Tuesday, February 15, 2005

Comment to Not-Even-Wrong

The discovery that strings in a fixed flat background could describe gravitation without any need to make the background dynamical was really momentous. The discovery should have raised an obvious question: How to generalize the theory to the physical 4-dimensional case by replacing string orbits with 4-surfaces? Instead, the extremely silly idea of making also imbedding space dynamical emerged and brought back and magnified all the problems of general relativity, which one had hoped to get rid of. I have tried for more than two decades to communicate simple core ideas about an alternative approach but have found that theoretical physicists are too arrogant to listen to those without name or position. a) The fusion of special relativity with general relativity is achieved by assuming that space-times are 4-surfaces in M^4xCP_2. The known quantum numbers pop out elegantly from this framework. The topological complexity of space-time surfacse allows to circumvent objection that the induced metrics are too restricted. Light-like 3-D causal determinants allow generalization of super-conformal invaraince by their metric 2-dimensionality and dimension 4 for space-time is the only possibility. b) The maximal symmetries of H=M^4xCP_2 have an excellent justification when quantum theory is geometrized by identifying physical states of the Universe as classical configuration space spinor fields, configuration space being defined as the space of 3-surfaces in H. The only hope of geometrizing this infinite-dimensional space is as union of infinite-dimensional symmetric spaces labelled by zero modes having interpretation as non-quantum fluctuating classical degrees of freedom. Infinite-dimensional variant of Cartan's problem of classifying symmetric spaces emerges as the challenge of finding TOE. Mathematical existence fixes physical existence. Just as in the case of loop space, and with even better reasons, one expects that there are very few choices of H allowing internally consistent Kaehler geometry. Fermion numbers and super-conformal symmetries find an elegant geometrization and generalization in terms of complexified gamma matrices representing super-symmetry generators. c) M^4xCP_2 follows also from purely number theoretical considerations as has now become clear. The theory can be formulated in two equivalent manners. *4-surfaces can be regarded as hyper-quaternionic 4-surfaces in M^8 possessing what I call hyper-octonionic tangent space structure (octonionic imaginary units are multiplied by commutative sqrt(-1) to make number theoretical norm Minkowskian). *Space-times can be regarded also as 4-surfaces in M^4xCP_2 identified as extrema of so called Kaehler action in M^4xCP_2. Spontaneous compactification has thus purely number theoretical analog but has nothing to do with dynamics. The surprise was that under some additional conditions (essentially hyper-octonion real-analyticity for the dynamical variables in M^8 picture) the theory can be coded by WZW action for two-dimensional string like 2-surfaces in M^8. These strings not super-strings but generalizations of braid/ribbon diagrams allowing n-vertices in which string orbits are glued together at their ends like pages of book. Vertices can be formulated in terms of octonionic multiplication. Both classical and quantum dynamics reduce to number theory and the dimensions of classical division algebras reflect the dimensions of string, string orbit, space-time surface, and imbddding space. The conclusion is that both particle data table, the vision about physics as free, classical dynamics of spinor fields in the infinite-dimensional configuration space of 3-surfaces, and physics as a generalized number theory, lead to the same identification: space-time can be regarded as 4-surfaces in M^4xCP_2. In the case that someone is more interested of learning about real progress instead of wasting time to heated arguments at the ruins M theory, he/she can read the chapter http://www.helsinki.fi/~matpitka/tgd.html#visionb summarizing part of the number theoretical vision, and also visit my blog at http://matpitka.blogspot.com/ where I have summarized the most recent progress and great ideas of TGD. With Best Regards, Matti Pitkanen

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