Space-time engineering from space-time 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.

1. Physical objects correspond to space-time surfaces of finite size – we see directly the non-trivial topology of space-time in everyday length scales.

2. 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.
3. 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.

Simplest legos

What could be the simplest surfaces of this kind – legos?

1. 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.

2. 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.

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.

1. 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 S²_I resp. S²_II. The geodesic sub-manifolds cen be products G₃ =G_D1× G_D2, D₂=3-D₁ of geodesic manifolds G_D1, D₁=1,2,3 for E³ and G_D2, D₂=0,1,2 for CP₂.

2. 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.

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.

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 S²_I resp. S²_II can be used so that also non-vanishing Kähler forms are obtained.

The basic legos are now D× S²_i, 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).

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× S²_i. 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.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.