Once Before Time - Martin Bojowald [53]
Not just solving these equations but even formulating them is difficult. For consistency, results obtained from the solutions must not depend on what time is used for the ordering. It may be natural to use the volume for arranging the spatial configurations of an expanding universe, but that is not the only possibility. Instead, we might use the temperature, or any other physical parameter with a sufficiently strong trend. If what we derive depended significantly on what quantity we choose to order events, no predictions could be made. This leads to an important principle, known as covariance in general relativity; it must also be obeyed by the atomic equations of quantum gravity. While no fully covariant formulation of loop quantum gravity has yet been achieved, Thiemann’s important work had provided some early hints as to how this may eventually be done.
In addition to the stubborn consistency issue, the equations are not uniquely specified; their final form remains unclear. Several of their general properties are known, based on the unique quantization rules for areas and curvature, which distinguish this type of equation from others in gravitational physics. But some of the terms in the dynamic equations are still affected by undetermined parameters. In contrast to string theory, the present status of knowledge does not at all indicate a unique dynamics even though the underlying quantum rules are unique. There is no choice as to what discrete values the steps on the ladder of volume must take; the growing universe can climb up on it in many different ways. Still, loop quantum gravity has a major advantage: With it one can unravel some properties of the universe at the big bang, perhaps not with exact solutions but at least with approximations and model systems. Thiemann’s original construction, though often modified since then7—an ongoing process that indicates the arduous development of quantum gravity—plays a decisive role in applications of the theory.
DISCRETE UNIVERSE: WEAVING THE STORY OF THE WORLD
Loops build space in a dense mesh. Even empty space is full of them; a count in just one cubic meter would result in a number with about a hundred decimal places. This mesh is ever-shifting, for the change of loops is what encodes time. Moreover, this is a fuzzy mesh, subject to quantum fluctuations. On a fundamental level, space is rather agitated. One might visualize it as a mass of fluttering heated air above a dark road exposed to the glaring sun of a hot summer day.
Space is commonly thought of as the stage on which matter moves or signals propagate. Now, all that does happen on a mesh of loops, but nothing jumps from loop to loop (for there is nothing between the loops). Rather, as the ever-fluttering loops change their connectivity, they carry matter or signals with them, moving them ahead one spatial atom at a time; everything is a-changing. In this way, matter can, as it were, diffuse through the mesh of loops and move around. When we move an arm, all its atoms or even elementary particles must move in concert. Similarly, when an elementary particle moves, all its spatial atoms flow in concert. Even though performing the resulting motions may seem trivial to us, describing them in exact mathematical detail is in both cases an enormously complex task.
In cosmology, the stage of change is the whole universe, whose volume in loop quantum gravity is subject to the same quantum theoretical uncertainty as is known for matter. On average, it expands uniformly; but if one could view this expansion at close range, one would notice tiny variations. In addition, quantum jumps occur just as they do in the energies of atoms and molecules: The