### 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.

## 4 Comments:

There is an heretical branch of physics devoted to adding "proper time" to the geometry of space-time. It's sometimes called "Euclidean Relativity", but it seems that everyone who gets involved with it discovers it on their own and then calls it something else. A good set of of links is at the Euclidean Relativity website: Euclidean Relativity.

It means that the metric becomes:

dt^2 = dx^2 + dy^2 + dz^2 + ds^2,

and this means that all particles, massive or not, travel at c in the four spatial dimensions (x,y,z,s). My attraction to this is that it simplifies certain aspects of quantum mechanics. Most people like it because it makes certain things in relativity nicer.

Anyway, when you do this, there are a bunch of things you can choose to do with the s coordinate. My own preference is to make it cyclic, small and constant. When you do this, you end up with a fundamental length.

Interesting interpretation. This would look like adding additional dimension (coordinate s) and interpreting dt as Euclidian line element length. Or alternatively interpreting dt^2-ds^2-dx^2...=0 as a lightlike geodesic, which is probably what you mean.

If one interprets ds as a coordinate of some compact space-like dimension one would obtain something which one might call fundemental unit 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.Hi Matti

Nice post. Now mathematically, what do you mean by a 'many-sheeted space-time'? Are the sheets copies of the imbedding space H=M^4xCP^2?

Your S=CP^2 version of TGD has a structure similar to the d=4 N=2 SUGRA with scalar fields lying on the symmetric space G_4/K_4=SU(3,3)/(SU(3)xSU(3)xU(1)). Here, SU(3,3) is the 'conformal group' of the CP^2 extremal black hole, arising from the 26D Freudenthal triple system over J(3,C). (See fig. 1 of the Gunaydin et al paper hep-th/0512296 for a comprehensive list of such symmetric spaces.)

What's nice about CP^2 is that it permits an elegant matrix description. Specifically, CP^2 permits a three-dimensional projective (matrix) basis. The J(3,C) reduced structure group G=SL(3,C) acts via collineations on this projective basis. Gunaydin often refers to SL(3,C) as the 'Lorentz group' for CP^2 and describes non-BPS orbits with non-zero central charge via the symmetric space SU(3,3)/SL(3,C) (see table 3, hep-th/0606209). Perhaps you can find an application of these new SUGRA structures in TGD, e.g., wormhole transformation groups.

Thank you for the link. This SL(3,C) action in CP_2 look interesting. It would be analogous to the action of SL(2,C) at r=contant sphere of lightcone cone boundary which appears naturally in the construction of quantum TGD. SL(3,C) might organize color partial waves in CP_2 to infinite-D representations of SL(3,C) just like SL(2,C) organizes representations of SU(2).

There are two different notions of multi-sheetednes.

1. Many-sheeted space-time

Many-sheeted space-time can be visualized in 2-D case as a hierarchy of 2-D sheets glued to larger sheets by topological sum contacts (tiny wormhole contacts). The size of wormhole contact and the scale for the distance between sheets is given by CP_2 size, about 10^4 Planck lengths so that the sheets are very near to each other.

The many-sheetedness is an outcome of what I call topological field quantization: one can imbed any em field locally to M4xcP_2 as induced gauge field but the imbedding usually fails globally and boundaries are generated. At boundaries wormhole contacts must feed conserved gauge fluxes to larger space-time sheets.

All physical objects beginning from elementary particles to macroscopic objects that we see around us to astronomical structures correspond to this kind of sheets. The outer boundaries of these sheets correspond to the boundaries of these sub-Universe. Also biological body defines this kind of mini-Universe and we indeed directly experience asa division of world to body and external world.

Sheets interact via these wormhole contacts and by bonds connecting their boundaries. For instance, chemical bonds corresponds to this kind of contacts.

2. Multisheetedness in the sense of multiple covering

I speak about multi-sheetedness also in the sense of multiple coverings but try to distinguish betwee the two notions. The quantization of Planck constant leads to the generalization of imbedding space H=M^4xCP_2.

a) One first gives imbedding space H an additional bundle structure H= M^4xCP_2-->M^4x CP_2/G_axG_b where G_a are and G_b are subgroups of rotation group and of SU(3) acting as isometries of CP_2. One can imagine alternative choices of G_a and G_b and I have not been able to decide which choice is the correct one. The most stringent condition would be that G_a and G_b are Abelian groups leaving invariant given choice of quantization axis of spin and color quantum numbers. G_a= Z_(n_a) and G_b= Z(n_b) would be the choice. Each sector would correspond to is own Planck constant determined by orders n_a and n_b of the groups G_a and G_b.

b) One then glues all copies of H together along common points of spaces H/G_axG_b. The common set for two copies contains at least the 4-D space M^2xCP_2/U(1)xU(1). The copies of H relate like pages of book glued together along this 4-D common set.

c) For ordinary matter would correspond to trivial choice of G_a and G_b. Various dark matters would correspond to H/G_axG_b so that various kinds of matters would live at different pages of book and phase transitions changing character of matter would mean leakage between pages of book via this 4-D "rim" (I am not sure whether this is an appropriate term) of the book. 4-D "rim" would correspond to maximal quantum criticality: in this state one can assign to particle any value of Planck constant. Elementary particles would correspond to maximally quantum critical states but their "field bodies" at the pages of the big book would correspond to definite value of Planck constant.

d) Space-time surfaces in a given sector must be consistent with the discrete bundle structure and this means multi-sheetedness analogous to that of Riemann surfaces having nothing to do with many-sheetedness.

Matti

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