Towards the consistent perturbative expansion in discrete gravity

RESUMO

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 $\Delta^{(s)}_\lambda=i\sin p_\lambda$ in the finite-difference action, the propagator has a graviton pole at $\sin^2p_0=\sum^3_{\alpha=1}\sin^2p_\alpha$, or, at small $p_\alpha$, at $p_0$ close to 0 or $\pm\pi$. This pole doubling means doubling the result of integration over d$p_0$ compared to the continuum. The usual derivative $\Delta_\lambda=\exp(ip_\lambda)-1$ leads to a tricky analytical structure of the propagator, since $\Delta_\lambda\neq-\bar{\Delta}_\lambda$, and again to a discrepancy with the continuum. The way out is to use an action $\check{S}_{\rm g}$ with both $\Delta^{(s)}_\lambda$ and $\Delta_\lambda$ and the synchronous gauge $g_{0\lambda}=g_{0\lambda}^{(0)}$ (implemented by adding a term bilinear in $n^\lambda(g_{\lambda\mu}-g_{\lambda \mu}^{(0)})$, $n^\lambda=[1,-\varepsilon(\Delta^{(s)\alpha}\Delta^{(s)}_\alpha)^{-1}\Delta^{(s)\beta}]$, $\varepsilon\to0$, 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 $p_0^{-j}\to[(p_0+i\varepsilon)^{-j}+(p_0-i\varepsilon)^{-j}]/2$ leading to separate diagram finiteness at $\varepsilon\to0$. We analyze a 1-parameter family of actions differing in using $\Delta_\lambda$ vs $\Delta^{(s)}_\lambda$, 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.