Once Before Time - Martin Bojowald [144]
Theoretical physics is a long way from infiltrating the stock markets, as seductive as such an enlargement of research funds would be. This is a consequence not least of the second part of our condition for an explicit applicability, namely, the strong control over the mathematical evaluation of a possibly unique wave function. Even if one could prove the uniqueness mathematically, a concrete computation would be too complex to provide predictions of the slightest use in everyday life. Large-scale predictions of cosmological interest would, however, be conceivable, and thus the uniqueness question is of importance. Intellectually, this outlook represents an opportunity just as enticing for an understanding of the universe. But even if we are concerned only with the clarification of uniqueness rather than its greedy exploitation, many questions remain open once we leave the range of the simplest models.
To describe the real world, many extensions of the currently understood models must be undertaken. Those models are isotropic—they look the same in all spatial directions—in contrast to the real world. Isotropy, realized around each point in space, comes combined with homogeneity. Such a universe looks the same at each place, clearly different from the real world. In loop quantum cosmology, one can formulate models including anisotropy and inhomogeneity. One then has to deal with a huge number of equations in which not only are changes of the size of the universe described in successive time steps, but also simultaneous spatial changes. When completely formulated even in compact notation, such equations occupy the space of several pages, and computers of currently available technologies would fail at a numerical solution.
In such situations, one must rely on abstract investigation independent of explicit solutions, something not unusual in theoretical physics and mathematics. But the question of uniqueness of solutions for such systems of equations remains complicated and is, unfortunately, far from being clarified. Certainly the decoupling of singular states still implies constraints on the wave function; but it is not guaranteed that this suffices for uniqueness, or perhaps that it may not be, as with DeWitt’s initial attempt, too strong a restriction for the wave function’s own good.
Of at least some help is the supporting fact that the decoupling of singular states, as well as the number of constraints on the wave function, is independent of the precise form of matter in the universe. These are, instead, pure effects of space-time geometry. A uniqueness analysis would not be impeded by open questions as to the exact material content of the universe, such as dark energy or the form of matter at the high energies of the big bang. To be sure, the exact form of the wave function does certainly depend on matter, causing the evolution of the quantum universe just as it does for classical space-time. But whether or not one has a unique solution is unaffected.
An interesting indication of the soundness of dynamic initial conditions exists in the form of a statement of consistency between cosmology and black holes. Black holes, after all, host a singularity, too, and they are described in quantum gravity by a wave function. As in cosmology, singular states decouple from the rest in the temporal evolution and so imply constraints. These conditions can be evaluated in the simplest model of a black hole, surrounded by no matter and not rotating—a black hole as it is described in general relativity by Schwarzschild’s solution. The exterior of such a black