International Journal of Modern Physics A

If one assumes the validity of conventional quantum field theory in the vicinity of the horizon of a black hole, one does not find a quantum-mechanical description of the entire black hole that even remotely resembles that of conventional forms of matter; in contrast with matter made out of ordinary particles one finds that, even if embedded in a finite volume, a black hole would be predicted to have a strictly continuous spectrum.

Dissatisfied with such a result, which indeed hinges on assumptions concerning the horizon that may well be wrong, various investigators have now tried to formulate alternative approaches to the problem of “quantizing” the black hole. We here review the approach based on the assumption of quantum-mechanical purity and unitarity as a starting point, as has been advocated by the present author for some time, concentrating on the physics of the states that should live on a black hole horizon. The approach is shown to be powerful in producing not only promising models for the quantum black hole, but also new insights concerning the dynamics of physical degrees of freedom in ordinary flat space–time.

In this paper we discuss the connection on a space of N = 2 TCFT's that appears in the context of background (in)dependence. We formulate a family of target space field theories with a similar connection on it. Each theory is a gauge theory (with the gauge group being Diff in the case of three-fold). It describes deformations of Kähler structures much like Kodaira Spencer theory describes deformations of the complex structures. It is manifestly background-independent. It appears to be a target space field theory for supersymmetric quantum mechanics.

Some new modified Runge–Kutta methods with minimal phase lag are developed for the numerical solution of the eigenvalue Schrödinger equation and related problems with oscillating solutions. These methods are based on the very well-known Runge–Kutta method of order 4. For the numerical solution of the eigenvalue Schrödinger equation, we investigate two cases: (i) the specific case in which the potential V(x) is an even function with respect to x; it is assumed, also, that the wave functions tend to zero for x → ±∞; (ii) the general case for the well-known cases of the Morse potential and Woods–Saxon or optical potential. Also, we have applied the new methods to some well-known problems with oscillatory solutions. Numerical and theoretical results show that this new approach is more efficient than the well-known classical fourth order Runge–Kutta method and the Numerov method.

We analyze the possible soft breaking of (N = 2)-supersymmetric Yang–Mills theory with and without matter flavor preserving the analyticity properties of the Seiberg–Witten solution. For a small supersymmetry-breaking parameter with respect to the dynamical scale of the theory we obtain an exact expression for the effective potential. We describe in detail the onset of the confinement transition and some of the patterns of chiral symmetry breaking. If we extrapolate the results to the limit where supersymmetry decouples, we obtain hints indicating that perhaps a description of the QCD vacuum will require the use of Lagrangians containing simultaneously mutually nonlocal degrees of freedom (monopoles and dyons).