Mathematical Tools for Calculation of the Effective Action in Quantum Gravity

We review the status of covariant methods in quantum field theory and quantum gravity, in particular, some recent progress in the calculation of the effective action via the heat kernel method. We study the heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold without boundary. We develop a manifestly covariant method for computation of the heat kernel asymptotic expansion as well as new algebraic methods for calculation of the heat kernel for covariantly constant background, in particular, on homogeneous bundles over symmetric spaces, which enables one to compute the low-energy non-perturbative effective action.Comment: 71 pages, 2 figures, submitted for publication in the Springer book (in preparation) "Quantum Gravity", edited by B. Booss-Bavnbek, G. Esposito and M. Lesc

Absence of classical and quantum mixing

It is shown, under mild assumptions, that classical degrees of freedom dynamically coupled to quantum ones do not inherit their quantum fluctuations. It is further shown that, if the assumptions are strengthen by imposing the existence of a canonical structure, only purely classical or purely quantum dynamics are allowed.Comment: REVTeX, 4 page

Convenient Versus Unique Effective Action Formalism in 2D Dilaton-Maxwell Quantum Gravity

The structure of one-loop divergences of two-dimensional dilaton-Maxwell quantum gravity is investigated in two formalisms: one using a convenient effective action and the other a unique effective action. The one-loop divergences (including surface divergences) of the convenient effective action are calculated in three different covariant gauges: (i) De Witt, (ii) $\Omega$-degenerate De Witt, and (iii) simplest covariant. The on-shell effective action is given by surface divergences only (finiteness of the $S$-matrix), which yet depend upon the gauge condition choice. Off-shell renormalizability is discussed and classes of renormalizable dilaton and Maxwell potentials are found which coincide in the cases of convenient and unique effective actions. A detailed comparison of both situations, i.e. convenient vs. unique effective action, is given. As an extension of the procedure, the one-loop effective action in two-dimensional dilaton-Yang-Mills gravity is calculated.Comment: 25 pages, LaTeX file, HUPD-93-0

Renormalized Kaluza-Klein theories

Using six-dimensional quantum electrodynamics ($QED_6$) as an example we study the one-loop renormalization of the theory both from the six and four-dimensional points of view. Our main conclusion is that the properly renormalized four dimensional theory never forgets its higher dimensional origin. In particular, the coefficients of the neccessary extra counterterms in the four dimensional theory are determined in a precise way. We check our results by studying the reduction of $QED_4$ on a two-torus.Comment: LaTeX, 36 pages. A new section added; references improved, typos fixe

QFT, String Temperature and the String Phase of De Sitter Space-time

The density of mass levels \rho(m) and the critical temperature for strings in de Sitter space-time are found. QFT and string theory in de Sitter space are compared. A `Dual'-transform is introduced which relates classical to quantum string lengths, and more generally, QFT and string domains. Interestingly, the string temperature in De Sitter space turns out to be the Dual transform of the QFT-Hawking-Gibbons temperature. The back reaction problem for strings in de Sitter space is addressed selfconsistently in the framework of the `string analogue' model (or thermodynamical approach), which is well suited to combine QFT and string study.We find de Sitter space-time is a self-consistent solution of the semiclassical Einstein equations in this framework. Two branches for the scalar curvature R(\pm) show up: a classical, low curvature solution (-), and a quantum high curvature solution (+), enterely sustained by the strings. There is a maximal value for the curvature R_{\max} due to the string back reaction. Interestingly, our Dual relation manifests itself in the back reaction solutions: the (-) branch is a classical phase for the geometry with intrinsic temperature given by the QFT-Hawking-Gibbons temperature.The (+) is a stringy phase for the geometry with temperature given by the intrinsic string de Sitter temperature. 2 + 1 dimensions are considered, but conclusions hold generically in D dimensions.Comment: LaTex, 24 pages, no figure

Unitarity Restoration in the Presence of Closed Timelike Curves

A proposal is made for a mathematically unambiguous treatment of evolution in the presence of closed timelike curves. In constrast to other proposals for handling the naively nonunitary evolution that is often present in such situations, this proposal is causal, linear in the initial density matrix and preserves probability. It provides a physically reasonable interpretation of invertible nonunitary evolution by redefining the final Hilbert space so that the evolution is unitary or equivalently by removing the nonunitary part of the evolution operator using a polar decomposition.Comment: LaTeX, 17pp, Revisions: Title change, expanded and clarified presentation of original proposal, esp. with regard to Heisenberg picture and remaining in original Hilbert spac

On the complete analytic structure of the massive gravitino propagator in four-dimensional de Sitter space

With the help of the general theory of the Heun equation, this paper completes previous work by the authors and other groups on the explicit representation of the massive gravitino propagator in four-dimensional de Sitter space. As a result of our original contribution, all weight functions which multiply the geometric invariants in the gravitino propagator are expressed through Heun functions, and the resulting plots are displayed and discussed after resorting to a suitable truncation in the series expansion of the Heun function. It turns out that there exist two ranges of values of the independent variable in which the weight functions can be divided into dominating and sub-dominating family.Comment: 21 pages, 9 figures. The presentation has been further improve

N=2 Super-Higgs, N=1 Poincare' Vacua and Quaternionic Geometry

In the context of N=2 supergravity we explain the occurrence of partial super-Higgs with vanishing vacuum energy and moduli stabilization in a model suggested by superstring compactifications on type IIB orientifolds with 3-form fluxes. The gauging of axion symmetries of the quaternionic manifold, together with the use of degenerate symplectic sections for special geometry, are the essential ingredients of the construction.Comment: 18 page

Gauged Gravity via Spectral Asymptotics of non-Laplace type Operators

We construct invariant differential operators acting on sections of vector bundles of densities over a smooth manifold without using a Riemannian metric. The spectral invariants of such operators are invariant under both the diffeomorphisms and the gauge transformations and can be used to induce a new theory of gravitation. It can be viewed as a matrix generalization of Einstein general relativity that reproduces the standard Einstein theory in the weak deformation limit. Relations with various mathematical constructions such as Finsler geometry and Hodge-de Rham theory are discussed.Comment: Version accepted by J. High Energy Phys. Introduction and Discussion significantly expanded. References added and updated. (41 pages, LaTeX: JHEP3 class, no figures

How much time does a measurement take?

We consider the problem of measurement using the Lindblad equation, which allows the introduction of time in the interaction between the measured system and the measurement apparatus. We use analytic results, valid for weak system-environment coupling, obtained for a two-level system in contact with a measurer (Markovian interaction) and a thermal bath (non-Markovian interaction), where the measured observable may or may not commute with the system-environment interaction. Analysing the behavior of the coherence, which tends to a value asymptotically close to zero, we obtain an expression for the time of measurement which depends only on the system-measurer coupling, and which does not depend on whether the observable commutes with the system-bath interaction. The behavior of the coherences in the case of strong system-environment coupling, found numerically, indicates that an increase in this coupling decreases the measurement time, thus allowing our expression to be considered the upper limit for the duration of the process.Comment: REVISED VERSION: 17 pages, 2 figure