International Meeting on

"Quantum Gravity and Spectral Geometry"

July 2 - July 7, 2001


(last revised on June 29th, 2001)




Lecture 1. "Effective Action Approach in Gauge Field Theories and Quantum Gravity"

Abstract: An overview of the effective action approach in quantum field theory and quantum gravity.

Lecture 2. "Calculation of the Heat Kernel Asymptotic Expansion"

Abstract: Review of basic techniques for calculation of the heat kernel asymptotic expansion for Laplace type partial diffferential operators acting o smooth sections of a vector bundle over a compact manifold without boundary.

Lecture 3. "Covariant Approximation Schemes for Calculation of the Heat Kernel"

Abstract: Development of various approximation schemes for the heat kernel based on the behavior of the "background fields". In the "high-energy" limit, which is determined by rapidly varying background fields, the highest derivative terms are computed and a partial summation is carried out leading to closed formulas for the effective action. In the "low-energy" limit, which is determined by slowly varying background fields, an algebraic approach is developed that enables one to obtain closed non-perturbative formulas for the heat kernel that generate the whole asymptotic expansion.

Lecture 4. "Heat-kernel Asymptotics for Non-Laplace Type Operators"

Abstract: The most general class of second-order operators (non-Laplace type operators), acting on sections of a vector bundle, with positive definite leading symbol, is studied and the calculation of the heat-kernel asymptotics is presented. The heat kernel and the resolvent are constructed explicitly in the leading order.

Lecture 5. "Heat-Kernel Asymptotics for the Grubb - Gilkey - Smith Boundary-Value Problem"

Abstract: Laplace type partial differential operator acting on sections of a vector bundle over a compact Riemannian manifold with smooth boundary with the Grubb - Gilkey - Smith boundary conditions, which involve both normal and tangential derivatives on the boundary. The criterion of strong ellipticity is discussed and the first non-trivial coefficient of the asymptotic expansion of the trace of the heat kernel is computed.

Lecture 6. "Heat-Kernel Asymptotics for Non-Smooth Boundary Conditions"

Abstract: The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with nonstandard singular boundary conditions, which include Dirichlet conditions on one part of the boundary and Neumann ones on another part of the boundary, is studied. The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed and the leading heat-kernel asymptotics are computed.


Abstract: The aim of these lectures is to discuss mathematical methods in statistical mechanics in external stationary gravitational and gauge fields, as well as their applications in black hole physics. We will also discuss how summation over the modes in the partition function on a stationary background can be related to the path integral in Euclidean gravity.

Preliminary plan of the lectures:

Part I. Introduction

Part II. Canonical and Euclidean Methods in Statistical Mechanics on Stationary Backgrounds

This part includes discussion of the following topics:

Part III. Some Applications

One of the applications which will be discussed here is thermodynamics and statistical mechanics of black holes.


Abstract: The lecture, based on joint work with Kirsten, Park and Vassilevich, describes heat-trace asymptotics with spectral, or Dirichlet/Neumann, or transmittal boundary conditions.


Abstract: Trace expansions for functions of partial differential and pseudodifferential operators, with or without boundary conditions, are of interest for example in the study of the index and other geometric invariants, as well as the noncommutative residue. The lectures will give an exposition of the main techniques and some applications.


Lecture 1. In this lecture I follow Segal's notes and show how the Grasmannian of the boundary conditions and the determinant defined on this Grassmannian appear in the Topological Quantum Field Theory.

Lecture 2. I discuss zeta determinant of the Dirac operators and elliptic boundary problems.

Lecture 3. In this lecture I prove the pasting formula for the phase of the zeta-determinant.

Lecture 4. In the final lecture I discuss pasting of the modulus of the zeta-determinant. This is the most difficult case from the analytical point of view and at present we only have an adiabatic formula.


Colloquium Lectures:



Abstract: We consider the heat kernel and the zeta function associated with Laplace type operators acting on a general irreducible rank-one locally symmetric space. The set of Minakshisundaram-Pleijel coefficients in the small-t asymptotic expansion of the kernel is calculated explicitly. The Chern-Simons invariants of irreducible U(n)-flat connections on compact hyperbolic 3-manifolds of the form G H3 are derived. The explicit formula for the Chern-Simons functional is given in terms of Selberg type zeta functions related to the twisted eta invariants of Atiyah-Patodi-Singer. The semiclassical limit for the partition function is presented.



Abstract: A non technical discussion of some aspects of Noncommutative Geometry. With the title of the workshop as a guide.



B. Booss-Bavnbek, J.S. Dowker, E. Elizalde, S. Fulling, G. Imponente, E. Gozzi, A.Yu. Kamenshchik, K.A. Kazakov, V.N. Maratchevski, D. Mauro, E. Melas, I.G. Moss, A. Mostafazadeh, L. Nesic, V.V. Nesterenko, J.H. Park, O.V. Pavlovsky, I.G. Pirozhenko, E.M. Santangelo, R. Seeley, D. Seminara, M. Seriu, O. Timofeevskaya, D.V. Vassilevich, F. Williams, R.P. Woodard, S. Zerbini

Details available so far:

Abstract: I shall explain why I consider the widely applied intuitive definition of the spectral flow of continuous families of unbounded self-adjoint Fredholm operators (with possibly varying domain) to be insufficient and provide a rigorous definition. This is joint work with Matthias Lesch, Cologne University, Germany, and John Phillips, Victoria University, Canada.



Abstract: The zeta function method, for the cases when the spectrum of the relevant operator is known either implicitly or explicitly (making use in this case of the asymptotics of the function determining the spectrum), will be described.


Abstract: Oscillations in eigenvalue density are associated with closed orbits of the corresponding classical (or geometrical optics) system.  Although invisible in the heat-kernel expansion, these features determine the nonlocal parts of propagators, including the Casimir energy.  I'll review the classic work (Balian & Bloch; Gutzwiller; Duistermaat & Guillemin), discuss the connection with vacuum energy, and show that when the coupling constant is varied with the energy, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it has always had for the wave equation in a cavity.


Abstract: In 1931 Koopman and von Neumann  extended previous work of Liouville and provided an operatorial version of Classical Mechanics (CM). In this talk we will review a path-integral formulation of this operatorial version of CM. In particular we will study the geometrical nature of the many auxiliary variables present and of the unexpected universal symmetries generated by the functional technique.


Abstract: Within a cosmological framework, we provide a Hamiltonian analysis of the Mixmaster Universe dynamics on the base of a standard Arnowitt-Deser-Misner approach, showing how the chaotic behavior characterizing the evolution of the system near the cosmological singularity can be obtained as the semiclassical limit of the canonical quantization of the model in the same dynamical representation. The relation between this intrinsic chaotic behavior and the indeterministic quantum dynamics is inferred through the coincidence between the microcanonical probability distribution and the semiclassical quantum one.


Abstract: We consider zeta regularization of the basic commutators in string theories suggested by Hwang, Marnellius and Saltsidis and apply it to string theories with non-trivial vacuums. It is shown that the application of this regularization allows one to restore the Jacobi identities in these theories.


Abstract: The problem of self-consistent transition to the classical limit in quantum gravity is considered. It is shown that the usual definition of the gravitational potential through the two-particle scattering amplitude is inconsistent with the classical result given by the Schwarzschild solution. A new interpretation of the correspondence principle is then given directly in terms of the effective action. Gauge independence of the proposed interpretation is proved to the leading order in the Planck constant. Application to the black holes is discussed.


Abstract: Path-integral and other approaches to the calculation of Casimir energy of dielectrics will be analysed with emphasis on the nature of divergences and comparison of different regularization schemes. Dielectric ball and perfectly conducting wedge will be considered as illustrative examples of general methods.


Abstract: Sometime ago it was shown that the operatorial approach to classical mechanics, pioneered in the 30's by Koopman and von Neumann, can have a functional version. In this talk we will extend this functional approach to the case of classical field theories and in particular to the Yang-Mills ones. We shall show that the issues of gauge-fixing and Faddeev-Popov determinant arise also in this classical formalism. We shall also indicate how to quantize the system by freezing to zero the Grassmannian partners of time.


Abstract: The BMS group is the asymptotic isometry for a large class of asymptotically flat Einstein-Maxwell space-times. A possible relation of the unitary irreducible representations of this group and gravitational instantons is discussed.



Abstract: We introduce a class of generalizations of supersymmetry that involve a set of integer-valued topological invariants. These symmetries are defined in terms of certain conditions on the spectral properties of the corresponding quantum systems. We identify the operator algebras of the generators of these symmetries and show that the algebras of ordinary and fractional supersymmetric quantum mechanics and certain parasupersymmetric quantum mechanics are special cases of these operator algebras. We also comment on the mathematical interpretation of the corresponding topological invariants.


Abstract: p-Adic quantum cosmology is an application of p-Adic quantum theory to the universe as a whole. p-Adic quantum theory is a p-adic generalization of standard quantum theory. In p-adic quantum theory the argument of the wave function is a  p-adic variable. Geometry of p-adic spaces has a nonarchimedean structure. Under some restrictions product of the ordinary wave function and its p-adic counterparts gives adelic wave function. In adelic Feynman's path integral approach, integration over nonarchimedean geometries is also taken into account. As an illustration, adelic wave function for some minisuperspace models is calculated and discussed.


Abstract: The high temperature asymptotics of the thermodynamic characteristics (free energy, internal energy, and entropy) of electromagnetic field subjected to boundary conditions with spherical and cylindrical symmetries are constructed by making use of the relevant heat kernel coefficients


Abstract: Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on manifolds M which are defined by a time-dependent metric, by a time dependent heat source, by  time dependent boundary conditions, and by a time-dependent specific heat.


Abstract: In this report a gravity representation of Yang-Mills theory is given. Using such approach, one obtain new information about solutions of classical YM theory, classifications of such solutions and so on. Singular solutions (black-hole-like solutions) of YM equations and problem of fixing of such solutions in all space-time are discussed in context of connection with bimetrical gravity. Behavior of these solutions in a theory with a "cosmological" Lambda-part is investigated also. Physical interpretation of such solutions is given. Using an effective field theory approach we try to show that quantum fluctuations and vacuum polarization effects lead to generation of finite-energy objects in QCD.


Abstract: We obtain, through zeta function methods, the one-loop effective action for massive Dirac fields in the presence of a uniform, but otherwise general, electromagnetic background. After discussing renormalization, we compare our zeta-function result with Schwinger's proper-time approach.




Abstract: Based on the spectral representation of spatial geometry, we construct an analysis scheme for Spacetime Physics and Cosmology, which enables us to discuss not only the dynamics of a spatial geometry, but also the relation of more than two spatial geometries. This scheme is especially useful for analyzing problems in which more than two geometries are involved, such as the model-fitting problem in cosmology.


Abstract: Quantization of the gravitational field on a classical background is described by covariant collective coordinates method. The group of isometries of the classical metric (exact solution of the Einstein Equations) determines the conservation laws. The collective coordinates method consists in extraction of collective variables that play the role of the symmetry group parameters and makes it possible to take into account the conservation laws. Perturbation theory is developed, expressions of the generating functional and Green's functions are represented.


Abstract: I give a short guide into applications of the heat kernel technique to the string/brane physics with emphasis on the resulting boundary-value problems.


Abstract: Using the trace formula we analyze the spectral zeta function for twisted p-forms on a compact real hyperbolic space. As an application we obtain explicit small-time asymptotics for the corresponding heat kernel.


Abstract: Quantum gravitational back-reaction offers a simultaneous explanation for why the cosmological constant is so small and a natural model of inflation in which scalars play no role. The idea is that the bare cosmological constant is actually GUT scale and that this causes inflation during the early universe. Inflation slows due to the accumulated gravitational attraction between virtual pairs of gravitons (and possibly also scalars) which are ripped apart by the rapid expansion. In this talk I first review work on the mechanism and then give simple physical answers to the following three questions:

(1) Why does the induced energy density go like       -H2/G (G H2)2 (H t)power ?

(2) Why does the induced pressure go like minus the induced energy density?

(3) How can the effect be causal when it involves super-horizon modes?


Abstract: The multiplicative anomaly of functional determinants will be introduced and evaluated in some simple cases. Some physical applications will be discussed.