Weak Field Expansion of Gravity: Graphs, Matrices and Topology

Details Weak Field Expansion of Gravity: Graphs, Matrices and Topology Weak Field Expansion of Gravity: Graphs, Matrices and Topology

1998-03-28

arxiv.org/abs/hep-th/9803238v2

Int.J.Mod.Phys. A14 (1999) 2705-2744

Physics

High Energy Physics

High Energy Physics - Theory

41pages, 26 figures, Latex, epsf.sty

Scientific paper

10.1142/S0217751X99001366

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO(n) tensor $(\pl)^d h_\ab$, which generally appears in the weak field expansion around the flat space: $g_\mn=\del_\mn+h_\mn$. Making use of this representation, we explain 1) Generating function of graphs (Feynman diagram approach), 2) Adjacency matrix (Matrix approach), 3) Graphical classification in terms of "topology indices" (Topology approach), 4) The Young tableau (Symmetric group approach). We systematically construct the global SO(n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of $\pl\pl h-, {and} (\pl\pl h)^2-$ invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions).

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