Thursday, October 26, 2006

Is the Kaluza-Klein idea conceptually compatible with quantum theory?

This post is a continuation of a conversation started at Peter Woit's blog.

For background: String theory is presently the most widely accepted candidate theory for fundamental physics beyond what is already confirmed experimentally. Some see the widespread acceptance of string theory as validation of its correctness. Others claim there are fundamental conceptual problems with string theory.

In this case, the question relates to the Kaluza-Klein idea - the idea that there are extra dimensions to space which wrap around within a tiny distance (often presumed to be close to the Planck length - a tiny length around 10^-33 metres which one can get by combining the fundamental physical constants in a particular way). The original Kaluza-Klein idea was to add a single extra "compactified" dimension to spacetime. When you study Einstein's equation (which describe gravity) in this 5-dimensional spacetime, you find that the equations describing electromagnetism appear automatically. This idea has been adopted by string theorists who claim that this way of tucking extra dimensions away and having them cause the physics we are familiar with can explain the discrepancy between the number of dimensions needed for string theory (or rather, critical string theory, which is more or less what everybody has been calling "string theory" until now) with the number of dimensions we actually see.

String theory says there are 10, 11, or 26 dimensions, depending on which version of string theory we are talking about. We observe fewer than this number experimentally. The explanation given is that the extra dimensions are wrapped up in the way described by Kaluza and Klein, and that the details of the complicated way that they are wrapped up (called a six-dimensional Calabi-Yau manifold for 10 dimensions and a seven dimensional G2 manifold for 11 dimensions) is responsible for the details of the physics we observe.

The suggestion has been made that the Kaluza-Klein idea, while compatible with the classical theory of general relativity, is not compatible with quantum theory, due to subtleties in the difference between diffeomorphism invariance (invariance of the physical state of affairs under continuous, infinitely differentiable, mappings of the universe to itself) and gauge invariance (invariance of the physical state of affairs under changes of the mathematical representation of the configuration of physics fields which leave the field configurations themselves unchanged).

The purpose of this post is to open the comment area for discussion of the topic in the hope of bringing some much needed light to a dark and shady area.