“Quantum field theory has emerged as the most successful physical framework describing the subatomic world. Both its computational power and its conceptual scope are remarkable. Its predictions for the interactions between electrons and photons have proved to be correct to within one part in 108. Furthermore, it can adequately explain the interactions of three of the four known fundamental forces in the universe. The success of quantum field theory as a theory of subatomic forces is today embodied in what is called the Standard Model. In fact, at present, there is no known experimental deviation from the Standard Model (excluding gravity). This impressive list of successes, of course, has not been without its problems. In fact, it has taken several generations of the world's physicists working over many decades to iron out most of quantum field theory's seemingly intractable problems. Even today, there are still several subtle unresolved problems about the nature of quantum field theory.”

In “Quantum Field Theory - A Modern Introduction International Student Edition” by Michio Kaku

“One of the main problems in superstring research has been to find the true vacuum of the theory, either perturbatively or nonperturbatively. Therefore, intense research over the years has been spent trying to catalog the various possible four-dimensional compactified strings.

A few classes of these solutions include:

1. Calabi-Yau manifolds, which are highly nonlinear, nontrivial manifolds studied by mathematicians;
2. Orbifolds, which are certain manifolds which have fixed points on them (e.g., a cone is an orbifold);
3. Free fermion/free boson solutions.

Unfortunately, we now know millions upon millions of possible string vacua. In fact, it is conjectured that the complete set of all possible string vacua is the totality of possible conformal field theories (CFTs). Although there are an enormous number of possible four-dimensional string vacua, the surprising feature of string theory is that, with a few rather mild assumptions, one can come fairly close to describing the physical universe. Earlier, we saw that Kaluza-Klein theory was too restrictive to describe the physical universe. In particular, the Standard Model's gauge group and complex fermion representations could not be accommodated. However, the string model, because it is not based on Riemannian space, does not suffer from these problems.”

In “Quantum Field Theory - A Modern Introduction International Student Edition” by Michio Kaku

Is the universe fond of pi? Seriously, that is indeed strange. Since 3+1-space and 4-space are not necessarily identical in how things radiate, it seems sensible to ask if the extra pi applies in either cases or just one.

If there is a difference, then things get interesting further up. There are, at most, two time dimensions within the 11 defined dimensions in modern M-theory. If 3+1 acts differently to 4, then you should be able to make predictions and observations that distinguish 10+1, 9+2 and a pure 11. Much more importantly, though, given that we're looking at stuff that would be entirely visible from any 3-dimensional cross-section of a higher-dimensional radiating property, we're looking at stuff that proves whether these dimensions exist at all. In other words, experimental evidence that could test string theory, supergravity, holographic universe theory (since you run into limits on what you can compress in what way), etc (If leptons of any given type are indeed the visible protrusion of a higher-dimensional object, then provided the geometry of that object is fixed, there are other experiments you can perform, since you cannot rotate an n-dimensional object around n+1 axes. That's an aside, though.) At the same time, we're told that such theories are closer to philosophy and untestable.

Any thoughts on this Kaku?

At such short distances, space is not smooth. Because it is not smooth, the effective radius (the radius you get by creating a perfect sphere of equal surface area) is not the apparent radius (the radius you get from naive assumptions about spacetime). If you apply the effective radius, the inverse square law is a perfect fit. This means that at Planck lengths, you have to consider the fact that spacetime is nothing more than geodesics. Now, obviously flattened spacetime is not going to be identical to curved spacetime. But That Does Not Matter. It doesn't matter because if you know the geometry, you know the space/area/line that the force is occupying. If you double the size of the container, you halve what is in any given part of it. This is so incredibly elementary. Anyone who knows about Compton wavelengths knows about geometry and knows that for every convex hull there is a circle of equal circumference. Why should I have to explain this? Kaku, step forward! ( )