### Precise definition of fundamental length as a royal road to TOE

The visit to Kea's blog inspired a little comment about the notion of fundamental length. During morning walk I realized that a simple conceptual analysis of this notion, usually taken as granted, allows to say something highly non-trivial about the basic structure of quantum theory of gravitation.

** 1. Stringy approach to Planck length as fundamental length fails**

In string models in their original formulation Planck length is identified as the fundamental length and appears in string tension. As such this means nothing unless one defines Planck length operationally. This means that one must refer to standard meter stick in Paris or something more modern. This is very very ugly if one is speaking about theory of everything. Perhaps this is why the writers of books about strings prefer not to go the details about what one means with Planck length;-). One could of course mumble something about the failure of Riemannian geometry at Planck lengths but this means going outside the original theory and the question remains why this failure at a length scale which is just this particular fraction of the meter stick in Paris.

** 2. Fundamental length as length of closed geodesic**

What seems clear that theory of everything should identify the fundamental length as something completely inherent to the structure of theory, not something in Paris. The most natural manner to define the fundamental length is as the length of a closed geodesic. Unfortunately, flat Minkowski space does not have closed geodesics. If one however adds compact dimensions to Minkowski space, problem disappears. If one requires that the unit is unique this leaves only symmetric spaces for which all geodesics have unit length into consideration.

** 3. Identification of fundamental length in terms of spontaneous compactification fails**

The question is how to do this. In super-string models and M-theory spontaneous compactification brings in the desired closed geodesics. Usually the lengths of geodesics however vary. Even worse, the length is defined by a particular solution of the theory rather than something universal so that we are in Paris again. It seems that the compact dimensions cannot be dynamical but in some sense God given.

** 4. The approach based on non-dynamical imbedding space works**

In TGD imbedding space is not dynamical but fixed by the requirement that the "world of classical worlds" consisting of light-like 3-surfaces representing orbits of partons (size can be actually anything) has a well-defined Kähler geometry. Already the Kähler geometry of loop spaces is unique and possesses Kac-Moody group as isometries. Essentially a constant curvature space is in question. The reason for the uniqueness is that the mathematically respectable existence of Riemann connection is not at all obvious in infinite-dimensional context. The fact that the curvature scalar is infinite however strongly suggests that strings are not fundamental objects after all.

In 3-dimensional situation the existential constraints are much stronger. A generalization of Kac-Moody symmetries is expected to be necessary for the existence of the geometry of the world of classical worlds. The uniqueness of infinite-dimensional Kähler geometric existence would thus become the Mother of All Principles of Physics.

This principle seems to work in the desired manner. The generalization of conformal invariance is possible only for 4-dimensional space-time surfaces and imbedding space of form H=M^{4}×S, and number theoretical arguments involving octonions and quaternions fix the symmetric space S (also Kähler manifold) to S=CP_{2}. Standard model symmetries lead to the same identification.

The conclusion is that in TGD the universal unit of length is explicitly present in the definition of the theory and that TGD in principle predicts the length of the meter stick in Paris using CP_{2} geodesic length as unit rather than expressing Planck length in terms of the length of the meter stick in Paris. It is actually enough to predict the Compton lengths of particles correctly (that is their masses) in terms of CP_{2} size. p-Adic mass calculations indeed predict particle masses correctly and p-adic length scale hypothesis brings in p-adic length scales as a hierarchy of derived and much more practical units of length. In particular, the habitant of many-sheeted space-time can measure distances at a given space-time sheet using smaller space-time sheet as a unit.