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This Concept Map, created with IHMC CmapTools, has information related to: Manysheeted space-time.cmap, MANY-SHEETED SPACE-TIME AND TOPOLOGICAL CONDENSATION 4. Many-sheetedness and CP_2 geo- metry allow to over-come the basic objections against induced gauge field concept. a) Induced gauge fields are expressib- le in terms of four CP_2 coordinates and their gradients. Hence linear super- position is lost except for massless ext- remals for Fourier components in the direction of massless wavevector. b) The problem is solved by the repla- cement of the superposoition of classi- cal fields with the superposition of their effects. A particle topologically condensing simultaneously to several space-time sheets experiences super- position of forces caused by the clas- sical fields at the space-time sheets. c) Second problem is that different classical gauge fields are expressible in terms of CP_2 coordinates and are therefore not independent. How the multiply topologically condensed partic- le knows that particular space-time sheet carries particular gauge field? Or are the classical fields always em fields? d) The condition that em charge of spinor modes is well-defined implies that the spinor modes are in the gene- ric case localized to string world sheets, which carry no W fields so that only classical em and Z^0 fields remain. One can pose also additional condition stating that classical Z^0 field vanishes so that only classical em field remains., MANY-SHEETED SPACE-TIME AND TOPOLOGICAL CONDENSATION 2. p-Adic length scale hypo- thesis and many-sheeted space-time. a) p-Adic length scale hypo- thesis allows quantitative for- mulation of the notion. p-Adic mass calculations plus Uncer- tainty principle allow to assign to space-time sheet of elemen- mentary particle p-adic length scale as L_p = sqrt(p)hbar_eff/R, R is CP_2 radius. p-Adic length scales would come as half-oc- taves if p= about 2^k holds true. b) The building bricks of ele- mentary particles correspond to wormhole throats. What is scale of M^4 projection of wormhole throat. For cosmic strings projection is point but one expects that it widens when cosmic strings widen to topologically condensed mag- netic flux tubes. Naive guess would be CP_2 scale but this would suggest mass of order CP_2 mass. The size of the projection might even corre- spond to Compton length. c) The length of the flux tube connecting wormhole contacts appearing as building bricks of particle would be naturally of the order of Compton length length. The other end would contain pair of right-handed and left handed neutrino to neutralize the weak isospin in long length scales., MANY-SHEETED SPACE-TIME AND TOPOLOGICAL CONDENSATION 1. How the notion of many- sheeted space-time has evolved? a) Observation: the global imbeddings of field configu- rations possible in M^4 are not possible as space-time surfacs due to the compact topology of CP_2: CP_2 effectively replaces the non- compact field space of Min- kowskian case. b) What happens that the ansatz for imbedding beco- mes ill-defined because real CP_2 coordinates become complex and one is led out- side of imbedding space. c) The belief was that im- bedding is possible only for regions with boundary but here boundary conditions become problematic. It would be nice to have M^4 vacuum extremal at boun- dary but this would mean that there is no need for bo- undary. d) The solution comes from the observation that one can glue to extremals developing singularity (in the sense that real CP_2 coordinates would become complex) together along the boundaries. One obtains two-sheeted cove- ring. All space-time sheets have this character so that "sheet" is not quite appro- priate term but is used for "historical" reasons., MANY-SHEETED SPACE-TIME AND TOPOLOGICAL CONDENSATION 3. Hierarchy of Planck constants and multi- sheetedness. a) The hierarchy of Planck constants h_eff=nh brings additional delicacy. The proposal is that h_eff=nh results from n-furcation of space-time sheet such that different sheets have same Kähler action. This n-furcation is reflects the non-determinism of Kähler action is closely related to the fractal hierarchy of sub-algebras of conformal algebras isomorphic to the algebra itself. b) This many-sheetedness differs from that for 3-sur- faces since the different branches of the multifurca- tion coincide at 3-surfaces at the ends of space-time sheet at boundaries of CD. c) In hope of avoiding con- fusion I have called the many-sheetedness of pre ferred extremals as oppo- to that of 3-surfaces multi- sheetedness.
