Once Before Time - Martin Bojowald [140]
ONE UNIVERSE—NONE UNIVERSE: THE BABY IN THE BATHTUB
How to interpret the wave function of the universe is already one question pertinent to quantum cosmology, but no other quantum system. Perhaps, then, it is possible to tackle the wave function’s uniqueness in a special way. Even quantum cosmology relies on the same type of mathematical equations as the rest of physics, with initial values at a fixed time to be selected for a unique solution. Whether the time selected is indeed the absolute beginning of the world, or merely the beginning of some phase under theoretical consideration, a decision is, apparently, always to be made.
The first proposal to deal with this problem in quantum cosmology was put forward by its founder, Bryce DeWitt, as early as 1967. Aiming to connect the question of uniqueness with the singularity problem, he postulated that the wave function of the universe should be zero for a space of vanishing volume. According to general relativity, the spatial volume is zero just at the singularity, and so DeWitt’s condition corresponds to an interpretation of the singularity as a beginning where one would, as elsewhere in physics, fix initial values.
But as a condition at the singularity, this procedure would be more powerful than using ordinary initial conditions: DeWitt here attempts to play the big problems of quantum cosmology—the classical singularity and uniqueness—against each other. At first, his condition in a sense eliminates the singularity in quantum cosmology. If the wave function vanishes there, the universe will, according to the interpretation of the wave function, never assume the state of the singularity. This would be a consequence of the condition chosen for the wave function, rather than a physical phenomenon such as a repulsive force in loop quantum cosmology. Still, if successfully implemented, it would have far-reaching importance. Further, independent implications, testable at least in principle, would result from the specific form of the wave function as the solution of a differential equation with DeWitt’s initial condition.
Uniqueness of the wave function would be just such a consequence, important enough to promote the singularity avoidance achieved by DeWitt’s condition to a wide and elegantly formulated principle. But there is a catch when different cosmological equations are considered. Indeed, DeWitt’s condition often implies a unique wave function, naturally able to describe the oneness of the universe. In most cases, however—especially more realistic ones of less symmetry than in exactly isotropic models—this unique wave function vanishes not just at the singularity but everywhere. Here, the baby is thrown out with the bathwater: Such a universe would avoid not only the singularity, but any geometrical state—it would not exist at all. The wave function, after all, gives the likelihood of measurement results—here, of the size of the universe; if the wave function is completely zero, there is no possibility of measurements, and thus no universe either.
This disastrous contradiction with most elementary observations quickly led to the downfall of DeWitt’s condition. Later, in 1991, Heinz-Dieter Conradi and Dieter Zeh made an attempt to prevent the failure of the condition based on postulated changes in the equations of the universe at small volume, modifications expected anyway from a general quantum theory of gravity in the form of quantum corrections. But without progress in the general development of quantum gravity, the situation turned out to be too complex to achieve the correct form by anything more than guessing. In the framework of loop quantum gravity, now available, we will shortly come back to this issue.
PHYSICAL PICTURES: NOTHINGNESS WITHOUT BORDERS
Two alternative approaches gained more popularity than DeWitt’s condition: the tunneling condition of Alex Vilenkin and the no-boundary condition of Jim Hartle and Stephen Hawking. Both conditions, as