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 M4 projection this gives rise to what I have called many-sheeted spacetime. Sheets are deformations of canonically imbedded M4 extremely near to each other (the maximal distance is determined by CP2 size scale about 104 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.
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 CP2: 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 E3. In CP2 one has points, circles, geodesic spheres, and CP2 itself.
- CP2 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.
Geodesic minimal surfaces with vanishing induced gauge fields
Consider first static objects with 1-D CP2 projection having thus vanishing induced gauge fields. These objects are of form M1× X3, X3⊂ E3× CP2. M1 corresponds to time-like or possible light-like geodesic (for CP2 type extremals). I will consider mostly Minkowskian space-time regions in the following.
- Quite generally, the simplest legos consist of 3-D geodesic sub-manifolds of E3× CP2. For E3 their dimensions are D=1,2,3 and for CP2, D=0,1,2. CP2 allows both homologically non-trivial resp. trivial geodesic sphere S2I resp. S2II. The geodesic sub-manifolds cen be products G3 =GD1× GD2, D2=3-D1 of geodesic manifolds GD1, D1=1,2,3 for E3 and GD2, D2=0,1,2 for CP2.
- It is also possible to have twisted geodesic sub-manifolds G3 having geodesic circle S1 as CP2 projection corresponding to the geodesic lines of S1⊂ CP2, whose projections to E3 and CP2 are geodesic line and geodesic circle respectively. The geodesic is characterized by S1 wave vector. One can have this kind of geodesic lines even in M1× E3× S1 so that the solution is characterized also by frequency and is not static in CP2 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 X2× D as local structures (thin layer in E3). One can perform twisting by replacing D with twisted closed geodesics in D× S1: this gives valued map from D to S1 (subset CP2) representing geodesic line of D× S1. This geodesic sub-manifold is trivially a minimal surface and defines a two-sheeted cover of X2× D. Wormhole contact pairs (elementary particles) between the sheets can be used to stabilize this structure.
- Structures of form D2× S1, where D2 is polygon, are perhaps the simplest building bricks for more complex structures. There are continuity conditions at vertices and edges at which polygons D2i 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.
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 CP2 combined with 1-D geodesic lines of E2. Actually both homologically non-trivial resp. trivial geodesic spheres S2I resp. S2II can be used so that also non-vanishing Kähler forms are obtained.
The basic legos are now D× S2i, i=I,II and they can be combined with the basic legos constructed above. These legos correspond to two kinds of magnetic flux tubes in the ideal infinitely thin limit. There are good reasons to expected that these infinitely thin flux tubes can be thickened by deforming them in E3 directions orthogonal to D. These structures could be used as basic building bricks assignable to the edges of the tensor networks in TGD.
Static minimal surfaces, which are not geodesic sub-manifolds
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 M1× X2× S1, S1 ⊂ CP2 a geodesic line and X2 minimal surface in E3.
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 E3 suggests the role of minimal surface for instance in bio-chemistry (see this).
The surfaces with constant positive curvature do not allow imbedding as minimal surfaces in E3. Corals provide an example of surface consisting of pieces of 2-D hyperbolic space H2 immersed in E3 (see this). Minimal surfaces have negative curvature as also H2 but minimal surface immersions of H2 do not exist. Note that pieces of H2 have natural imbedding to E3 realized as light-one proper time constant surface but this is not a solution to the problem.
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 M4 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 E3× S1 rather than E3. 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 H2 and S2 to E3 to minimal surfaces in E3× S1? 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.
Dynamical minimal surfaces: how space-time manages to engineer itself?
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 X2× S2i. As a matter fact, S2 can be replaced with complex sub-manifold of CP2.
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.
See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? of "Towards M-matrix" or article with the same title.
For a summary of earlier postings see Latest progress in TGD.
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