## How to build TGD space-time from legos?
TGD predicts shocking simplicity of both quantal and classical dynamics at space-time level. Could one imagine a construction of more complex geometric objects from basic building bricks - space-time legos? Let us list the basic ideas. - Physical objects correspond to space-time surfaces of finite size - we see directly the non-trivial topology of space-time in everyday length scales.
- There is also a fractal scale hierarchy: 3-surfaces are topologically summed to larger surfaces by connecting them with wormhole contact, which can be also carry monopole magnetic flux in which one obtains particles as pairs of these: these contacts are stable and are ideal for nailing together pieces of the structure stably.
- In long length scales in which space-time surface tend to have 4-D M
^{4}projection this gives rise to what I have called many-sheeted spacetime. Sheets are deformations of canonically imbedded M^{4}extremely near to each other (the maximal distance is determined by CP_{2}size scale about 10^{4}Planck lengths. The sheets touch each other at topological sum contacts, which can be also identified as building bricks of elementary particles if they carry monopole flux and are thus stable. In D=2 it is easy to visualize this hierarchy.
Simplest legos
What could be the simplest surfaces of this kind - legos? - Assume twistor lift so that action contain volume term besides Kähler action: preferred extremals can be seen as non-linear massless fields coupling to self-gravitation. They also simultaneously extremals of Kähler action. Also hydrodynamical interpretation makes sense in the sense that field equations are conservation laws. What is remarkable is that the solutions have no dependence on coupling parameters: this is crucial for realizing number theoretical universality. Boundary conditions however bring in the dependence on the values of coupling parameters having discrete spectrum by quantum criticality.
- The simplest solutions corresponds to Lagrangian sub-manifolds of CP
_{2}: induced Kähler form vanishes identically and one has just minimal surfaces. The energy density defined by scale dependent cosmological constant is small in cosmological scales - so that only a template of physical system is in question. In shorter scales the situation changes if the cosmological constant is proportional the inverse of p-adic prime.The simplest minimal surfaces are constructed from pieces of geodesic manifolds for which not only the trace of second fundamental form but the form itself vanishes. Geodesic sub-manifolds correspond to points, pieces of lines, planes, and 3-D volumes in E ^{3}. In CP_{2}one has points, circles, geodesic spheres, and CP_{2}itself. - CP
_{2}type extremals defining a model for wormhole contacts, which can be used to glue basic building bricks at different scales together stably: stability follows from magnetic monopole flux going through the throat so that it cannot be split like homologically trivial contact. Elementary particles are identified as pairs of wormhole contacts and would allow to nail the legos together to from stable structures.
Consider first - Quite generally, the simplest legos consist of 3-D geodesic sub-manifolds of E
^{3}× CP_{2}. For E^{3}their dimensions are D=1,2,3 and for CP_{2}, D=0,1,2. CP_{2}allows both homologically non-trivial resp. trivial geodesic sphere S^{2}_{I}resp. S^{2}_{II}. The geodesic sub-manifolds cen be products G_{3}=G_{D1}× G_{D2}, D_{2}=3-D_{1}of geodesic manifolds G_{D1}, D_{1}=1,2,3 for E^{3}and G_{D2}, D_{2}=0,1,2 for CP_{2}. - It is also possible to have twisted geodesic sub-manifolds G
_{3}having geodesic circle S^{1}as CP_{2}projection corresponding to the geodesic lines of S^{1}⊂ CP_{2}, whose projections to E^{3}and CP_{2}are geodesic line and geodesic circle respectively. The geodesic is characterized by S^{1}wave vector. One can have this kind of geodesic lines even in M^{1}× E^{3}× S^{1}so that the solution is characterized also by frequency and is not static in CP_{2}degrees of freedom anymore.These parameters define a four-D wave vector characterizing the warping of the space-time surface: the space-time surface remains flat but is warped. This effect distinguishes TGD from GRT. For instance, warping in time direction reduces the effective light-velocity in the sense that the time used to travel from A to B increases. One cannot exclude the possibility that the observed freezing of light in condensed matter could have this warping as space-time correlate in TGD framework. For instance, one can start from 3-D minimal surfaces X ^{2}× D as local structures (thin layer in E^{3}). One can perform twisting by replacing D with twisted closed geodesics in D× S^{1}: this gives valued map from D to S^{1}(subset CP_{2}) representing geodesic line of D× S^{1}. This geodesic sub-manifold is trivially a minimal surface and defines a two-sheeted cover of X^{2}× D. Wormhole contact pairs (elementary particles) between the sheets can be used to stabilize this structure. - Structures of form D
^{2}× S^{1}, where D^{2}is polygon, are perhaps the simplest building bricks for more complex structures. There are continuity conditions at vertices and edges at which polygons D^{2}_{i}meet and one could think of assigning magnetic flux tubes with edes in the spirit of homology: edges as magnetic flux tubes, faces as 2-D geodesic sub-manifolds and interiors as 3-D geodesic sub-manifolds.Platonic solids as 2-D surfaces can be build are one example of this and are abundant in biology and molecular physics. An attractive idea is that molecular physics utilizes this kind of simple basic structures. Various lattices appearing in condensed matter physics represent more complex structures but could also have geodesic minimal 3-surfaces as building bricks. In cosmology the honeycomb structures having large voids as basic building bricks could serve as cosmic legos. - This lego construction very probably generalizes to cosmology, where Euclidian 3-space is replaced with 3-D hyperbolic space SO(3,1)/SO(3). Also now one has pieces of lines, planes and 3-D volumes associated with an arbitrarily chosen point of hyperbolic space. Hyperbolic space allows infinite number of tesselations serving as analogs of 3-D lattices and the characteristic feature is quantization of redshift along line of sight for which empirical evidence is found.
- These basic building bricks can glued together by wormhole contact pairs defining elementary particles so that matter emerges as stabilizer of the geometry: they are the nails allowing to fix planks together, one might say.
Geodesic minimal surfaces with non-vanishing gauge fields
What about minimal surfaces and geodesic sub-manifolds carrying non-vanishing gauge fields - in particular em field (Kähler form identifiable as U(1) gauge field for weak hypercharge vanishes and thus also its contribution to em field)? Now one must use 2-D geodesic spheres of CP
The basic legos are now D× S
One can consider also more complex static basic building bricks by allowing bricks which are not anymore geodesic sub-manifolds. The simplest static minimal surfaces are form M
Could these structures represent higher level of self-organization emerging in living systems? Could the flexible network formed by living cells correspond to a structure involving more general minimal surfaces - also non-static ones - as basic building bricks? The Wikipedia article about minimal surfaces in E
The surfaces with constant positive curvature do not allow imbedding as minimal surfaces in E Does this mean that the proposal fails? - One can build approximately spherical surfaces from pieces of planes. Platonic solids represents the basic example. This picture conforms with the notion of monadic manifold having as a spine a discrete set of points with coordinates in algebraic extension of rationals (preferred coordinates allowed by symmetries are in question). This seems to be the realistic option.
- The boundaries of wormhole throats at which the signature of the induced metric changes can have arbitrarily large M
^{4}projection and they take the role of blackhole horizon. All physical systems have such horizon and the approximately boundaries assignable to physical objects could be horizons of this kind. In TGD one has minimal surface in E^{3}× S^{1}rather than E^{3}. If 3-surface have no space-like boundaries they must be multi-sheeted and the sheets co-incide at some 2-D surface analogous to boundary. Could this 3-surface give rise to an approximately spherical boundary. - Could one lift the immersions of H
^{2}and S^{2}to E^{3}to minimal surfaces in E^{3}× S^{1}? The constancy of scalar curvature, which is for the immersions in question quadratic in the second fundamental form would pose one additional condition to non-linear Laplace equations expressing the minimal surface property. The analyticity of the minimal surface should make possible to check whether the hypothesis can make sense. Simple calculations lead to conditions, which very probably do not allow solution.
At even higher level of self-organization emerge dynamical minimal surfaces. Here string world sheets as minimal surfaces represent basic example about a building block of type X One can also ask about how to perform this building process. Also massless extremals (MEs) representing TGD view about topologically quantized classical radiation fields are minimal surfaces but now the induced Kähler form is non-vanishing. MEs can be also Lagrangian surfaces and seem to play fundamental role in morphogenesis and morphostasis as a generalization of Chladni mechanism. One might say that they represent the tools to assign material and magnetic flux tube structures at the nodal surfaces of MEs. MEs are the tools of space-time engineering. Here many-sheetedness is essential for having the TGD counterparts of standing waves. For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?. |