One of the concepts in this summary of physics has not yet been described. It is such a basic assumption to it that it went without saying. But it may not be obvious except in hindsight. There is the idea that a solution to quantum mechanics, QM equations leads to a travel path that is a geodesic. Geodesics are a minimum action path. The unit of QM, planck's constant is also a unit of action. The solution of QM is for all practical purposes a solution of least action. Higher probabilities occur along least action trajectories or locations. With the idea of least action in both spacetime and quantum theory, it is assumed that a solution of one, QM, automatically meets the requirements for the other, General Relativity, GR.
The link between the two can be shown that an element of momentum on a closed space can be solved either of two ways, fitting it like Bohr's orbits on the surface, or use Schroedinger's equation. To fit a path on a surface one assumes it is only allowed to follow geodesic paths on the surface. If a closed geodesic path can be plotted, a corresponding Eigenvalue to the path length is obtained as a solution to Schroedinger's equation. A closed path is one that eventually returns to its starting point. A path that crosses itself or cycles many times before returning will have a low probability of existence since the negative value portions of the wave will probably destructively interfere with the positive. The ideal case is where the waves negative portion is as distant as possible from the positive.
There is a one to one correspondence between each eigenfunction solution and a closed geodesic path on the surface. A mathematical proof of this theory would lead one to conclude that GR geodesic paths are QM effects. That is why one of the first solutions to obtain is for a QM solution on a torus.
Last Updated on December 6, 2000 by Bob Rutkiewicz