EUCLIDEAN QUANTUM GRAVITY: BOUNDARY CONDITIONS AND SPECTRAL GEOMETRY
DAMTP Relativity Group; 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
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.
BASIC REFERENCES (66, in direct chronological order)
 V. A. Rohlin, A Three-Dimensional Manifold is the Boundary of a Four-Dimensional One (Doklady Akad. Nauk SSSR, Vol. 81, pp. 355--357, 1951).
 C. Misner, Feynman Quantization of General Relativity (Reviews of Modern Physics, Vol. 29, pp. 497--509, 1957).
 G. Grubb, Properties of Normal Boundary Problems for Elliptic Even-Order Systems (Annali Scuola Normale Superiore Pisa, Ser. IV, Vol. 1, pp. 1-61, 1974).
 D. Deutsch and P. Candelas, Boundary Effects in Quantum Field Theory (Physical Review D, Vol. 20, pp. 3063--3080, 1979).
 S. W. Hawking, The Path-Integral Approach to Quantum Gravity, in: General Relativity, an Einstein Centenary Survey, eds. S. W. Hawking and W. Israel, pp. 746--789 (Cambridge University Press, Cambridge, 1979).
 S. W. Hawking, The Boundary Conditions of the Universe (Pontificiae Academiae Scientiarum Scripta Varia, Vol. 48, pp. 563--574, 1982).
 S. W. Hawking, The Boundary Conditions for Gauged Supergravity (Physics Letters B, Vol. 126, pp. 175--177, 1983).
 J. B. Hartle and S. W. Hawking, Wave Function of the Universe (Physical Review D, Vol. 28, pp. 2960--2975, 1983).
 S. W. Hawking, The Quantum State of the Universe (Nuclear Physics B, Vol. 239, pp. 257-276, 1984).
 J. W. York, Boundary Terms in the Action Principles of General Relativity (Foundations of Physics, Vol. 16, pp. 249--257, 1986).
 I. G. Moss, Boundary Terms in the Heat-Kernel Expansion (Classical and Quantum Gravity, Vol. 6, pp. 759--765, 1989).
 M. Schroder, On the Laplace Operator with Non-Local Boundary Conditions and Bose Condensation (Reports in Mathematical Physics, Vol. 27, pp. 259-269, 1989).
 I. G. Moss and S. Poletti, Boundary Conditions for Quantum Cosmology (Nuclear Physics B, Vol. 341, pp. 155--161, 1990).
 P. D. D'Eath and G. Esposito, Local Boundary Conditions for the Dirac Operator and One-Loop Quantum Cosmology (Physical Review D, Vol. 43, pp. 3234--3248, 1991).
 P. D. D'Eath and G. Esposito, Spectral Boundary Conditions in One-Loop Quantum Cosmology (Physical Review D, Vol. 44, pp. 1713--1721, 1991).
 H. C. Luckock, Mixed Boundary Conditions in Quantum Field Theory (Journal of Mathematical Physics, Vol. 32, pp. 1755--1766, 1991).
 D. M. McAvity and H. Osborn, A DeWitt Expansion of the Heat Kernel for Manifolds with a Boundary (Classical and Quantum Gravity, Vol. 8, pp. 603--638, 1991).
 D. M. McAvity and H. Osborn, Asymptotic Expansion of the Heat Kernel for Generalized Boundary Conditions (Classical and Quantum Gravity, Vol. 8, pp. 1445--1454, 1991).
 A. O. Barvinsky, A. Yu. Kamenshchik and I. P. Karmazin, One-Loop Quantum Cosmology: Zeta-Function Technique for the Hartle--Hawking Wave Function of the Universe (Annals of Physics, Vol. 219, pp. 201--242, 1992).
 B. Booss--Bavnbek and K. P. Wojciechowski, Elliptic Boundary Problems for Dirac Operators (Birkhauser, Boston, 1993).
 G. Esposito, Quantum Gravity, Quantum Cosmology and Lorentzian Geometries (Springer Lecture Notes in Physics, Vol. m12, pp. 1--349, 1994).
 G. Esposito, Gauge-Averaging Functionals for Euclidean Maxwell Theory in the Presence of Boundaries (Classical and Quantum Gravity, Vol. 11, pp. 905--926, 1994).
 G. Esposito, A. Yu. Kamenshchik, I. V. Mishakov and G. Pollifrone, Euclidean Maxwell Theory in the Presence of Boundaries. II (Classical and Quantum Gravity, Vol. 11, pp. 2939--2950, 1994).
 I. G. Moss and S. Poletti, Conformal Anomalies on Einstein Spaces with Boundary (Physics Letters B, Vol. 333, pp. 326--330, 1994).
 G. Esposito, A. Yu. Kamenshchik, I. V. Mishakov and G. Pollifrone, One-Loop Amplitudes in Euclidean Quantum Gravity (Physical Review D, Vol. 52, pp. 3457--3465, 1995).
 G. Esposito and A. Yu. Kamenshchik, Mixed Boundary Conditions in Euclidean Quantum Gravity (Classical and Quantum Gravity, Vol. 12, pp. 2715--2722, 1995).
 D. V. Vassilevich, Vector Fields on a Disk with Mixed Boundary Conditions (Journal of Mathematical Physics, Vol. 36, pp. 3174--3182, 1995).
 P. B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah--Singer Index Theorem (CRC Press, Boca Raton, 1995).
 M. Bordag, E. Elizalde and K. Kirsten, Heat-Kernel Coefficients of the Laplace Operator on the d-Dimensional Ball (Journal of Mathematical Physics, Vol. 37, 895--916, 1996).
 G. Grubb, Functional Calculus of Pseudo-Differential Boundary Problems (Birkhauser, Boston, 1996).
 G. Esposito and A. Yu. Kamenshchik, One-Loop Divergences in Simple Supergravity: Boundary Effects (Physical Review D, Vol. 54, pp. 3869--3881, 1996).
 D. V. Vassilevich and V. N. Marachevsky, A Diffeomorphism-Invariant Eigenvalue Problem for Metric Perturbations in a Bounded Region (Classical and Quantum Gravity, Vol. 13, pp. 645--652, 1996).
 I. G. Avramidi, G. Esposito and A. Yu. Kamenshchik, Boundary Operators in Euclidean Quantum Gravity (Classical and Quantum Gravity, Vol. 13, pp. 2361--2373, 1996).
 P. D. D'Eath, Supersymmetric Quantum Cosmology (Cambridge University Press, Cambridge, 1996).
 G. Esposito, A. Yu. Kamenshchik and G. Pollifrone, Euclidean Quantum Gravity on Manifolds with Boundary (Fundamental Theories of Physics, Vol. 85, pp. 1--319, Kluwer, Dordrecht, 1997).
 I. G. Moss and P. J. Silva, BRST Invariant Boundary Conditions for Gauge Theories (Physical Review D, Vol. 55, pp. 1072--1078, 1997).
 G. Esposito, Dirac Operators and Spectral Geometry (Cambridge Lecture Notes in Physics, Vol. 12, pp. 1--209, Cambridge University Press, Cambridge, 1998).
 K. Kirsten, The a5 Coefficient on a Manifold with Boundary (Classical and Quantum Gravity, Vol. 15, pp. L5--L12, 1998).
 I. G. Avramidi and G. Esposito, New Invariants in the One-Loop Divergences on Manifolds with Boundary (Classical and Quantum Gravity, Vol. 15, pp. 281--297, 1998).
 I. G. Avramidi and G. Esposito, Gauge Theories on Manifolds with Boundary (Communications in Mathematical Physics, Vol. 200, pp. 495--543, 1999).
 G. Esposito and A. Yu. Kamenshchik, Fourth-Order Operators on Manifolds with a Boundary (Classical and Quantum Gravity, Vol. 16, pp. 1097--1111, 1999).
 G. Esposito, Non-Local Boundary Conditions in Euclidean Quantum Gravity (Classical and Quantum Gravity, Vol. 16, pp. 1113--1126, 1999).
 G. Esposito, New Kernels in Quantum Gravity (Classical and Quantum Gravity, Vol. 16, pp. 3999--4010, 1999).
 E. Witten and S. T. Yau, Connectedness of the Boundary in the AdS/CFT Correspondence (Advances in Theoretical and Mathematical Physics, Vol. 3, pp. 1635--1655, 1999).
 G. Esposito, Boundary Operators in Quantum Field Theory (International Journal of Modern Physics A, Vol. 15, pp. 4539--4555, 2000).
 G. Tsoupros, Radiative Contributions to the Effective Action of Selfinteracting Scalar Field on a Manifold with Boundary (Classical and Quantum Gravity, Vol. 17, pp. 2255--2266, 2000).
 G. Esposito, Quantum Gravity in Four Dimensions (Nova Science, New York, 2001).
 K. Kirsten, Spectral Functions in Mathematics and Physics (CRC Press, Boca Raton, 2001).
 G. Esposito, G. Miele and B. Preziosi, Quantum Gravity and Spectral Geometry (Nuclear Physics B Proceedings Supplement, Vol. 104, pp. 1--263, 2002).
 G. Esposito and K. Kirsten, Chiral Bag Boundary Conditions on the Ball (Physical Review D, Vol. 66, 085014, 2002).
 B. S. DeWitt, The Global Approach to Quantum Field Theory (International Series of Monographs on Physics, Vol. 114, Oxford University Press, Oxford, 2003).
 G. Grubb, Spectral Boundary Conditions for Generalizations of Laplace and Dirac Operators (Communications in Mathematical Physics, Vol. 240, pp. 243--280, 2003).
 D. V. Vassilevich, Heat Kernel Expansion: User's Manual (Physics Reports, Vol. 388, pp. 279--360, 2003).
 M. Asorey, A. Ibort and G. Marmo, Global Theory of Quantum Boundary Conditions and Topology Change (International Journal of Modern Physics A, Vol. 20, pp. 1001--1025, 2005).
 I. G. Avramidi, Heat Kernel Asymptotics of Zaremba Boundary Value Problem (Mathematical Physics, Analysis and Geometry, Vol. 7, pp. 9--46, 2004).
 J. Cardy, Boundary Conformal Field Theory (HEP-TH/0411189).
 G. T. Horowitz, Spacetime in String Theory (New Journal of Physics, Vol. 7, 201, 2005).
 G. Tsoupros, Conformal Anomalies for Interacting Scalar Fields on Curved Manifolds with Boundary (International Journal of Modern Physics A, Vol. 20, pp. 1027--1064, 2005).
 G. Tsoupros, Conformal Anomaly for Free Scalar Propagation on Curved Bounded Manifolds (General Relativity and Gravitation, Vol. 37, pp. 399--406, 2005).
 D. V. Vassilevich, Spectral Problems from Quantum Field Theory (Contemporary Mathematics, Vol. 366, pp. 3--22, 2005)
 G. Esposito, Euclidean Quantum Gravity in Light of Spectral Geometry (Contemporary Mathematics, Vol. 366, pp. 23--42, 2005).
 G. Esposito, G. Fucci, A. Yu. Kamenshchik and K. Kirsten, Spectral Asymptotics of Euclidean Quantum Gravity with Diff-Invariant Boundary Conditions (Classical and Quantum Gravity, Vol. 22, pp. 957--974, 2005).
 G. Esposito, G. Fucci, A. Yu. Kamenshchik and K. Kirsten, A Non-Singular One-Loop Wave Function of the Universe From a New Eigenvalue Asymptotics in Quantum Gravity (JHEP, 0509:063, 2005).
 Z. Haba, Quantum Field Theory on Manifolds With a Boundary (Journal of Physics A, Vol. 38, pp. 10393--10402, 2005).
 P. van Nieuwenhuizen and D. V. Vassilevich, Consistent Boundary Conditions for Supergravity (Classical and Quantum Gravity, Vol. 22, pp. 5029--5051, 2005).
 A. O. Barvinsky and D. V. Nesterov, Quantum Effective Action in Spacetimes with Branes and Boundaries (Physical Review D, Vol. 73. 066012, 2006).