**EUCLIDEAN QUANTUM GRAVITY: BOUNDARY
CONDITIONS AND SPECTRAL GEOMETRY**

Links to related topics and authors:

DAMTP Relativity Group; MPG Potsdam; SIGRAV Bulletin; Quantum Gravity on INSPIRE; Quantum Gravity on Google Scholar

Bryce DeWitt (A) (B); Roger Penrose; Robert Geroch; Stephen Hawking; Gary Gibbons; Edward Witten; Abhay Ashtekar; Carlo Rovelli; Gregori Vilkovisky; Ivan Avramidi

**Introduction**:

The aim of theoretical physics is to provide a clear conceptual framework for the wide variety of natural phenomena, so that not only are we able to make accurate predictions to be checked against observations, but the underlying mathematical structures of the world we live in can also become sufficiently well understood by the scientific community. What are therefore the key elements of a mathematical description of the physical world? Can we derive all basic equations of theoretical physics from a few symmetry principles? What do they tell us about the origin and evolution of the universe? Why is gravitation so peculiar with respect to all other fundamental interactions?

The above questions have received careful consideration over the last decades, and have led, in particular, to several approaches to a theory aiming at achieving a synthesis of quantum physics on the one hand, and general relativity on the other hand. This remains, possibly, the most important task of theoretical physics. The need for a quantum theory of gravity is already suggested from singularity theorems in classical cosmology. Such theorems prove that the Einstein theory of general relativity leads to the occurrence of space-time singularities in a generic way.

At first sight one might be tempted to conclude that a breakdown of all physical laws occurred in the past, or that general relativity is severely incomplete, being unable to predict what came out of a singularity. It has been therefore pointed out that all these pathological features result from the attempt of using the Einstein theory well beyond its limit of validity, i.e. at energy scales where the fundamental theory is definitely more involved. General relativity might be therefore viewed as a low-energy limit of a richer theory, which achieves the synthesis of both the basic principles of modern physics and the fundamental interactions in the form presently known.

Within the framework just outlined it remains however true that the various approaches to quantum gravity developed so far suffer from mathematical inconsistencies, or incompleteness in their ability of accounting for some basic features of the laws of nature. From the point of view of general principles, the space-time approach to quantum mechanics and quantum field theory, and its application to the quantization of gravitational interactions, remains indeed of fundamental importance. When one tries to implement the Feynman sum over histories one discovers that, already at the level of non-relativistic quantum mechanics, a well defined mathematical formulation is only obtained upon considering a heat-equation problem. The measure occurring in the Feynman representation of the Green kernel is then meaningful, and the propagation amplitude of quantum mechanics in flat Minkowski space-time is obtained by analytic continuation. This is a clear indication that quantum-mechanical problems via path integrals are well understood only if the heat-equation counterpart is mathematically well posed.

In quantum field theory one then deals with the Euclidean approach, and its application to quantum gravity relies heavily on the theory of elliptic operators on Riemannian manifolds. To obtain a complete picture one has then to specify the boundary conditions of the theory, i.e. the class of Riemannian geometries with their topologies involved in the sum, and the form of boundary data assigned on the bounding surfaces.

In particular, work in the late nineties has shown that the only set of local boundary conditions on metric perturbations which are completely invariant under infinitesimal diffeomorphisms is incompatible with the request of a good elliptic theory. More precisely, while the resulting operator on metric perturbations can be made of Laplace type and elliptic in the interior of the Riemannian manifold under consideration, the property of strong ellipticity is violated. This is a precise mathematical expression of the requirement that a unique smooth solution of the boundary-value problem should exist which vanishes at infinite geodesic distance from the boundary. This opens deep interpretive issues, since only for gravity does the request of complete gauge invariance of the boundary conditions turn out to be incompatible with a good elliptic theory.

We have been therefore led to consider in our research non-local boundary conditions for the quantized gravitational field at one-loop level. On the one hand, such a scheme already arises in simpler problems, i.e. the quantum theory of a free particle subject to non-local boundary data on a circle. One then finds two families of eigenfunctions of the Hamiltonian: surface states which decrease exponentially as one moves away from the boundary, and bulk states which remain instead smooth and non-vanishing.

The generalization to an Abelian gauge theory such as Maxwell theory can fulfill non-locality, ellipticity and complete gauge invariance of boundary conditions providing one learns to work with pseudo-differential operators in one-loop quantum theory. On the other hand, in the application to quantum gravity, since the boundary operator acquires new kernels responsible for the pseudo-differential nature of the boundary-value problem, one might hope to be able to recover a good elliptic theory under a wider variety of conditions.

In our latest research, we have however proved that, on the Euclidean four-ball, local and diff-invariant boundary conditions still lead to a generalized zeta-function which is regular at the origin, by virtue of a peculiar spectral identity obtained by us for the first time in the literature. Boundary-value problems that are not strongly elliptic remain therefore viable in Euclidean quantum gravity, at least on some particular backgrounds with boundary.

These spectral properties are very important for quantum cosmology, quantum gravity and the foundations of quantized gauge theories, and have deep roots in global analysis and spectral geometry.

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