Once Before Time - Martin Bojowald [50]
These are the Wilson loops used by Rovelli and Smolin in their conception of loop quantum gravity. Ashtekar’s work initially remained classical; there were no fluctuations and no quantum uncertainty, and Wilson loops as well as the areas of surfaces could take on arbitrarily precise values. Rovelli and Smolin realized that Ashtekar’s connection can, much more easily than the traditional one used in general relativity, be described by a wave function, thus quantizing it. Here, one can directly combine the central concepts of general relativity and quantum theory: space-time and the wave function. A quantum theoretical description of the interplay of space and time now requires some kind of “wave function of sizes” rather than position as in quantum mechanics.
Not only, then, are the position and velocity of matter particles such as an electron imprecise, but so are the geometrical dimensions of the stage on which an electron moves. It is no longer possible to measure areas and curvature independently at the same time; to measure the area of a surface, we must align its boundaries with measuring rods and compare the marks. A very precise comparison requires strong resolutions, such as may be provided by a powerful microscope, to ensure that the corners of the surface indeed line up with the marks on the rod. Just as for position measurements in quantum mechanics, a microscope uses signals that must interact with the surface, making its edges move slightly. The stronger the resolution, the more rapid the movement. A measured surface seems to have fluctuating expansion or contraction rates, and expansion or contraction, as in cosmology, is related to the curvature—changing velocities due to position changes—of its surrounding space-time. Precise area measurements lead to more fluctuation and less precise knowledge of curvature, which is the uncertainty principle of loop quantum gravity. (Analogous uncertainty principles could be formulated for distances or volumes instead of areas; the latter are appealing because they turn out to lead to more elegant mathematics.)
Ashtekar’s reformulation combined with Rovelli and Smolin’s quantization has the flavor of unification, although not of the form seen in string theory. It is, as it were, a mathematical rather than a physical unification: Different forces are not reduced to a single principle, but the mathematical description of the forces is made uniform. The additional forces—the electromagnetic, the strong, and the weak—are in fact based on similar connections on fences. The fibers are not orientations or angles in physical space as in Ashtekar’s formulation of general relativity, but angles in abstract mathematical spaces. For the theories, this difference plays hardly any role, and thus a multitude of methods and insights, gained for instance in quantum electrodynamics—the quantum theory of the electromagnetic field—can be applied also to gravity. But there are, to be sure, important special properties of relativity that required a wider advancement of mathematics before a complete quantum theory of gravity could be developed.
The initial mathematical work of the early 1990s had provided suitable quantization rules of loop quantum gravity which Ashtekar’s areas and angles must obey, including their quantum mechanical uncertainty. At that time, it remained unknown whether there could be other forms of such rules, possibly leading to different quantum theories. In other words, the uniqueness question here, in contrast to string theory, was not yet clarified. Other quantization rules would imply different predictions. In principle one could select the correct ones by comparison with observations, but those do not yet exist for quantum gravity. Threatened by the possible existence of widely differing quantizations, the usefulness and predictivity of loop quantum gravity could seriously be