Once Before Time - Martin Bojowald [62]
Luckily, complete control is not always required. When we are interested only in, say, the large-scale behavior of cosmic expansion, crucial simplifications result. Once they are realized, the spectrum of all possible volumes becomes calculable concretely and in detail. Moreover, for every value of the volume there are only a few states—two, as it turns out (except for the vanishing volume of the singularity, granted a single unique state).
When I first investigated these relevant structures in loop quantum cosmology, I was puzzled by this doubling. But there is a clear and far-reaching explanation: A state is characterized not only by volume, but also by orientation; space and its inside-out replica are distinct from each other. In two dimensions, one can visualize this as a balloon, which can be blown up to a certain size in two ways: with or without being initially upended, turning its inside out. This is in agreement with the fact that vanishing volume has a unique state, for a space devoid of extension has no inside to be turned out. Mathematically, this degree of freedom is called the orientation of space.
15. The ladder of volume, changing when asymmetry is continuously switched on along the horizontal direction. On the left, the ladder is regular and easily computable; on the right, it looks more and more messy. The behavior of this level splitting is analogous to an effect in the energy spectra of atoms, whose levels change and split if new forces such as a magnetic one are switched on, breaking initially realized symmetries.
Even more complicated than the allowed volume values is their behavior in time, the dynamics: With the volume values we know the ladder a universe can grow on, but how precisely does it climb up? How fast must it expand, and can this be accelerated? Or, winding back time to approach the big bang singularity, how does the universe climb down to small sizes into its potential demise? Reaching the unique state of vanishing volume, do we touch down hard on the singular ground, as if using too short a ladder, which, if we lose the grip of the lowest rung, makes us fall abruptly into the abyss?
After I had analyzed the doubling of states due to orientation, I had already noticed its possible meaning for the dynamics of a universe. Instead of considering the set of all volumes as a twin pair of two unidirectional positive axes, each starting at the singularity, we can arrange them as a single axis made of positive and negative numbers, the singular zero now lying in the middle rather than at a boundary. Orientation is indeed described by a sign factor, determining whether a state is counted positively or negatively. We have not two separate ladders, but a single one stretching right through the singularity.
It quickly turned out that the temporal evolution of the universe does indeed take place on this long ladder. With the symmetrical conditions of the large-scale universe, I finally managed to simplify the dynamic equation—provided by Thiemann for general states—sufficiently to find solutions. It has a compact form, of a difference equation, by now used in numerous scientific works:
Here, ψn symbolizes the state of the universe—the wave function—at different values n representing the ladder rungs: the volume values, together with a sign for orientation to indicate the part of the ladder we are on. Moreover, there are coefficients C1, C0, and C2, whose form ensures the correct correspondence to Einstein’s equation, and Ĥ describes the matter content of the universe. The equation connects states at different sizes with each other, telling us about the quantum mechanical growth of the universe: One can successively solve the equation for ψn +1, the value at the n +1st rung, by relating it to the values at the preceding