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This Concept Map, created with IHMC CmapTools, has information related to: 4-D spin glass degeneracy.cmap, 4-DIMENSIONAL SPIN GLASS DEGENERACY 2. d) The space of maxima of Kähler function would repla- ce free energy landscape and p-adic metrics strongly sug- gests them as topologies for regions of the space of the maxima. The non-determism of preferred extremals would make the spin glass property 4-D and correlate with the non-determinism of p-adic differential equations. e) Note that in real topology the space of maxima is disc- rete just as the space of possibly infinitely long series of binary digits is discrete. In p-adic topology this space is however smooth just like 2-adic numbers formed by these series and pinary cut- off which can be arbitrary good approximation brings in discreteness. f) The non-determinism ref- lected as possibility of having several maxima of Kähler function connecting given 3-surface at boundary of CD CD to several alternative 3- surface such that non-deter- minism appears in the inter- ior as multifurcation of pre- ferred extremal could serve as space-time correlate for quantum non-determinism., 4-DIMENSIONAL SPIN GLASS DEGENERACY 2. In TGD vacuum degeneracy of Kähler action is the sour- ceof 4-D spin glass degene- racy. a) Vacuum extremals repre- sentable as graphs of a map from M^4→CP_2 have huge degeneracy. Entire sub-theo- ry with space-time surfaces having at most 2-D Lagran- gian manifold as CP_2 pro- jection are vacuum extremals with vanishing induce Kähler form so that Kähler gauge po- tential is pure gauge. Time evolution is non-determinism for the vacuum extremals. Symplectic transformati- ons of CP_2 give new Lagran- gian manifolds defining this kind of degeneracy. b) For non-vacuum extre- mal symplectic transforma- tions do not leave action in- variant but act as isometries of world of classical worlds (WCW). Classical gravitation breaks the gauge symmetry to dynamical symmetry ac- ting as isometries of WCW. c) Large class of preferred ex- tremals are obtained as small deformations of vacuum extremals. 4-D spin glass landscape for the maxima of the exponent of Kähler func- tion defining vacuum functio- nal "square root" of exponent of free energy in square root of thermodynamics inspi- red by ZEO., 4-DIMENSIONAL SPIN GLASS DEGENERACY 1. Ordinary spin glass: a) Examples: spin glass as mag- netic system. Consists of magne- tized regions with varying direc- tion of static magnetization. b) Large number of almost dege- nerate ground states are minima of free energy. The notion of frust- ration catches this property. Spin glass energy land scape. Free energy has valleys inside valleys. Fractal structure. c) Parisi understood spin glasses mathematically. The space for the minima of free energy allows ult- rametric topology: also p-adic to- pologies are ultrametric. In ultra- metric topology the length d(A-B) smaller than Min(d(A),d(B): topo- logy is non-Archimedean. Concre- te definition of distance: define the distance along given path between A and B as height of the highest valley along path. Defi- ne the distance between A and B as minimum of this distance over all paths, just like ordinary geodesic distance. This metric is ultrametric. d) Parisi's model was based on replica method. One assumes that also the parameters of the Hamiltonian defining a model of magnetic system have have thermodynamical distribution. One obtains formally large num- ber of replicas of the original thermodynamical systems. Pro- bability distributions for proba- bility distributions. e) Breaking of ergodicity as out- come: time evolution of single member of ensemble does not define probability distribution of states equal to that of ensemble., 4-DIMENSIONAL SPIN GLASS DEGENERACY 3. Is the 4-D spin glass dege- neracy visible at the level of effective space-time of GRT? a) Obtained by replacing ma- ny-sheeted space-time with M^4 and its metric with sum of M^4 metric and deviations from it for various space-time sheet having non-empty pro- jection to the region of M^4 considered and performing qu- antum average. b) To test this picture experi- mentally, one must be able to avoid many-sheetednes. c) In living matter the visibi- lity of 4-D spin glass property is especially suggestive. Consi- der only non-deterministic be- havior of living systems., 4-DIMENSIONAL SPIN GLASS DEGENERACY 2. g) This p-adicity is in princip- independent of fundamental p-adicity postulated at space- time level to server as correla- te for cognition by generalizing the notion of number so that also p-adic space-time sheets become possible in the genera- lization of imbedding space ob- tained by gluing together real and various p-adic imbedding spaces along common rationals. e) An interesting question is whether p-adic thermodyna- mics used to calculated elemen- tary particle masses could be seen as application made pos- sible by 4-D spin glass degene- racy or by fundamental p-adic- ity.
