Tadashi Yano

A high-speed method for eigenvalue problems. IV. Sturm-Liouville-type differential equations

We present a new version MEV4 of the program package MEV3 by Milnes method generalized for the eigenvalue problem of the linear differential equation of the Sturm-Liouville-type. This version MEV4 enables one to calculate eigenvalues for various types of boundary conditions.

A canonical formalism for a higher-curvature gravity

Following the method of Buchbinder and Lyakhovich, we carry out a canonical formalism for a higher-curvature gravity theory in which the Lagrangian density is given in terms of a function of the scalar curvature as , where f is a nonlinear differentiable function of . We comment on the physical significance of the metric taking the Robertson-Walker metric as an example.

Brown—Feynman reduction of one-loop Feynman diagrams to scalar integrals with orthonormal basis tensors

Abstract Brown—Feynman reduction formulas for one-loop amplitudes in gauge theories are presented in the form of decomposition in terms of orthonormal basis tensors. Use of the orthonormal basis tensors greatly simplifies algebraic manipulation on computers, and allows us to write down the full formulas explicitly.

A canonical formalism of f(R)-type gravity in terms of Lie derivatives

We propose a canonical formalism of f(R)-type gravity using a set of phase variables which is partly different from that used in the formalism of Buchbinder and Lyakhovich (BL). The new coordinates corresponding to the time derivatives of the metric are taken to be the Lie derivatives of the metric, which is the same as in BL. The momenta canonically conjugate to the new coordinates and Hamiltonian density are defined similarly to the formalism of Ostrogradski. It is shown that, in our formalism, the Hamiltonian is invariant under the change of the original coordinates, which is not the case in the formalism of BL.A canonical formalism of f(R)-type gravity is proposed, resolving the problem in the formalism of Buchbinder and Lyakhovich(BL). The new coordinates corresponding to the time derivatives of the metric are taken to be its Lie derivatives which is the same as in BL. The momenta canonically conjugate to them and Hamiltonian density are defined similarly to the formalism of Ostrogradski. It is shown that our method surely resolves the problem of BL.

Can Milne's method work well for the Coulomb-like potentials ?

We extend the improved Milnes (Milne-spline) method for obtaining eigenvalues and eigenfunctions to the cases of long-range and singular potentials, for which we have conjectured that it is difficult to apply the method. Contrary to our conjecture it turned out that the method is valid also for Coulomb potential and repulsive 1/xn(n = 2,3,...) type potential. Further we applied the method for two cases, for which the solutions are not known, in order to investigate the stability of the multi-dimensional universe. It has been shown that the extra-dimensional (internal) space of our universe is not stable in classical Einstein gravity as well as canonically quantized one. Two possibilities for stabilization were investigated: (i) noncanonically quantized Einstein gravity and (ii) canonically quantized higher curvature gravity. It has been suggested that the space is stable by qualitative and approximate methods. Exact analytical treatment is very difficult, so that numerical investigation is highly desirable. Numerical investigation shows that the space is stable with sufficient reliability.

A high-speed method for eigenvalue problems. II Calculation of the eigenfunction in Milne's method

Abstract A new version of the program MEIGEN is presented for the eigenvalue problem of Sturm-Liouville-type linear equations in Milnes method. Use of the spline function and the WKB approximation provide a high-speed method for calculating eigenvalues and eigenfunctions avoiding divergence problems.

A high-speed method for solving eigenvalue problems: use of the spline function in Milne's method

Abstract Use of the spline function in Milnes method for eigenvalue problems of Sturm-Liouville-type linear equations is found to provide a high-speed method for calculating eigenvalues.

A high-speed method for eigenvalue problems. III: Case of unsymmetrical potentials in Milne's method

Abstract A new version of MEV2 by Milnes method is presented for eigenvalue problems of linear differential equations of Sturm-Liouville-type. This version MEV3 enables one to calculate eigenvalues and eigenfunctions, for instance, for the case of unsymmetrical potentials in Schrodinger equations.

Semiclassical approach to stability of the extra-dimensional spaces in higher-curvature gravity theories

The stability of the extra-dimensional spaces in higher-curvature gravity theories is investigated using the semiclassical approximation to the Wheeler-DeWitt equation. It is shown that, if there exists only one internal space, the space is stable under a certain condition.

A possible way of stabilizing a higher-dimensional universe

SummaryWe examine the stability of the extra-dimensional compactified spaces necessary for,e.g., the string theory. The analysis of a simplified model indicates that these spaces collapse classically and quantum-mechanically. To resolve this serious problem, we propose a new Planckian-scale physics which suggests the non-commutative nature of the space-time.