Twodimensional illustrations related to the manysheeted spacetime conceptMatti Pitkänen (January 20 2003)

B. Elementary particles as 3surfaces of size of order R=10^{4} Planck lengths: CP_{2} extremals. Elementary particles have geometric representation as so called CP_{2} type extremals. Instead of standard imbedding of CP_{2} as a surface of M^{4}_{+}×CP_{2} obtained by putting Minkowski coordinates m^{k} constant m^{k}=const., one considers 'warped' imbedding m^{k} =f^{k}(u) u is arbitrary function of CP_{2} coordinates with the property that the M^{4}_{+} projection of the surface is random light like curve: m_{kl} dm^{k}/du dm^{l}/du =0, m_{kl} is flat M^{4} metric. (A) The condition implies that induced metric is just CP_{2} metric, which is Euclidian! The curve is random and therefore one has classical nondeterminism: this makes sense since the solution is vacuum extremal. Fig. 7. The projection of CP_{2} type extremal to M^{4}_{+} is lightlike curve. Elementary particles correspond to CP_{2} type extremals with holes: the intersection of bound with m^{0}=const hyperplane is sphere, torus, sphere with two handles, etc...: shortly a surface with genus g=0,1,2,... . Different fermion families correspond to different genera. Bosons are also predicted to have family replication phenomenon. Fig. 8. Different fermion families correspond to different genera for the boundary component of CP_{2} type extremal. Feynman diagrams correspond to topological sums of CP_{2} type extremals: the lines of diagram being thickened to CP_{2} type extremals: Fig. 9. Feynman diagrams correspond to connected sums of CP_{2} type extremals: each line of Feynman diagram is thickened to CP_{2} type extremal. The quantum version of the condition (A) stating that the M^{4}_{+} projection is light like curve leads to Super Virasoro conditions and it turns out that elementary particles together with their 10^{4} Planck mass excitations belong to representation of padic Super Virasoro and Kac Moody. The padic mass calculations lead to excellent predictions for particle masses. 
D. Matter as topology Since many sheetedness is encountered in all length scales a very attractive manner to reinterpret our visual experience about world suggests itself. Material objects having macroscopic boundaries correspond actually to sheets of 3space and 3space literally ends at the boundary of object. The 3space outside the object corresponds to the 'lower' spacetime sheet. Actually we can see this wild 3topology every moment!! The following 2dimensional illustration should make clear what the generalization really means. Fig. 18. Matter as topology 
E. Join along boundaries contacts and join along boundaries condensate The recipe for constructing manysheeted 3space is simple. Take 3surfaces with boundaries, glue them by topological sum to larger 3surfaces, glue these 3surfaces in turn on even larger 3surfaces, etc.. The smallest 3surfaces correspond to CP_{2} type extremals that is elementary particles and they are at the top of hierarchy. In this manner You get quarks, hadrons, nuclei, atoms, molecules,... cells, organs, ..., stars, ..,galaxies, etc... Besides this one can also glue different 3surfaces together by tubes connecting their BOUNDARIES : this is just connected sum operation for boundaries. Take disks D_{2} on the boundaries of two objects and connect these disks by cylinder D_{2}×D^{1} having D_{2}:s as its ends. Or more concretely: let the two 3surfaces just touch each other. Fig. 19. Join along boundaries bond a): in two dimensions and b): in 3dimensions for solid balls. Depending on the scale join along boundaries bonds are identified as color flux tubes connecting quarks, bonds giving rise to strong binding between nucleons inside nuclei, bonds connecting neutrons inside neutron star, chemical bonds between atoms and molecules, gap junctions connecting cells, the bond which is formed when You touch table with Your finger, etc. One can construct from a group of nearby disjoint 3surfaces so called join along boundaries condensate by allowing them to touch each other here and there. Fig. 20. Join along boundaries condensate in 2 dimensions. The formation of join along boundaries condensates creates clearly strong correlation between two quantum systems and it is assumed that the formation of join along boundaries condensate is necessary prequisite for the formation of MACROSCOPIC QUANTUM SYSTEMS. 
F. pAdic numbers and vacuum degeneracy pAdic length scale hypothesis derives from the analogy between SPIN GLASS and TGD. Kähler action allows enormous VACUUM DEGENERACY: ANY spacetime surface, which belongs to M^{4}_{+}×Y_{2}, where Y_{2} is so called Lagrangian submanifold of CP_{2} is vacuum due to the vanishing of induced Kähler form (recall that Kähler action is just Maxwell action for induced Kähler form which can be regarded as U(1) gauge field). Lagrangian submanifolds can be written in the canonical coordinates P^{i},Q^{i}, i=1,2 for CP_{2} as P^{i} =f^{i}(Q^{1},Q_{2}) f^{i} =∂_{i} f(Q^{1},Q_{2}) where partial_{i} means partial derivative with respect to Q^{i}. f is arbitrary function of Q^{i}! Lagrangian submanifolds are 2dimensional. The topology of vacuum space time is restricted only by the imbeddability requirement. Vacuum spacetimes can have also finite extend in time direction(!!): charge conservation does not force infinite duration. Fig. 21. Vacuum extremals can have finite time duration. This enormous vacuum degeneracy resembles the infinite ground state degeneracy of spin glasses. In case of spin glasses the space of free energy minima obeys ultrametric topology. This raises the question whether the effective topology of the real spacetime sheets could be also ultrametric in some length scale range so that the distance function would satisfy d(x,y) <= Max(d(x),d(y)) rather than d(x)+d(y) pAdic topologies are ultrametric and there is padic topology for each prime p=2,3,5,7,... The classical nondeterminism of the vacuum extremals implies also classical nondeterminism of field equations (but not complete randomness of course). pAdic differential equations are also inherently nondeterministic. This suggests that the nondeterminism of Kähler action is effectively like padic nondeterminism in some length scale range, so that that the topology of the real spacetime sheet is effectively padic for some value p. The lower cutoff length scale could be CP_{2} length scale. Of course, cutoff length scales could be dynamical. Standard representation of padic number is defined as generalization of decimal expansion x= ∑_{n≥n0} x_{n}p^{n} pAdic norm reads as N(x)_{p} = p^{n0} , and clearly depends on the lowest pinary digit only and is thus very rough: for reals norm is same only for x and x. Note that integers which are infinite as real numbers are finite as padic numbers: padic norm of any integer is at most one. Essential element is the so called CANONICAL CORRESPONDENCE between padics and reals pAdic number x= ∑_{n≥n0} x_{n}p^{n} is mapped to real number y = ∑_{n≥n0} x_{n}p^{n} Note that only the signs of powers of p are changed. Second natural correspondence between padics and reals is based on the fact that both reals and padics are completions of rational numbers. Hence rational numbers can be regarded as common to both padic and real numbers. This defines a correspondence in the set of rationals. Allowing algebraic extensions of padic numbers, one can regard also algebraic numbers as common to reals and algebraic extensions of padics. pAdic and real transcendentals do not have anything in common. Note that rationals have pinary expansion in powers of p, which becomes periodic for high pinary digits (predictability) whereas transcendentals have nonperiodic pinary expansions (nonpredictability). One could say that the numbers common to reals and padics are like islands of order in the middle of real and padic seas of chaos. Both correspondences are important in the recent formulation of padic physics. 
G. pAdic length scale hypothesis pAdic mass calculations force to conclude that the length scale below which padic effective topology is satisfied is given L_{p} ≈ p^{1/2}R, R= 10^{4}× G^{1/2} (CP_{2} length scale). One has also good reasons to guess that pAdic effective topology makes sense only above CP_{2} length scale. One can also define nary padic length scales L_{p}(n) =p^{(n1)/2}L_{p} It is very natural to assume that the spacetime sheets of increasing size have typical sizes not too much larger than L_{p}(n). The following figure illustrates the situation. Fig. 22. pAdic length scale hierarchy The obvious question is 'Are there some physically favored padic primes?'. pAdic mass calculations encourage the following hypothesis The most interesting padic primes p correspond to primes near prime powers of two p ≈ 2^{k}, k prime Especially important are physically Mersenne primes M_{k} for which this condition is optimally satisfied p= 2^{k}1 Examples: M_{127}= 2^{127} 1, M_{107} = 2^{107} 1, M_{89}= 2^{89} 1: electron, hadrons, intermediate gauge bosons. A real mathematical justification for this hypothesis is still lacking: probably the padic dynamics depends sensitively of p and this selects certain padic primes via some kind of 'natural selection'. < 
H. Generalization of spacetime concept One can wonder whether padic topology is only an effective topology or whether one could speak about a decomposition of the spacetime surface to real and genuinely padic regions, and what might be the interpretation of the padic regions (note that also real spacetime regions would still be characterized by some prime characterizing their effective topology). The development of TGD inspired theory of consciousness led finally to what seems to be a definite answer to this question. pAdic physics is physics of cognition and intention. pAdic nondeterminism is the classical spacetime correlate for the nondeterminism of imagination and cognition. pAdic spacetime sheets represent intentions and quantum jump in which padic spacetime sheet is transformed to real one can be seen as a transformation of intention to action. This forces to generalize the notion of the imbedding space. The basic idea is that rational numbers are in a welldefined sense common to both real number field R and all padic number fields R_{p}. The generalized imbedding space results when the real H and all padic versions H_{p} of the imbedding space are glued together along rational points. One can visualize real and padic imbedding spaces as planes, which intersect along a common axis representing rational points of H. Real and padic spacetime region are glued together along the boundaries of the real spacetime sheet at rational points. The construction of padic quantum physics and the fusion of real physics and padic physics for various primes to a larger scheme is quite a fascinating challenge. For instance, a new number theoretic view about information emerges. pAdic entropy can be negative, which means that system carries genuine information rather than entropy.
