Towards the consistent perturbative expansion in discrete gravity

Abstract

We consider correctly defining the perturbative expansion in a discrete gravity (simplicial or Regge calculus) needed to study physical effects like graviton loop corrections to Newton's potential. For the symmetric derivative $\hat{I}{”}_{\hat{I}}^{(s)}»=i\sin p_{\hat{I}}»$ in the finite-difference action, the propagator has a graviton pole at ${\sin }^2{p}_0={\sum }_{\hat{I}\pm =1}^3{\sin }^2{p}_{\hat{I}}\pm$, or, at small $p_{\hat{I}}\pm$, at $p_0$ close to 0 or $\pm \ddot{I}€$. This pole doubling means doubling the result of integration over d$p_0$ compared to the continuum. The usual derivative $\hat{I}{”}_{\hat{I}}»=\exp (i{p}_{\hat{I}}»)-1$ leads to a tricky analytical structure of the propagator, since $\hat{I}{”}_{\hat{I}}»\ne -\bar{\hat{I}}{”}_{\hat{I}}»$, and again to a discrepancy with the continuum. The way out is to use an action ${\check{S}}_{\mathrm{g}}$ with both $\hat{I}{”}_{\hat{I}}^{(s)}»$ and $\hat{I}{”}_{\hat{I}}»$ and the synchronous gauge $g_{0\hat{I}»}=g_{0\hat{I}»}^{(0)}$ (implemented by adding a term bilinear in $n^{\hat{I}}»(g_{\hat{I}»\hat{I}¼}-g_{\hat{I}»\hat{I}¼}^{(0)})$, $n^{\hat{I}}»=[1,-\epsilon (\hat{I}{”}^{(s)\hat{I}\pm }\hat{I}{”}_{\hat{I}}^{(s)}\pm {)}^{-1}\hat{I}{”}^{(s){\hat{I}}^2}]$, $\epsilon \to 0$, thus removing singularities at $p_0=0$). Given the propagator $\check{G}(n,\bar{n})$, we form a principal value propagator $[\check{G}(n,n)+\check{G}(\bar{n},\bar{n})]/2$ by analytically continuing from real $n=\bar{n}$. Singularities are resolved like $p_0^{-j}\to [(p_0+i\epsilon {)}^{-j}+(p_0-i\epsilon {)}^{-j}]/2$ leading to separate diagram finiteness at $\epsilon \to 0$. We analyze a 1-parameter family of actions differing in using $\hat{I}{”}_{\hat{I}}»$ vs $\hat{I}{”}_{\hat{I}}^{(s)}»$, find the only one reproducing convergent continuum diagrams for small external momenta (which is natural to demand from discretization), consider finiteness of the principal value gauge-fixing term and vanishing ghost contribution. The analysis is illustrated by the electromagnetic (Yang-Mills) case.

59 pages, 3 figures. v3: Intro&Conclusion&Biblio expanded