Positive Semi-Definite Action in a Kaluza-Klein Theory with Compactification onto Time-Like Extra Dimensions

摘要

We consider the (4+N)-dimensional theory whose Lagrangian function is Lsub(4+N)= sqrt -g-circumflex alpha R-circumflex sup 2 , where R-circumflex is the Ricci scalar and alpha is a positive constant. The metric is g-circumflexsub(AB)= diag(gsub(ab), phi sup -1 g-barsub(mn)). Dimensional reduction leads to an effective four-dimensional Lagrangian of induced-gravity type. The positive semi-definiteness of L avoids the difficulties, pointed out recently by Horowitz and by Rubakov, which can arise in quantum cosmology when the (Euclidean) action becomes negative. The compactification is onto a time-like internal space g-barsub(mn), as suggested by Aref'eva and Volovich, giving a four-dimensional de Sitter space-time with phi=constant, which however is classically unstable on a time scale approx. H sup -1 . Decrease of the radius phisup(-1/2) of the internal space is ultimately halted by quantum effects, via some V(phi), and L sub 4 then includes the usual Hilbert term and a cosmological constant. 26 refs. (Atomindex citation 17:063342)