Once Before Time - Martin Bojowald [143]
At this juncture the final fate of quantum gravity is to be decided: Will it constitute a complete and consistent theory extending general relativity; or will it remain singular, itself to be extended in some as yet unknown way?
Loop quantum cosmology here proceeds as follows, as I observed in 2001: First, it does allow the existence of a state with vanishing volume, where the classical theory would assume its singularity. It approaches the problem in an unbiased yet daring way; it does not, as attempted by DeWitt without success, pose the absence of a singular state as a condition from the outset. Then it lets the dynamic equations, used to describe how the wave function of the universe evolves, decide for themselves which role this state shall play: the apocalypse of a singularity, or merely a harmless transition point.
In simple systems containing the usual symmetry assumptions of cosmology—the cosmological principle—the mathematical equations lend themselves to relatively easy analysis. These are not differential but difference equations, as on this page, realizing a time that is discrete rather than continuous. While a differential equation at each place provides the direction to be followed by a solution curve, as in figure 3 or figure 4, a difference equation determines steps by which a solution has to change in a fixed time interval. The astounding result: The solutions of these equations in loop quantum cosmology remain entirely uninfluenced by the value of the wave function at the singular state. Unflinchingly, the wave function of the universe wends its way before and behind the big bang, without even taking notice of the potential singularity. In particular, the temporal evolution does not break down. It leaves the singularity isolated, standing still, separated from the world’s course of the wave function before and after it—or around it.
Decoupling the singularity from the evolution of the universe has a further consequence. The dynamics of loop quantum cosmology is provided by a collection of mathematical equations, one for each change of state when transiting from one discrete time step to the next. If one of the states—the singularity—is decoupled, one equation too many remains, unused for the evolution. It is to be solved nonetheless, and gives exactly the condition desired for a unique wave function.
In this way, initial conditions are dynamically imposed in loop quantum cosmology: They are not independent of dynamic equations—natural laws—but can be derived from them. Here we have the ultimate triumph of discrete time, for the decoupling of the singularity and the related constraints on the wave function would not occur if continuous time, as in old quantum cosmology, would allow one to reach arbitrarily closely to the singularity. Firmly tied to uniqueness is the final prevention of the singularity: Even if the wave function can tunnel through the mighty barrier of repulsive forces, this subversion does not cause the breakdown of the theory.
ONE WORLD?:
THE LAST IDEAL
What is the status of the wave function’s uniqueness in more complicated situations that do not exactly exhibit the cosmological symmetries? A completely general rule for the uniqueness of the wave function of the universe, together with a powerful mathematical method to compute it, would surely have numerous applications by virtue of the predictions it would make. In jest, the physicist Murray Gell-Mann (who won the 1969 Nobel Prize in Physics for his contributions to the quark model of particle physics) posed it thus in a question to Jim Hartle: “If you know the