arXiv subjects

Giampiero Esposito

Publications and source records attributed to Giampiero Esposito.

At least 181 records · Page 10Linked to original sources

arXiv1999

Exact Solutions of the Schrodinger Equation with Inverse-Power Potential in Two Dimensions

The Schrodinger equation for stationary states is studied in a central potential V(r) proportional to the inverse power of r of degree beta in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrodinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case beta=4 is elucidated. In general, whenever the parameter beta is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.

quant-ph↗

arXiv1999

Nonlocal Ghost Formalism for Quantum Gravity

The quantum theory of a free particle on a portion of two-dimensional Euclidean space bounded by a circle and subject to non-local boundary conditions gives rise to bulk and surface states. Starting from this well known property, a counterpart for gravity is here considered. In particular, if spatial components of metric perturbations are set to zero at the boundary, invariance of the full set of boundary conditions under infinitesimal diffeomorphisms is compatible with non-local boundary conditions on normal components of metric perturbations if and only if both the gauge-field operator and the ghost operator are pseudo-differential operators in one-loop quantum gravity.

hep-th↗

arXiv1999

New Supplementary Conditions for a Non-Linear Field Theory: General Relativity

The Einstein theory of general relativity provides a peculiar example of classical field theory ruled by non-linear partial differential equations. A number of supplementary conditions (more frequently called gauge conditions) have also been considered in the literature. In the present paper, starting from the de Donder gauge, which is not conformally invariant but is the gravitational counterpart of the Lorenz gauge, we consider, led by geometric structures on vector bundles, a new family of gauges in general relativity, which involve fifth-order covariant derivatives of metric perturbations. A review of recent results by the authors is presented: restrictions on the general form of the metric on the vector bundle of symmetric rank-two tensor fields over space-time; admissibility of such gauges in the case of linearized theory about flat Euclidean space; generalization to a suitable class of curved Riemannian backgrounds, by solving an integral equation. Eventually, the applications to Euclidean quantum gravity are discussed.

gr-qc↗

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

Casimir Effect in Problems with Spherical Symmetry: New Perspectives

Since the Maxwell theory of electromagnetic phenomena is a gauge theory, it is quite important to evaluate the zero-point energy of the quantized electromagnetic field by a careful assignment of boundary conditions on the potential and on the ghost fields. Recent work by the authors has shown that, for a perfectly conducting spherical shell, it is precisely the contribution of longitudinal and normal modes of the potential which enables one to reproduce the result first due to Boyer. This is obtained provided that one works with the Lorenz gauge-averaging functional, and with the help of the Feynman choice for a dimensionless gauge parameter. For arbitrary values of the gauge parameter, however, covariant and non-covariant gauges lead to an entangled system of three eigenvalue equations. Such a problem is crucial both for the foundations and for the applications of quantum field theory.

hep-th↗

arXiv1998

Non-Local Boundary Conditions in Euclidean Quantum Gravity

Non-local boundary conditions for Euclidean quantum gravity are proposed, consisting of an integro-differential boundary operator acting on metric perturbations. In this case, the operator P on metric perturbations is of Laplace type, subject to non-local boundary conditions; by contrast, its adjoint is the sum of a Laplacian and of a singular Green operator, subject to local boundary conditions. Self-adjointness of the boundary-value problem is correctly formulated by looking at Dirichlet-type and Neumann-type realizations of the operator P, following recent results in the literature. The set of non-local boundary conditions for perturbative modes of the gravitational field is written in general form on the Euclidean four-ball. For a particular choice of the non-local boundary operator, explicit formulae for the boundary-value problem are obtained in terms of a finite number of unknown functions, but subject to some consistency conditions. Among the related issues, the problem arises of whether non-local symmetries exist in Euclidean quantum gravity.

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

Complex Parameters in Quantum Mechanics

The Schrodinger equation for stationary states in a central potential is studied in an arbitrary number of spatial dimensions, say q. After transformation into an equivalent equation, where the coefficient of the first derivative vanishes, it is shown that in such equation the coefficient of the second inverse power of r is an even function of a parameter, say lambda, depending on a linear combination of q and of the angular momentum quantum number, say l. Thus, the case of complex values of lambda, which is useful in scattering theory, involves, in general, both a complex value of the parameter originally viewed as the spatial dimension and complex values of the angular momentum quantum number. The paper ends with a proof of the Levinson theorem in an arbitrary number of spatial dimensions, when the potential includes a non-local term which might be useful to understand the interaction between two nucleons.

quant-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

Scattering from Singular Potentials in Quantum Mechanics

In non-relativistic quantum mechanics, singular potentials in problems with spherical symmetry lead to a Schrodinger equation for stationary states with non-Fuchsian singularities both as r tends to zero and as r tends to infinity. In the sixties, an analytic approach was developed for the investigation of scattering from such potentials, with emphasis on the polydromy of the wave function in the r variable. The present paper extends those early results to an arbitrary number of spatial dimensions. The Hill-type equation which leads, in principle, to the evaluation of the polydromy parameter, is obtained from the Hill equation for a two-dimensional problem by means of a simple change of variables. The asymptotic forms of the wave function as r tends to zero and as r tends to infinity are also derived. The Darboux technique of intertwining operators is then applied to obtain an algorithm that makes it possible to solve the Schrodinger equation with a singular potential containing many negative powers of r, if the exact solution with even just one term is already known.

hep-th↗

arXiv1998

New Results in Heat-Kernel Asymptotics on Manifolds with Boundary

A review is presented of some recent progress in spectral geometry on manifolds with boundary: local boundary-value problems where the boundary operator includes the effect of tangential derivatives; application of conformal variations and other functorial methods to the evaluation of heat-kernel coefficients; conditions for strong ellipticity of the boundary-value problem; fourth-order operators on manifolds with boundary; non-local boundary conditions in Euclidean quantum gravity. Many deep developments in physics and mathematics are therefore in sight.

hep-th↗

arXiv1998

On the Zero-Point Energy of a Conducting Spherical Shell

The zero-point energy of a conducting spherical shell is evaluated by imposing boundary conditions on the potential, and on the ghost fields. The scheme requires that temporal and tangential components of perturbations of the potential should vanish at the boundary, jointly with the gauge-averaging functional, first chosen of the Lorenz type. Gauge invariance of such boundary conditions is then obtained provided that the ghost fields vanish at the boundary. Normal and longitudinal modes of the potential obey an entangled system of eigenvalue equations, whose solution is a linear combination of Bessel functions under the above assumptions, and with the help of the Feynman choice for a dimensionless gauge parameter. Interestingly, ghost modes cancel exactly the contribution to the Casimir energy resulting from transverse and temporal modes of the potential, jointly with the decoupled normal mode of the potential. Moreover, normal and longitudinal components of the potential for the interior and the exterior problem give a result in complete agreement with the one first found by Boyer, who studied instead boundary conditions involving TE and TM modes of the electromagnetic field. The coupled eigenvalue equations for perturbative modes of the potential are also analyzed in the axial gauge, and for arbitrary values of the gauge parameter. The set of modes which contribute to the Casimir energy is then drastically changed, and comparison with the case of a flat boundary sheds some light on the key features of the Casimir energy in non-covariant gauges.

hep-th↗

arXiv1998

Two-boundary problems in Euclidean quantum gravity

Recent work in the literature has studied a new set of local boundary conditions for the quantized gravitational field, where the spatial components of metric perturbations, and ghost modes, are subject to Robin boundary conditions, whereas normal components of metric perturbations obey Dirichlet boundary conditions. Such boundary conditions are here applied to evaluate the one-loop divergence on a portion of flat Euclidean four-space bounded by two concentric three-spheres.

gr-qc↗

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↗

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↗

arXiv1997

Spectral Geometry and Quantum Gravity

Recent progress in quantum field theory and quantum gravity relies on mixed boundary conditions involving both normal and tangential derivatives of the quantized field. In particular, the occurrence of tangential derivatives in the boundary operator makes it possible to build a large number of new local invariants. The integration of linear combinations of such invariants of the orthogonal group yields the boundary contribution to the asymptotic expansion of the integrated heat-kernel. This can be used, in turn, to study the one-loop semiclassical approximation. The coefficients of linear combination are now being computed for the first time. They are universal functions, in that are functions of position on the boundary not affected by conformal rescalings of the background metric, invariant in form and independent of the dimension of the background Riemannian manifold. In Euclidean quantum gravity, the problem arises of studying infinitely many universal functions.

hep-th↗

arXiv1997

New Results in One-Loop Quantum Cosmology

A crucial problem in quantum cosmology is a careful analysis of the one-loop semiclassical approximation for the wave function of the universe, after an appropriate choice of mixed boundary conditions. The results for Euclidean quantum gravity in four dimensions are here presented, when linear covariant gauges are implemented by means of the Faddeev-Popov formalism. On using zeta-function regularization and a mode-by-mode analysis, one finds a result for the one-loop divergence which agrees with the Schwinger-DeWitt method only after taking into account the non-trivial effect of gauge and ghost modes. For the gravitational field, however, the geometric form of heat-kernel asymptotics with boundary conditions involving tangential derivatives of metric perturbations is still unknown. Moreover, boundary effects are found to be responsible for the lack of one-loop finiteness of simple supergravity, when only one bounding three-surface occurs. This work raises deep interpretative issues about the admissible backgrounds and about quantization techniques in quantum cosmology.

hep-th↗

arXiv1997

Green Functions for Classical Euclidean Maxwell Theory

Recent work on the quantization of Maxwell theory has used a non-covariant class of gauge-averaging functionals which include explicitly the effects of the extrinsic-curvature tensor of the boundary, or covariant gauges which, unlike the Lorentz case, are invariant under conformal rescalings of the background four-metric. This paper studies in detail the admissibility of such gauges at the classical level. It is proved that Euclidean Green functions of a second- or fourth-order operator exist which ensure the fulfillment of such gauges at the classical level, i.e. on a portion of flat Euclidean four-space bounded by three-dimensional surfaces. The admissibility of the axial and Coulomb gauges is also proved.

hep-th↗