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

An application of Green-function methods to 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: 18 pages, Revtex4. Submitted to Lecture Notes of S.I.M., volume edited by D. Cocolicchio and S. Dragomir, with kind permission by IOP to use material in Ref. [12]. arXiv admin note: substantial text overlap with arXiv:gr-qc/010107

SO(10) GUT Models and Cosmology

$SO(10)$ grand unified models have an intermediate symmetry group in between $SO(10)$ and $SU(3)_{C} \otimes SU(2)_{L} \otimes U(1)_{Y}$. Hence they lead to a prediction for proton lifetime in agreement with the experimental lower limit. This paper reviews the recent work on the tree-level potential and the one-loop effective potential for such models, with application to inflationary cosmology. The open problems are the use of the most general form of tree-level potential for $SO(10)$ models in the reheating stage of the early universe, and the analysis of non-local effects in the semiclassical field equations for such models in Friedmann-Robertson-Walker backgrounds.Comment: 7 pages, Latex, talk prepared for the Second International Sakharov Conference on Physics, Moscow (May 1996

A new application of non-canonical maps in quantum mechanics

A proof is given that an invertible and a unitary operator can be used to reproduce the effect of a q-deformed commutator of annihilation and creation operators. In other words, the original annihilation and creation operators are mapped into new operators, not conjugate to each other, whose standard commutator equals the identity plus a correction proportional to the original number operator. The consistency condition for the existence of this new set of operators is derived, by exploiting the Stone theorem on 1-parameter unitary groups. The above scheme leads to modified equations of motion which do not preserve the properties of the original first-order set for annihilation and creation operators. Their relation with commutation relations is also studied.Comment: 13 pages, plain Tex. In the revised version, section 3 contains new calculation

Quantum cosmology from three different perspectives

Our review is devoted to three promising research lines in quantum cosmology and the physics of the early universe. The nonperturbative renormalization programme is making encouraging progress that we here assess from the point of view of cosmological applications: Lagrangian and Hamiltonian form of pure gravity with variable G and Lambda; power-law inflation for pure gravity; an accelerating universe without dark energy. In perturbative quantum cosmology, on the other hand, diffeomorphism-invariant boundary conditions lead naturally to a singularity-free one-loop wave function of the universe. Last, but not least, in the braneworld picture one discovers the novel concept of cosmological wave function of the bulk space-time. Its impact on quantum cosmology and singularity avoidance is still, to a large extent, unexplored.Comment: 8 pages, Latex file, invited brief review prepared for the COT5 Session of the MG11 Conference in Berli

Boundary-Value Problems for the Squared Laplace Operator

The squared Laplace operator acting on symmetric rank-two tensor fields is studied on a (flat) Riemannian manifold with smooth boundary. Symmetry of this fourth-order elliptic operator is obtained provided that such tensor fields and their first (or second) normal derivatives are set to zero at the boundary. Strong ellipticity of the resulting boundary-value problems is also proved. Mixed boundary conditions are eventually studied which involve complementary projectors and tangential differential operators. In such a case, strong ellipticity is guaranteed if a pair of matrices are non-degenerate. These results find application to the analysis of quantum field theories on manifolds with boundary.Comment: 22 pages, plain Tex. In the revised version, section 5 has been amende