Applications to Gravity, Strings and p-Branes

Abstract

In this chapter two more examples of applications of the method of zeta function regularization are investigated, again in two very different, although related, contexts. The first of them is the spontaneous compactification that occurs in two-dimensional quantum gravity. This theory is considered to be an adequate toy model for a more fundamental formulation of quantum gravity. As an interesting physical result, with the help of the method of zeta-function regularization advocated here it can be proven that this compactification is stable, in contradistinction to what happens in multidimensional quantum gravity on a corresponding background (in more than two dimensions), which is known to be one-loop unstable. In the second section of the chapter we obtain the effective potential for strings and p-branes, in general, and specifically study the stability of the rigid membrane. A careful analysis of the zeta-functions relevant for the calculation of the effective potential for fixed-end and toroidal rigid p-branes at one-loop order and in the one-dimensional approximation (which are of inhomogeneous Epstein type) is performed. Asymptotic formulas which give accurate results (allowing only for exponentially decreasing errors) are presented, for the general case of p-branes, which carry all the dependencies on the basic parameters of the theory explicitly. The behavior of the corresponding effective potential is investigated too. Finally, the extrema of this effective potential are obtained.