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This Concept Map, created with IHMC CmapTools, has information related to: Non-determinisms of TGD.cmap, THE THREE NON-DETERMINISMS OF TGD 4. p-Adic non-determinism is inherent to p-adic dif- ferential equations. a) The initial values of ordinary differential equ- ations are constants. For p-adic differential equa- tions they are replaced by pseudo-constants, which by definition depend on finite number of positive pinary digits of the coor- dinate variable. b) p-Adic non-determinism is independent of non-de- terminism of Kähler action and the proposal is that it could relate to the non-de- terminism of imagination and cognition. p-Adic spa- time sheets would serve as correlates of intentions and thoughts. c) It seems that there are connections with the p-adic non-terminism and non-de- terminism of Kähler action., THE THREE NON-DETERMINISMS OF TGD 2. The proposed hierarchy h_eff=n×h of Planck constants would define dark matter as hierarchy of phases of ordi- nary matter and would follow from the non-determinism of Kähler action. a) n would correspond to the number of preferred extremals connecting given 3-surfaces at boundaries of CD and having same Kähler action. One might wonder whether also conserved charges should be identical. This would reduce the degeneracy. b) Mathematical description could be in terms of covering space of imbedding space with n giving the number of sheets. Multi-sheeted would be the term to be not confused with many-sheeted used about 3-surfaces. A singular local covering would be in question and sin- gularities would be at the ends of CD. c) One could consider also stronger con- dition n=n_1×n_2, where n_1 and n_2 can be assigned to M^4 and CP_2 degrees of freedom. d) The assumption is that this degenera- cy brings in genuine dynamical degene- racy so that one can per- form the ana- log of second quantization at the sheets of resulting surface. This would give rise to discrete degrees of freedom. e) What the h_eff=nh really means? Kähler action is proportional to 1/α_K, α_K= g_K^2/2h. One obtains the sum of identical Kähler actions over n she- ets so that one effectively single sheet but with h_eff=nh. f) The original vision was that the failure of perturbative description means multi- sheetedness and the increase of Planck constant allows to achieve reducing α_K allows to achieve perturbative descript- ion. Mother Nature would be merciful for theoreticians., THE THREE NON-DETERMINISMS OF TGD 1. The non-determinism for preferred extremals of Kähler action. a) The huge vacuum de- generacy of Kähler acti- on means that any spa- ce-time surface with CP_2 projection belong- ing to Lagrange manifold of CP_2 is vacuum extre- mals since by definition the induced Kähler form is pure gauge. Lagrange sub-manifolds are at most 2-D and analogous to momentum of position space in phase phase of classical mechanics. b) Vacuum extremals are analogous to pure gauge configurations in gauge theory but now one does not have genuine gauge symmetry since the degeneracy is in general lost for non-vacuum ext- remals. c) One expects that the vacuum degeneracy gives rise to non-determinism for small deformations of vacuum extremals in the sense that two spa- ce-like 3- surfaces at the upper and lower bounda- ries of causal diamond (CD) are connected by several preferred extre- mals with the same Käh- ler action defining Kähler function/Morse function at Euclidian/Minkowskian regions of space-time surface., THE THREE NON-DETERMINISMS OF TGD 2. The proposed hierarchy h_eff=n×h of Planck constants would define dark matter as hierarchy of phases of ordi- nary matter and would follow from the non-determinism of Kähler action. a) n would correspond to the number of preferred extremals connecting given 3-surfaces at boundaries of CD and having same Kähler action. One might wonder whether also conserved charges should be identical. This would reduce the degeneracy. b) Mathematical description could be in terms of covering space of imbedding space with n giving the number of sheets. Multi-sheeted would be the term to be not confused with many-sheeted used about 3-surfaces. A singular local covering would be in question and sin- gularities would be at the ends of CD. c) One could consider also stronger con- dition n=n_1×n_2, where n_1 and n_2 can be assigned to M^4 and CP_2 degrees of freedom. d) The assumption is that this degenera- cy brings in genuine dynamical degene- racy so that one can per- form the ana- log of second quantization at the sheets of resulting surface. This would give rise to discrete degrees of freedom. e) What the h_eff=nh really means? Kähler action is proportional to 1/α_K, α_K= g_K^2/2h. One obtains the sum of identical Kähler actions over n she- ets so that one effectively single sheet but with h_eff=nh. f) The original vision was that the failure of perturbative description means multi- sheetedness and the increase of Planck constant allows to achieve reducing α_K allows to achieve perturbative descript- ion. Mother Nature would be merciful for theoreticians., THE THREE NON-DETERMINISMS OF TGD 3. Non-determinism of Käh- ler action could yield spa- ce-time correlate for qu- antal non-determinism. a) The preferred extremal connecting space-like 3- surface A at first end of CD CD to B at second end of CD can be modified if there is multifurcation associa- ted with its time evolution. As a consequence, one has new preferred extremal, which is same as before ne- ar A but is changed later so that B is replaced with C. b) This non-determinism could also define space- time correlate for the non- determinism of state func- tion reduction making pos- sible kind of symbolic rep- resentations for sequences of quantum jumps giving rise to conscious experien- ces. c) Summation over loops - that is path integral - is central in QFT. In TGD one sums only over 3-surfaces at the ends of CD. Could the summation over loops correspond to the non-de- terminism at space-time level.
