Green functions and Euclidean fields near the bifurcate Killing horizon

Abstract

We approximate a Euclidean version of a D + 1 dimensional manifold with a bifurcate Killing horizon by a product of the two dimensional Rindler space R2 and a D − 1 dimensional Riemannian manifold M. We obtain approximate formulas for the Green functions. We study the behaviour of Green functions near the horizon and their dimensional reduction. We show that if M is compact then the massless minimally coupled quantum field contains a zero mode which is a conformal invariant free field on R 2 . Then, the Green function near the horizon can be approximated by the Green function of the two-dimensional quantum field theory. The correction term is exponentially small away from the horizon. If the volume of a geodesic ball is growing to infinity with its radius then the Green function cannot be approximated by a two-dimensional one.

Green functions and dimensional reduction of quantum fields on product manifolds

“…In sec.5 we have estimated the difference G − G 0 , where G 0 is the D − d dimensional Green function. We have shown in some models that there is no dimensional reduction for a non-compact M [2]. Nevertheless, a study of Green functions with non-compact manifolds M can lead to some unexpected results.…”

“…In this paper we ask the question whether a higher dimensional quantum field theory can be approximated by a lower dimensional one (not necessarily from the point of view of the Kaluza-Klein program). We have shown in our earlier paper [2] (see also [3]) that such an approximation applies near the bifurcate Killing horizon when D dimensional quantum field theory can be approximated by a two-dimensional one. We are interested in the dimensional reduction from the point of view of Green functions,i.e., correlation functions of quantum fields.…”

“…In spite of a vanishing at zero (in the original x 0 coordinate) it can be checked by means of a calculation of the curvature tensor R that R is a continuous function (if M = R n then the formula for the curvature in Bianchi type space-times derived in [13][14] gives R = 0). We have discussed the model (38) in detail in [2]. It has been shown that the model serves as an approximation to the Green function on a space-time with the bifurcate Killing horizon.…”