry appears to be smooth. Like Berg, they advocated a fundamental-length scenario, where the dynamically determined average link length provides an effective UV-cutoff.



3.4 The phase structure3 Quantum Regge Calculus3.2 Higher-derivative terms

image Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
http://www.livingreviews.org/lrr-1998-13
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