404 research outputs found

### On the Green functions of gravitational radiation theory

Previous work in the literature has studied gravitational radiation in
black-hole collisions at the speed of light. In particular, it had been proved
that the perturbative field equations may all be reduced to equations in only
two independent variables, by virtue of a conformal symmetry at each order in
perturbation theory. The Green function for the perturbative field equations is
here analyzed by studying the corresponding second-order hyperbolic operator
with variable coefficients, instead of using the reduction method from the
retarded flat-space Green function in four dimensions. After reduction to
canonical form of this hyperbolic operator, the integral representation of the
solution in terms of the Riemann function is obtained. The Riemann function
solves a characteristic initial-value problem for which analytic formulae
leading to the numerical solution are derived.Comment: 15 pages, plain Tex. A misprint on the right-hand side of Eqs. (3.5)
and (3.6) has been amende

### 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.Comment: 17 pages, plain Tex. The revised version is much longer, and section
5 is entirely ne

### A parametrix for quantum gravity?

In the sixties, DeWitt discovered that the advanced and retarded Green
functions of the wave operator on metric perturbations in the de Donder gauge
make it possible to define classical Poisson brackets on the space of
functionals that are invariant under the action of the full diffeomorphism
group of spacetime. He therefore tried to exploit this property to define
invariant commutators for the quantized gravitational field, but the operator
counterpart of such classical Poisson brackets turned out to be a hard task. On
the other hand, the mathematical literature studies often an approximate
inverse, the parametrix, which is, strictly, a distribution. We here suggest
that such a construction might be exploited in canonical quantum gravity. We
begin with the simplest case, i.e. fundamental solution and parametrix for the
linear, scalar wave operator; the next step are tensor wave equations, again
for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear
Einstein equations are studied, relying upon the well-established
Choquet-Bruhat construction, according to which the fifth derivatives of
solutions of a nonlinear hyperbolic system solve a linear hyperbolic system.
The latter is solved by means of Kirchhoff-type formulas, while the former
fifth-order equations can be solved by means of well-established parametrix
techniques for elliptic operators. But then the metric components that solve
the vacuum Einstein equations can be obtained by convolution of such a
parametrix with Kirchhoff-type formulas. Some basic functional equations for
the parametrix are also obtained, that help in studying classical and quantum
version of the Jacobi identity.Comment: 27 page

### Quantization of Field Theories in the Presence of Boundaries

This paper reviews the progress made over the last five years in studying
boundary conditions and semiclassical properties of quantum fields about
4-real-dimensional Riemannian backgrounds. For massless spin-${1\over 2}$
fields one has a choice of spectral or supersymmetric boundary conditions, and
the corresponding conformal anomalies have been evaluated by using
zeta-function regularization. For Euclidean Maxwell theory in vacuum, the
mode-by-mode analysis of BRST-covariant Faddeev-Popov amplitudes has been
performed for relativistic and non-relativistic gauge conditions. For massless
spin-${3\over 2}$ fields, the contribution of physical degrees of freedom to
one-loop amplitudes, and the 2-spinor analysis of Dirac and Rarita-Schwinger
potentials, have been obtained. In linearized gravity, gauge modes and ghost
modes in the de Donder gauge have been studied in detail. This program may lead
to a deeper understanding of different quantization techniques for gauge fields
and gravitation, to a new vision of gauge invariance, and to new points of view
in twistor theory.Comment: 11 pages, plain-tex, to appear in Proceedings of the XI Italian
Conference on General Relativity and Gravitational Physics, Trieste (Italy),
September 26-30, 1994; 1995 World Scientific Publishing Compan

### 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.Comment: 31 pages, plain Tex. Paper prepared for the Fourth Workshop on
Quantum Field Theory under the Influence of External Conditions, Leipzig,
September 199

### 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.Comment: 6 pages, Latex, invited talk given at the Tomsk Conference: Quantum
Field Theory and Gravity (July-August 1997

### Local Supersymmetry in One-Loop Quantum Cosmology

The contribution of physical degrees of freedom to the one-loop amplitudes of
Euclidean supergravity is here evaluated in the case of flat Euclidean
backgrounds bounded by a three-sphere, recently considered in perturbative
quantum cosmology. The physical degrees of freedom (denoted by PDF) are picked
out by imposing the supersymmetry constraints and choosing a gauge condition.
Remarkably, for the massless gravitino field the PDF method and local boundary
conditions lead to a result for the trace anomaly which is equal to the PDF
value one obtains using spectral boundary conditions on a 3-sphere.Comment: 23 pages, plain-tex, recently appearing in Int. J. Mod. Phys. D,
volume 3, pages 593-607, September 1994. The Author would like to apologize
for the delay in circulating the paper, due to technical problems now fixe

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