• Choose a category
• All
• Classic-Fiction

By Root 644 0

development was complete in the sense that all four types were involved, which was extremely important for its progress. This was its great fortune; its misfortune was that all the types were independently present in different people. The rare complete physicist would embody all four types at once, making it possible to move ahead all the way from original ideas to sharply formulated results. Such a person would be the natural leader of a field, nurturing it from its inception and guiding it to maturity. When the types are all represented by different people, it is not intellectual qualities but secondary skills such as political ones or simply seniority that determine leadership.5 One can easily imagine that this can be harmful to the development of an emerging field and its unity. Different personalities push in different directions; well-intentioned checks can too easily be dismissed as self-righteous criticism (or self-righteous criticism camouflaged as well-intentioned checks). This is a difficult situation, full of tensions that the initiators of loop quantum gravity were not always able to resolve in an optimal way.

MORE DETAILS: FENCES, FIBERS, FILIGREE

A fence is to be painted in two different colors separated by a horizontal borderline. To do so, we need to compare positions on its different boards, normally by moving horizontally from one board to the next taking the ground as the base. Now imagine a fence whose boards are infinitely long, without a bottom where they would be affixed to the base. We can still easily say what it means to move “vertically”; we just move along the boards. But how do we determine the “horizontal” direction without any orientation from the base to refer to?

In such a situation, we need an extra structure, such as a collection of nonvertical arrows as road signs telling us what we should mean by the horizontal direction. Following such an arrow we would move uniquely from one point on a board to some point on the next board, just as we were able to do by referring to the conventional horizontal direction. Only such a collection of arrows defining horizontality tells us how the different boards are related to each other; it may be called a connection.

Modern physics and mathematics is full of fences. This is not meant merely in the sense of researchers keeping competitors out of their own turf, but in an abstract sense: The fence is a fiber bundle whose boards are called fibers, the ground is the base manifold of the bundle, and we compare different fibers by means of a connection. Moving along the horizontal direction as determined by the connection is called parallel transport or holonomy. A crucial property of general fences in mathematics is that in returning to the same board while moving horizontally along a circular fence, one need not end up at the same position on the board. (In situations where one does always end up at the same point on the fiber when moving along a circle in the base, the connection is called flat.) Depending on how horizontality is determined by the connection, there may be a displacement, which is called a Wilson loop.6

General relativity has a fence whose base consists of all of space-time and whose fibers are made up of all possible rotation angles (including changes of velocities, since we are talking about angles in space-time). Changing position along the fiber means that one would remain at the same space-time point but change one’s orientation, just as we are forced to do when moving on a curved sphere. As this example illustrates, Wilson loops on this fence are a measure of curvature, and thus of the gravitational force. In 1986, Abhay Ashtekar built a new fence together with a special connection in general relativity, based on earlier work by Amitabha Sen. Here, the base is not space-time but only space, and the fibers are made only from spatial rotations. There is thus a reduction in the amount of information, since time is initially disregarded. But it turns out that all information about the space-times of general relativity can be recovered