Calabi-Yau Manifold

Apparent and Real Problems

There are many problems with Superstring theory as mentioned in a section about "Problems and Future Development" elsewhere within this website. Some of them are rather technical, the followings summarize a few of those conceptual ones.


• In spite of the many proposed tests as mentioned above, critic keeps maintaining that the Superstring theory is not testable. The reason could be the outlandish objects often associated with the tests. However, the anti-particle proposed by Dirac in 1928 is also bordering on fantasy before the positron had finally been detected in about 4 years. On the other hand the confirmation of the Superstring theory seems to take much longer (more than 20 years and counting), people just don't have the patience to wait that long (troubling either by psychological preconception or by monetary consideration).



• Another sore point about the Superstring theory is the richness of solutions. Early on in the investigation, researchers were looking for an unique universe with all the derived fundamental constants in agreement with observations. Their dreams were shattered by so many possible configurations. Actually it is common for a theory to have even an infinite number of solutions. For example, according to Newtonian mechanics there are infinite number of possible orbits for objects moving around the Sun, yet there are only 9 planets each circulating around in a specific way. The difficulty with Superstring theory is that we can observe only one universe (the one we live in) and would never know if other universes exist beyond the cosmic horizon (with different configurations).



• On the other hand, propoents often cite mathematical beauty in the Superstring theory as supporting evidence for its truth. They maintain that successful theories in the past always appear that way. However, beauty is a subjective feeling, which should never be relied upon by scientists. They also marvel at such wonder of mathematics, which seems to be so unreasonably well adopted to describe the physical world. Actually, it is the other way round. Physicists instead compress the languages of physics into mathematical formulas. In other word, not all the mathematical formulas have a correspondence to physical reality (only the selected few do).




The real problem with the Calabi-Yau manifold is related to the fact that it is based on a smooth and differentiable space (top of Figure 27). At the Planck scale of ~ 10-33 cm, spacetime becomes foamy similar to the picture at the bottom of Figure 27 (courtesy of the Uncertainty principle in quantum theory). Thus the classical geometry should be replaced by the non-commutative geometry in which the Calabi-Yau manifold has to be either modified or become obsolete altogether. That's the process of scientific method in which a better theory always supersedes the old one (see more in "Space-time Background Dependence").

Figure 27 Quantum Foam [view large image]

Meanwhile, effort is under way to salvage the beloved Calabi-Yau manifold by equating topological variation with quantum fluctuation. Hopefully the string in the Superstring theory will provide gentler disturbance to enable the scheme.

Nevertheless, the Calabi-Yau manifold is embraced all the way. It may not be a part of the ultimate solution in quantum gravity; but the manifold will help further development of the Superstring theory, and there is still much to be learnt as a pure geometric object. To emphasize the importance of geometry in mathematics and science, the inscription : "Let no one ignorant of geometry enter here" at the entrance of Plato's Academy is quoted at the end of the book (Figure 28).

Figure 28 Plato's Inscription [view large image]

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