arXiv subjects

Ivan G. Avramidi

Publications and source records attributed to Ivan G. Avramidi.

At least 37 records · Page 2Linked to original sources

arXiv2008

Kinematics in Matrix Gravity

We develop the kinematics in Matrix Gravity, which is a modified theory of gravity obtained by a non-commutative deformation of General Relativity. In this model the usual interpretation of gravity as Riemannian geometry is replaced by a new kind of geometry, which is equivalent to a collection of Finsler geometries with several Finsler metrics depending both on the position and on the velocity. As a result the Riemannian geodesic flow is replaced by a collection of Finsler flows. This naturally leads to a model in which a particle is described by several mass parameters. If these mass parameters are different then the equivalence principle is violated. In the non-relativistic limit this also leads to corrections to the Newton's gravitational potential. We find the first and second order corrections to the usual Riemannian geodesic flow and evaluate the anomalous nongeodesic acceleration in a particular case of static spherically symmetric background.

gr-qc↗

arXiv2008

Non-perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles

We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group $G\times U(1)$ over a Riemannian manifold $M$ without boundary. The total connection on the vector bundle naturally splits into a $G$-connection and a U(1)-connection, which is assumed to have a parallel curvature $F$. We find a new local short time asymptotic expansion of the off-diagonal heat kernel $U(t|x,x')$ close to the diagonal of $M\times M$ assuming the curvature $F$ to be of order $t^{-1}$. The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the $G$-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature $F$, more precisely, on $tF$. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in $F$ in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature $F$. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion.

math-ph↗

arXiv2007

Noncommutative Einstein Equations

We study a noncommutative deformation of general relativity where the gravitational field is described by a matrix-valued symmetric two-tensor field. The equations of motion are derived in the framework of this new theory by varying a diffeomorphisms and gauge invariant action constructed by using a matrix-valued scalar curvature. Interestingly the genuine noncommutative part of the dynamical equations is described only in terms of a particular tensor density that vanishes identically in the commutative limit. A noncommutative generalization of the energy-momentum tensor for the matter field is studied as well.

gr-qc↗

arXiv2007

Heat Kernel Asymptotics on Homogeneous Bundles

math.DG↗

arXiv2007

Heat Kernel on Homogeneous Bundles over Symmetric Spaces

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtained result correctly reproduces the first non-trivial heat kernel coefficient as well as the exact heat kernel diagonals on two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.

math.AP↗

arXiv2005

Non-Laplace type Operators on Manifolds with Boundary

We study second-order elliptic partial differential operators acting on sections of vector bundles over a compact manifold with boundary with a non-scalar positive definite leading symbol. Such operators, called non-Laplace type operators, appear, in particular, in gauge field theories, string theory as well as models of non-commutative gravity theories, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays the role of a ``non-commutative'' metric. It is well known that there is a small-time asymptotic expansion of the trace of the corresponding heat kernel in half-integer powers of time. We initiate the development of a systematic approach for the explicit calculation of these coefficients, construct the corresponding parametrix of the heat equation and compute explicitly the first two heat trace coefficients.

math-ph↗

arXiv2005

Dirac Operator in Matrix Geometry

We review the construction of the Dirac operator and its properties in Riemannian geometry and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein-Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a ``non-commutative'' metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, that can be called ``non-commutative'' Dirac operator and non-commutative Laplace operator. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein-Hilbert action.

math-ph↗

arXiv2004

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.

hep-th↗

arXiv2003

Matrix General Relativity: A New Look at Old Problems

We develop a novel approach to gravity that we call `matrix general relativity' (MGR) or `gravitational chromodynamics' (GCD or GQCD for quantum version). Gravity is described in this approach not by one Riemannian metric (i.e. a symmetric two-tensor field) but by a multiplet of such fields, or by a matrix-valued symmetric two-tensor field that satisfies certain conditions. We define the matrix extensions of standard constructions of differential geometry including connections and curvatures, and finally, an invariant functional of the new field that reduces to the standard Einstein action functional in the commutative (diagonal) case. Our main idea is the analogy with Yang-Mills theory (QCD and Standard Model). We call the new degrees of freedom of gravity associated with the matrix structure `gravitational color' or simply `gravicolor' and introduce a new gauge symmetry associated with this degree of freedom. As in the Standard Model there are two possibilities. First of all, it is possible that at high energies (say at Planckian scale) this symmetry is exact (symmetric phase), but at low energies it is badly broken, so that one tensor field remains massless (and gives general relativity) and the other ones become massive with the masses of Planckian scale. Second possibilty is that the additional degrees of freedom of gravitational field are confined within the Planckian scale. What one sees at large distances are singlets (invariants) of the new gauge symmetry.

hep-th↗

arXiv2001

Heat Kernel Asymptotics of Operators with Non-Laplace Principal Part

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part $-\N^μ\N_μ$. Our objective is to obtain information on the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Green's function are constructed explicitly in the leading order. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators is constructed. Some particular cases are studied in more detail.

math-ph↗

arXiv1999

Foundational Problems in Quantum Gravity

Boundary conditions play a crucial role in the path-integral approach to quantum gravity and quantum cosmology, as well as in the current attempts to understand the one-loop semiclassical properties of quantum field theories. Within this framework, one is led to consider boundary conditions completely invariant under infinitesimal diffeomorphisms on metric perturbations. These are part of a general scheme, which can be developed for Maxwell theory, Yang-Mills Theory, Rarita-Schwinger fields and any other gauge theory. A general condition for strong ellipticity of the resulting field theory on manifolds with boundary is here proved, following recent work by the authors. The relevance for Euclidean quantum gravity is eventually discussed.

gr-qc↗

arXiv1998

Heat Kernel Asymptotics of the Gilkey-Smith Boundary Value Problem

The formulation of gauge theories on compact Riemannian manifolds with boundary leads to partial differential operators with Gilkey--Smith boundary conditions, whose peculiar property is the occurrence of both normal and tangential derivatives on the boundary. Unlike the standard Dirichlet or Neumann boundary conditions, this boundary-value problem is not automatically elliptic but becomes elliptic under certain conditions on the boundary operator. We study the Gilkey--Smith boundary-value problem for Laplace-type operators and find a simple criterion of ellipticity. The first non-trivial coefficient of the asymptotic expansion of the trace of the heat kernel is computed and the local leading asymptotics of the heat-kernel diagonal is also obtained. It is shown that, in the non-elliptic case, the heat-kernel diagonal is non-integrable near the boundary, which reflects the fact that the heat kernel is not of trace class. We apply this analysis to general linear bosonic gauge theories and find an explicit condition of ellipticity.

math-ph↗

arXiv1998

On Ellipticity and Gauge Invariance in Euclidean Quantum Gravity

Invariance principles determine many key properties in quantum field theory, including, in particular, the appropriate form of the boundary conditions. A crucial consistency check is the proof that the resulting boundary-value problem is strongly elliptic. In Euclidean quantum gravity, the appropriate principle seems to be the invariance of boundary conditions under infinitesimal diffeomorphisms on metric perturbations, and hence their BRST invariance. However, if the operator on metric perturbations is then chosen to be of Laplace type, the boundary-value problem for the quantized gravitational field fails to be strongly elliptic. A detailed proof is presented, and the corresponding open problems are discussed.

hep-th↗

arXiv1998

Gauge theories on manifolds with boundary

The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed. As a result, the previous work in the literature on heat-kernel asymptotics is shown to be a particular case of a more general structure. For a bosonic gauge theory on a compact Riemannian manifold with smooth boundary, the problem is studied of obtaining a gauge-field operator of Laplace type, jointly with local and gauge-invariant boundary conditions, which should lead to a strongly elliptic boundary-value problem. The scheme is extended to fermionic gauge theories by means of local and gauge-invariant projectors. After deriving a general condition for the validity of strong ellipticity for gauge theories, it is proved that for Euclidean Yang--Mills theory and Rarita--Schwinger fields all the above conditions can be satisfied. For Euclidean quantum gravity, however, this property no longer holds, i.e. the corresponding boundary-value problem is not strongly elliptic. Some non-standard local formulae for the leading asymptotics of the heat-kernel diagonal are also obtained. It is shown that, due to the absence of strong ellipticity, the heat-kernel diagonal is non-integrable near the boundary.

hep-th↗

arXiv1998

Green functions of higher-order differential operators

The Green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian manifold are investigated via the heat kernel methods. We study the resolvent of a special class of higher-order operators formed by the products of second-order operators of Laplace type defined with the help of a unique Riemannian metric but with different bundle connections and potential terms. The asymptotic expansion of the Green functions near the diagonal is studied in detail in any dimension. As a by-product a simple criterion for the validity of the Huygens principle is obtained. It is shown that all the singularities as well as the non-analytic regular parts of the Green functions of such high-order operators are expressed in terms of the usual heat kernel coefficients $a_k$ for a special Laplace type second-order operator.

hep-th↗

arXiv1998

Heat-Kernel Asymptotics with Generalized Boundary Conditions

The quantization of gauge fields and gravitation on manifolds with boundary makes it necessary to study boundary conditions which involve both normal and tangential derivatives of the quantized field. The resulting one-loop divergences can be studied by means of the asymptotic expansion of the heat kernel, and a particular case of their general structure is here analyzed in detail. The interior and boundary contributions to heat-kernel coefficients are written as linear combinations of all geometric invariants of the problem. The behaviour of the differential operator and of the heat kernel under conformal rescalings of the background metric leads to recurrence relations which, jointly with the boundary conditions, may determine these linear combinations. Remarkably, they are expressed in terms of universal functions, independent of the dimension of the background and invariant under conformal rescalings, and new geometric invariants contribute to heat-kernel asymptotics. Such technique is applied to the evaluation of the A(1) coefficient when the matrices occurring in the boundary operator commute with each other. Under these assumptions, the form of the A(3/2) and A(2) coefficients is obtained for the first time, and new equations among universal functions are derived. A generalized formula, relating asymptotic heat kernels with different boundary conditions, is also obtained.

hep-th↗

arXiv1997

A new explicit expression for the Korteweg-De Vries hierarchy

We derive an improved fully explicit expression for the right-hand sides of the matrix KdV hierarchy using the relation to the heat kernel of the one-dimensional Schrödinger operator. Our method of "matrix elements" produces, moreover, an explicit expression for the powers of a Schrödinger-like differential operator of any order.

solv-int↗

arXiv1997

Lack of strong ellipticity in Euclidean quantum gravity

Recent work in Euclidean quantum gravity has studied boundary conditions which are completely invariant under infinitesimal diffeomorphisms on metric perturbations. On using the de Donder gauge-averaging functional, this scheme leads to both normal and tangential derivatives in the boundary conditions. In the present paper, it is proved that the corresponding boundary value problem fails to be strongly elliptic. The result raises deep interpretative issues for Euclidean quantum gravity on manifolds with boundary.

hep-th↗