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Topological Geometrodynamics: an Overview

Note: Newest contributions are at the top!

Year 2016

Non-commutative imbedding space and strong form of holography

The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of non-commutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI).

Quantum group theorists have studied the idea that space-time coordinates are non-commutative and tried to construct quantum field theories with non-commutative space-time coordinates (see this). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor Jkl and uncertainty relation in linear M4 coordinates mk would look something like [mk, ml] = lP2Jkl, where lP is Planck length. This would be a direct generalization of non-commutativity for momenta and coordinates expressed in terms of symplectic form Jkl.

1+1-D case serves as a simple example. The non-commutativity of p and q forces to use either p or q. Non-commutativity condition reads as [p,q]= hbar Jpq and is quantum counterpart for classical Poisson bracket. Non-commutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian sub-manifold to which the projection of Jpq vanishes: coordinates become commutative in this sub-manifold. This condition can be formulated purely classically: wave function is defined in Lagrangian sub-manifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of "World of Classical Worlds" (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. "Quantization without quantization would have Wheeler stated it.

GCI poses however a problem if one wants to generalize quantum group approach from M4 to general space-time: linear M4 coordinates assignable to Lie-algebra of translations as isometries do not generalize. In TGD space-time is surface in imbedding space H=M4× CP2: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as space-time coordinates. The analog of symplectic structure J for M4 makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP2 has naturally symplectic form.

Could it be that the coordinates for space-time surface are in some sense analogous to symplectic coordinates (p1,p2,q1,q2) so that one must use either (p1,p2) or (q1,q2) providing coordinates for a Lagrangian sub-manifold. This would mean selecting a Lagrangian sub-manifold of space-time surface? Could one require that the sum Jμν(M4)+ Jμν(CP2) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2-surfaces. In special case also higher-D surfaces - even 4-D surfaces as products of Lagrangian 2-manifolds for M4 and CP2 are possible: they would correspond to homologically trivial cosmic strings X2× Y2⊂ M4× CP2, which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term.

But why this kind of restriction? In TGD one has strong form of holography (SH): 2-D string world sheets and partonic 2-surfaces code for data determining classical and quantum evolution. Could this projection of M4 × CP2 symplectic structure to space-time surface allow an elegant mathematical realization of SH and bring in the Planck length lP defining the radius of twistor sphere associated with the twistor space of M4 in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the non-uniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2-D surfaces.

The analog of brane hierarchy for the localization of spinors - space-time surfaces; string world sheets and partonic 2-surfaces; boundaries of string world sheets - is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian sub-manifolds of space-time in the sense that J(M4)+J(CP2)=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M4)+J(CP2)=0 at them. The vanishing of induced W boson fields is needed to guarantee well-defined em charge at string world sheets and that also this condition allow also 4-D solutions besides 2-D generic solutions. This condition is physically obvious but mathematically not well-understood: could the condition J(M4)+J(CP2)=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X2× Y2 would allow 4-D spinor modes. If the light-like 3-surface defining boundary between Minkowskian and Euclidian space-time regions is Lagrangian surface, the total induced Kähler form Chern-Simons term would vanish. The 4-D canonical momentum currents would however have non-vanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of space-time super-symmetries could be interpreted as addition of higher-D right-handed neutrino modes to the 1-fermion states assigned with the boundaries of string world sheets.

An alternative - but of course not necessarily equivalent - attempt to formulate this picture would be in terms of number theoretic vision. Space-time surfaces would be associative or co-associative depending on whether tangent space or normal space in imbedding space is associative - that is quaternionic. These two conditions would reduce space-time dynamics to associativity and commutativity conditions. String world sheets and partonic 2-surfaces would correspond to maximal commutative or co-commutative sub-manifolds of imbedding space. Commutativity (co-commutativity) would mean that tangent space (normal space as a sub-manifold of space-time surface) has complex tangent space at each point and that these tangent spaces integrate to 2-surface. SH would mean that data at these 2-surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2-surfaces intersecting partonic 2-surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD.

For background see the chapter Topological Geometrodynamics: Three Visions.

Antimatter as dark matter?

It has been found in CERN (see this ) that matter and antimatter atoms have no differences in the energies of their excited states. This is predicted by CPT symmetry. Notice however that CP and T can be separately broken and that this is indeed the case. Kaon is classical example of this in particle physics. Neutral kaon and anti-kaon behave slightly differently.

This finding forces to repeat an old question. Where does the antimatter reside? Or does it exist at all?

GUTs predicted that baryon and lepton number are not conserved separately and suggested a solution to the empirical absence of antimatter. GUTs have been however dead for years and there is actually no proposal for the solution of matter-antimatter asymmetry in the framework of mainstream theories (actually there are no mainstream theories after the death of superstring theories which also assumed GUTs as low energy limits!).

In TGD framework many-sheeted space-time suggests possible solution to the problem. Matter and antimatter are at different space-time sheets. One possibility is that antimatter corresponds to dark matter in TGD sense that is a phase with heff=n× h, n=1,2,3,... such that the value of n for antimatter is different from that for visible matter. Matter and antimatter would not have direct interactions and would interact only via classical fields or by emission of say photons by matter (antimatter) suffering a phase transition changing the value of heff before absorbtion by antimatter (matter). This could be rather rare process. Bio-photons could be produced from dark photons by this process and this is assumed in TGD based model of living matter.

What the value of n for ordinary visible matter could be? The naive guess is that it is n=1, the smallest possible value. Randell Mills has however claimed the existence of scaled down hydrogen atoms - Mills calls them hydrinos - with ground state binding energy considerably higher than for hydrogen atom. The experimental support for the claim is published in respected journals and the company of Mills is developing a new energy technology based on the energy liberated in the transition to hydrino state.

These findings can be understood in TGD framework if one has actually n=6 for visible atoms and n=1, 2, or 3 for hydrinos. Hydrino states would be stabilized in the presence of some catalysts. See this.

The model suggests a universal catalyst action. Among other things catalyst action requires that reacting molecule gets energy to overcome the potential barrier making reaction very slow. If an atom - say (dark) hydrogen - in catalyst suffers a phase transition to hydrino (hydrogen with smaller value of heff/h), it liberates binding energy, and if one of the reactant molecules receives it it can overcome the barrier. After the reaction the energy can be sent back and catalyst hydrino returns to the ordinary hydrogen state. The condition that the dark binding energy is above the thermal energy gives a condition on the value of heff/h=n as n≤ 32. The size scale of the dark largest allowed dark atom would be about 100 nm, 10 times the thickness of the cell membrane.

The notion of phosphate high energy bond is somewhat mysterious concept and manifests as the ability provide energy in ATP to ADP transition. There are claims that there is no such bond. I have spent considerable amount of time to ponder this problem. Could phosphate contain (dark) hydrogen atom able to go to the hydrino state (state with smaller value of heff/h) and liberate the binding energy? Could the decay ATP to ADP produce the original possibly dark hydrogen? Metabolic energy would be needed to kick it back to ordinary bond in ATP.

So: could it be that one has n=6 for stable matter and n is different from this for stable antimatter? Could the small CP breaking cause this?

For background see the chapter Classical TGD.

Minimal surface cosmology

Before the discovery of the twistor lift TGD inspired cosmology has been based on the assumption that vacuum extremals provide a good estimate for the solutions of Einstein's equations at GRT limit of TGD . One can find imbeddings of Robertson-Walker type metrics as vacuum extremals and the general finding is that the cosmological with super-critical and critical mass density have finite duration after which the mass density becomes infinite: this period of course ends before this. The interpretation would be in terms of the emergence of new space-time sheet at which matter represented by smaller space-time sheets suffers topological condensation. The only parameter characterizing critical cosmologies is their duration. Critical (over-critical) cosmologies having SO3× E3 (SO(4)) as isometry group is the duration and the CP2 projection at homologically trivial geodesic sphere S2: the condition that the contribution from S2 to grr component transforms hyperbolic 3-metric to that of E3 or S3 metric fixes these cosmologies almost completely. Sub-critical cosmologies have one-dimensional CP2 projection.

Do Robertson-Walker cosmologies have minimal surface representatives? Recall that minimal surface equations read as

Dα(gαββhkg1/2)= ∂α[gαββhk g1/2] + {αkm} gαββhm g1/2=0 ,

{αkm} ={l km} ∂αhl .

Sub-critical minimal surface cosmologies would correspond to X4⊂ M4× S1. The natural coordinates are Robertson-Walker coordinates, which co-incide with light-cone coordinates (a=[(m0)2-r2M]1/2, r= rM/a,θ, φ) for light-cone M4+. They are related to spherical Minkowski coordinates (m0,rM,θ,φ) by (m0=a(1+r2)1/2, rM= ar). β =rM/m0=r/(1+r2)1/20,rM). r corresponds to the Lorentz factor r= γ β=β/(1-β2)1/2

The metric of M4+ is given by the diagonal form [gaa=1, grr=a2/(1+r2), gθθ= a2r2, gφφ= a2r2sin2(θ)]. One can use the coordinates of M4+ also for X4.

The ansatz for the minimal surface reads is Φ= f(a). For f(a)=constant one obtains just the flat M4+. In non-trivial case one has gaa= 1-R2 (df/da)2. The gaa component of the metric becomes now gaa=1/(1-R2(df/da)2). Metric determinant is scaled by gaa1/2 =1 → (1-R2(df/da)21/2. Otherwise the field equations are same as for M4+. Little calculation shows that they are not satisfied unless one as gaa=1.

Also the minimal surface imbeddings of critical and over-critical cosmologies are impossible. The reason is that the criticality alone fixes these cosmologies almost uniquely and this is too much for allowing minimal surface property.

Thus one can have only the trivial cosmology M4+ carrying dark energy density as a minimal surface solution! This obviously raises several questions.

  1. Could Λ=0 case for which action reduces to Kähler action provide vacuum extremals provide single-sheeted model for Robertson-Walker cosmologies for the GRT limit of TGD for which many-sheeted space-time surface is replaced with a slightly curved region of M4? Could Λ=0 correspond to a genuine phase present in TGD as formal generalization of the view of mathematicians about reals as p=∞ p-adic number suggest. p-Adic length scale would be strictly infinite implying that Λ∝ 1/p vanishes.
  2. Second possibility is that TGD is quantum critical in strong sense. Not only 3-space but the entire space-time surface is flat and thus M4+. Only the local gravitational fields created by topologically condensed space-time surfaces would make it curved but would not cause smooth expansion. The expansion would take as quantum phase transitions reducing the value of Λ ∝ 1/p as p-adic prime p increases. p-Adic length scale hypothesis suggests that the preferred primes are near but below powers of 2 p≈ 2k for some integers k. This led for years ago to a model for Expanding Earth.
  3. This picture would explain why individual astrophysical objects have not been observed to expand smoothly (except possibly in these phase transitions) but participate cosmic expansion only in the sense that the distance to other objects increase. The smaller space-time sheets glued to a given space-time sheet preserving their size would emanate from the tip of M4+ for given sheet.
  4. RW cosmology should emerge in the idealization that the jerk-wise expansion by quantum phase transitions and reducing the value of Λ (by scalings of 2 by p-adic length scale hypothesis) can be approximated by a smooth cosmological expansion.
One should understand why Robertson-Walker cosmology is such a good approximation to this picture. Consider first cosmic redshift.
  1. The cosmic recession velocity is defined from the redshift by Doppler formula.

    z= (1+β)/(1-β)-1 ≈ β = v/c .

    In TGD framework this should correspond to the velocity defined in terms of the coordinate r of the object.

    Hubble law tells that the recession velocity is proportional to the proper distance D from the source. One has

    v= HD , H= (da/dt)/a= 1/(gaaa)1/2 .

    This brings in the dependence on the Robertson-Walker metric.

    For M4+ one has a=t and one would have gaa=1 and H=1/a. The experimental fact is however that the value of H is larger for non-empty RW cosmologies having gaa<1. How to overcome this problem?

  2. To understand this one must first understand the interpretation of gravitational redshift. In TGD framework the gravitational redshift is property of observer rather than source. The point is that the tangent space of the 3-surface assignable to the observer is related by a Lorent boost to that associated with the source. This implies that the four-momentum of radiation from the source is boosted by this same boost. Redshift would mean that the Lorentz boost reduces the momentum from the real one. Therefore redshift would be consistent with momentum conservation implied by Poincare symmetry.

    gaa for which a corresponds to the value of cosmic time for the observer should characterize the boost of observer relative to the source. The natural guess is that the boost is characterized by the value of gtt in sufficiently large rest system assignable to observer with t is taken to be M4 coordinate m0. The value of gtt fluctuates do to the presence of local gravitational fields. At the GRT limit gaa would correspond to the average value of gtt.

  3. There is evidence that H is not same in short and long scales. This could be understood if the radiation arrives along different space-time sheets in these two situations.
  4. If this picture is correct GRT description of cosmology is effective description taking into account the effect of local gravitation to the redshift, which without it would be just the M4+ redshift.
Einstein's equations for RW cosmology should approximately code for the cosmic time dependence of mass density at given slightly deformed piece of M4+ representing particular sub-cosmology expanding in jerkwise manner.
  1. Many-sheeted space-time implies a hierarchy of cosmologies in different p-adic length scales and with cosmological constant Λ ∝ 1/p so that vacuum energy density is smaller in long scale cosmologies and behaves on the average as 1/a2 where a characterizes the scale of the cosmology. In zero energy ontology given scale corresponds to causal diamond (CD) with size characterized by a defining the size scale for the distance between the tips of CD.
  2. For the comoving volume with constant value of coordinate radius r the radius of the volume increases as a. The vacuum energy would increase as a3 for comoving volume. This is in sharp conflict with the fact that the mass decreases as 1/a for radiation dominated cosmology, is constant for matter dominated cosmology, and is proportional to a for string dominated cosmology.

    The physical resolution of the problem is rather obvious. Space-time sheets representing topologically condensed matter have finite size. They do not expand except possibly in jerkwise manner but in this process Λ is reduced - in average manner like 1/a2.

    If the sheets are smaller than the cosmological space-time sheet in the scale considered and do not lose energy by radiation they represent matter dominated cosmology emanating from the vertex of M4+. The mass of the co-moving volume remains constant.

    If they are radiation dominated and in thermal equilibrium they lose energy by radiation and the energy of volume behaves like 1/a.

    Cosmic strings and magnetic flux tubes have size larger than that the space-time sheet representing the cosmology. The string as linear structure has energy proportional to a for fixed value of Λ as in string dominated cosmology. The reduction of Λ decreasing on the average like 1/a2 implies that the contribution of given string is reduced like 1/a on the average as in radiation dominated cosmology.

  3. GRT limit would code for these behaviours of mass density and pressure identified as scalars in GRT cosmology in terms of Einstein's equations. The time dependence of gaa would code for the density of the topologically condensed matter and its pressure and for dark energy at given level of hierarchy. The vanishing of covariant divergence for energy momentum tensor would be a remnant of Poincare invariance and give Einstein's equations with cosmological term.
  4. Why GRT limit would involve only the RW cosmologies allowing imbedding as vacuum extremals of Kähler action? Can one demand continuity in the sense that TGD cosmology at p→ ∞ limit corresponds to GRT cosmology with cosmological solutions identifiable as vacuum extremals? If this is assumed the earlier results are obtained. In particular, one obtains the critical cosmology with 2-D CP2 projection assumed to provide a GRT model for quantum phase transitions changing the value of Λ.
If this picture is correct, TGD inspired cosmology at the level of many-sheeted space-time would be extremely simple. The new element would be many-sheetedness which would lead to more complex description provided by GRT limit. This limit would however lose the information about many-sheetedness and lead to anomalies such as two Hubble constants.

See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title.

LIGO blackhole anomaly and minimal surface model for star

The TGD inspired model of star as a minimal surface with stationary spherically symmetric metric suggests strongly that the analog of blackhole metric as two horizons. The outer horizon is analogous to Scwartschild horizon in the sense that the roles of time coordinate and radial coordinate change. Radial metric component vanishes at Scwartschild horizon rather than divergence. Below the inner horizon the metric has Eucldian signature.

Is there any empirical evidence for the existence of two horizons? There is evidence that the formation of the recently found LIGO blackhole (discussed from TGD view point in is not fully consistent with the GRT based model (see this). There are some indications that LIGO blackhole has a boundary layer such that the gravitational radiation is reflected forth and back between the inner and outer boundaries of the layer. In the proposed model the upper boundary would not be totally reflecting so that gravitational radiation leaks out and gave rise to echoes at times .1 sec, .2 sec, and .3 sec. It is perhaps worth of noticied that time scale .1 sec corresponds to the secondary p-adic time scale of electron (characterized by Mersenne prime M127= 2127-1). If the minimal surface solution indeed has two horizons and a layer like structure between them, one might at least see the trouble of killing the idea that it could give rise to repeated reflections of gravitational radiation.

The proposed model (see this) assumes that the inner horizon is Schwarstchild horizon. TGD would however suggests that the outer horizon is the TGD counterpart of Schwartschild horizon. It could have different radius since it would not be a singularity of grr (gtt/grr would be finite at rS which need not be rS=2GM now). At rS the tangent space of the space-time surface would become effectively 2-dimensional: could this be interpreted in terms of strong holography (SH)?

One should understand why it takes rather long time T=.1 seconds for radiation to travel forth and back the distance L= rS-rE between the horizons. The maximal signal velocity is reduced for the light-like geodesics of the space-time surface but the reduction should be rather large for L∼ 20 km (say). The effective light-velocity is measured by the coordinate time Δ t= Δ m0+ h(rS)-h(rE) needed to travel the distance from rE to rS. The Minkowski time Δ m0-+ would be the from null geodesic property and m0= t+ h(r)

Δ m0-+ =Δ t -h(rS)+h(rE) ,

Δ t = ∫rErS(grr/gtt)1/2 dr== ∫rErS dr/c# .

The time needed to travel forth and back does not depend on h and would be given by

Δ m0 =2Δ t =2∫rErSdr/c# .

This time cannot be shorter than the minimal time (rS-rE)/c along light-like geodesic of M4 since light-like geodesics at space-time surface are in general time-like curves in M4. Since .1 sec corresponds to about 3× 104 km, the average value of c# should be for L= 20 km (just a rough guess) of order c#∼ 2-11c in the interval [rE,rS]. As noticed, T=.1 sec is also the secondary p-adic time assignable to electron labelled by the Mersenne prime M127. Since grr vanishes at rE one has c#→ ∞. c# is finite at rS.

There is an intriguing connection with the notion of gravitational Planck constant. The formula for gravitational Planck constant given by hgr= GMm/v0 characterizing the magnetic bodies topologically for mass m topologically condensed at gravitational magnetic flux tube emanating from large mass M. The interpretation of the velocity parameter v0 has remained open. Could v0 correspond to the average value of c#? For inner planets one has v0≈ 2-11 so that the the order of magnitude is same as for the the estimate for c#.

See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title.

Minimal surface counterpart of Reissner-Nordstöm solution

Occarm's razor have been used to debunk TGD. The following arguments provide the information needed by the reader to decide himself. Considerations at three levels.

The level of "world of classical worlds" (WCW) defined by the space of 3-surfaces endowed with Kähler structure and spinor structure and with the identification of WCW space spinor fields as quantum states of the Universe: this is nothing but Einstein's geometrization program applied to quantum theory. Second level is space-time level.

Space-time surfaces correspond to preferred extremals of Käction in M4× CP2. The number of field like variables is 4 corresponding to 4 dynamically independent imbedding space coordinates. Classical gauge fields and gravitational field emerge from the dynamics of 4-surfaces. Strong form of holography reduces this dynamics to the data given at string world sheets and partonic 2-surfaces and preferred extremals are minimal surface extremals of Kähler action so that the classical dynamics in space-time interior does not depend on coupling constants at all which are visible via boundary conditions only. Continuous coupling constant evolution is replaced with a sequence of phase transitions between phases labelled by critical values of coupling constants: loop corrections vanish in given phase. Induced spinor fields are localized at string world sheets to guarantee well-definedness of em charge.

At imbedding space level the modes of imbedding space spinor fields define ground states of super-symplectic representations and appear in QFT-GRT limit. GRT involves post-Newtonian approximation involving the notion of gravitational force. In TGD framework the Newtonian force correspond to a genuine force at imbedding space level.

For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?.

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.

  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.

  3. 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.
Amazingly, what emerges is the elementary geometry. My apologies for those who hated school geometry.

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 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.
  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.

  3. 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.

  4. 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.
  5. 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 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?

  1. 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.
  2. 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.
  3. 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.

For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?.

Can one apply Occam's razor as a general purpose debunking argument to TGD?

Occarm's razor have been used to debunk TGD. The following arguments provide the information needed by the reader to decide himself. Considerations at three levels.

The level of "world of classical worlds" (WCW) defined by the space of 3-surfaces endowed with Kählerstructure and spinor structure and with the identification of WCW space spinor fields as quantum states of the Universe: this is nothing but Einstein's geometrization program applied to quantum theory. Second level is space-time level.

Space-time surfaces correspond to preferred extremals of Kähler action in M4× CP2. The number of field like variables is 4 corresponding to 4 dynamically independent imbedding space coordinates. Classical gauge fields and gravitational field emerge from the dynamics of 4-surfaces. Strong form of holography reduces this dynamics to the data given at string world sheets and partonic 2-surfaces and preferred extremals are minimal surface extremals ofKähler action so that the classical dynamics in space-time interior does not depend on coupling constants at all which are visible via boundary conditions only. Continuous coupling constant evolution is replaced with a sequence of phase transitions between phases labelled by critical values of coupling constants: loop corrections vanish in given phase. Induced spinor fields are localized at string world sheets to guarantee well-definedness of em charge.

At imbedding space level the modes of imbedding space spinor fields define ground states of super-symplectic representations and appear in QFT-GRT limit. GRT involves post-Newtonian approximation involving the notion of gravitational force. In TGD framework the Newtonian force correspond to a genuine force at imbedding space level.

For background see the chapter Can one apply Occam's razor as a general purpose debunking argument to TGD?.

Critizing the view about elementary particles

The concrete model for elementary particles has developed gradually during years and is by no means final. In the recent model elementary particle corresponds to a pair of wormhole contacts and monopole flux runs between the throats of of the two contacts at the two space-time sheets and through the contacts between space-time sheets.

The first criticism relates to twistor lift of TGD. In the case of Kähler action the wormhole contacts correspond to deformations for pieces of CP2 type vacuum extremals for which the 1-D M4 projection is light-like random curve. Twistor lift adds to Kähler action a volume term proportional to cosmological constant and forces the vacuum extremal to be a minimal surface carrying non-vanishing light-like momentum (this is of course very natural): one could call this surface CP2 extremal. This implies that M4 projection is light-like geodesic: this is physically rather natural.

Twistor lift leads to a loss of the proposed space-time correlate of massivation used also to justify p-adic thermodynamics: the average velocity for a light-like random curve is smaller than maximal signal velocity - this would be a clear classical signal for massivation. One could however conjecture that the M4 projection for the light-like boundaries of string world sheets becomes light-like geodesic of M4× CP2 instead light-like geodesic of M4 and that this serves as the correlate for the massivation in 4-D sense.

Second criticism is that I have not considered in detail what the monopole flux hypothesis really means at the level of detail. Since the monopole flux is due to the CP2 topology, there must be a closed 2-surface which carries this flux. This implies that the flux tube cannot have boundaries at larger space-time surface but one has just the flux tube which closed cross section obtained as a deformation of a cosmic string like object X2× Y2, where X2 is minimal surface in M4 and Y2 a complex surface of CP2 characterized by genus. Deformation would have 4-D M4 projection instead of 2-D string world sheet.

Note: One can also consider objects for which the flux is not monopole flux: in this case one would have deformations of surfaces of type X2× S2, S2 homologically trivial geodesic sphere: these are non-vacuum extremals for the twistor lift of Kähler action (volume term). The net magnetic flux would vanish - as a matter fact, the induced Kähler form would vanish identically for the simplest situation. These objects might serve as correlates for gravitons since the induced metric is the only field degree of freedom. One could also have non-vanishing fluxes for flux tubes with disk-like cross section.

If this is the case, the elementary particles would be much simpler than I have though hitherto.

  1. Elementary particles would be simply closed flux tubes which look like very long flattened squares. Short sides with length of order CP2 radius would be identifiable as pieces of deformed CP2 type extremals having Euclidian signature of the induced metric. Long sides would be deformed cosmic strings with Minkowskian signature with apparent ends, which are light-like 3-surfaces at which the induced 4-metric is degenerate. Both Minkowskian and Euclidian regions of closed flux tubes would be accompanied by fermionic strings. These objects would topologically condense at larger space-time sheets with wormhole contacts that do not carry monopole flux: touching the larger space-time surface but not sticking to it.
  2. One could understand why the genus for all wormhole throats must be the same for the simplest states as the TGD explanation of family replication phenomenon demands. Of course, the change of the topology along string like object cannot be excluded but very probably corresponds to an unstable higher mass excitation.
  3. The basic particle reactions would include re-connections of closed string like objects and their reversals. The replication of 3-surfaces would remain a new element brought by TGD. The basic processes at fermionic level would be reconnections of closed fermionic strings. The new element would be the presence of Euclidian regions allowing to talk about effective boundaries of strings as boundaries between the Minkowskian or Euclidian regions. This would simplify enormously the description of particle reactions by bringing in description topologically highly analogous to that provided by closed strings.
  4. The original picture need not of course be wrong: it is only slightly more complex than the above proposal. One would have two space-time sheets connected by a pair of wormhole contacts between, which most of the magnetic flux would flow like in flux tube. The flux from the throat could emerges more or less spherically but eventually end up to the second wormhole throat. The sheets would be connected along their boundaries so that 3-space would be connected. The absence of boundary terms in the action implies this. The monopole fluxes would sum up to a vanishing flux at the boundary, where gluing of the sheets of the covering takes place.
There is a further question to be answered. Are the fermionic strings closed or not? Fermionic strings have certainly the Minkowskian portions ending at the light-like partonic orbits at Minkowskian-Euclidian boundaries. But do the fermionic strings have also Euclidian portions so that the signature of particle would be 2+2 kinks of a closed fermionic string? If strong for of holography is true in both Euclidian and Minkowskian regions, this is highly suggestive option.

If only Minkowskian portions are present, particles could be seen as pairs of open fermionic strings and the counterparts of open string vertices would be possible besides reconnection of closed strings. For this option one can also consider single fermionic open strings connecting wormhole contacts: now possible flux tube would not carry monopole flux.

For background see Particle Massivation in TGD Universe.

Does GRT really allow gravitational radiation?

In Facebook discussion Niklas Grebäck mentioned Weyl tensor and I learned something that I should have noticed long time ago. Wikipedia article lists the basic properties of Weyl tensor as the traceless part of curvature tensor, call it R. Weyl tensor C is vanishing for conformally flat space-times. In dimensions D=2,3 Weyl tensor vanishes identically so that they are always conformally flat: this obviously makes the dimension D=3 for space very special. Interestingly, one can have non-flat space-times with nonvanishing Weyl tensor but the vanishing Schouten/Ricci/Einstein tensor and thus also with vanishing energy momentum tensor.

The rest of curvature tensor R can be expressed in terms of so called Kulkarni-Nomizu product P• g of Schouten tensor P and metric tensor g: R=C+P• g, which can be also transformed to a definition of Weyl tensor using the definition of curvature tensor in terms of Christoffel symbols as the fundamental definition. Kulkarni-Nomizu product • is defined as tensor product of two 2-tensors with symmetrization with respect to first and second index pairs plus antisymmetrization with respect to second and fourth indices.

Schouten tensor P is expressible as a combination of Ricci tensor Ric defined by the trace of R with respect to the first two indices and metric tensor g multiplied by curvature scalar s (rather than R in order to use index free notation without confusion with the curvature tensor). The expression reads as

P= 1/(D-2)×[Ric-(s/2(D-1))×g] .

Note that the coefficients of Ric and g differ from those for Einstein tensor. Ricci tensor and Einstein tensor are proportional to energy momentum tensor by Einstein equations relate to the part.

Weyl tensor is assigned with gravitational radiation in GRT. What I see as a serious interpretational problem is that by Einstein's equations gravitational radiation would carry no energy and momentum in absence of matter. One could argue that there are no free gravitons in GRT if this interpretation is adopted! This could be seen as a further argument against GRT besides the problems with the notions of energy and momentum: I had not realized this earlier.

Interestingly, in TGD framework so called massless extremals (MEs) (see this and this) are four-surfaces, which are extremals of Kähler action, have Weyl tensor equal to curvature tensor and therefore would have interpretation in terms of gravitons. Now these extremals are however non-vacuum extremals.

  1. Massless extremals correspond to graphs of possibly multi-valued maps from M4 to CP2. CP2 coordinates are arbitrary functions of variables u=k• m and w= ε • m (here "•" denotes M4 inner product). k is light-like wave vector and ε space-like polarization vector orthogonal to k so that the interpretation in terms of massless particle with polarization is possible. ME describes in the most general case a wave packet preserving its shape and propagating with maximal signal velocity along a kind of tube analogous to wave guide so that they are ideal for precisely targeted communications and central in TGD inspired quantum biology. MEs do not have Maxwellian counterparts. For instance, MEs can carry light-like gauge currents parallel to them: this is not possible in Maxwell's theory.
  2. I have discussed a generalization of this solution ansatz so that the directions defined by light-like vector k and polarization vector ε orthogonal to it are not constant anymore but define a slicing of M4 by orthogonal curved surfaces (analogs of string world sheets and space-like surfaces orthogonal to them). MEs in their simplest form at least are minimal surfaces and actually extremals of practically any general coordinate invariance action principle. For instance, this is the case if the volume term suggested by the twistorial lift of Kähler action (see this) and identifiable in terms of cosmological constant is added to Kähler action.
  3. MEs carry non-trivial induced gauge fields and gravitational fields identified in terms of the induced metric. I have identified them as correlates for particles, which correspond to pairs of wormhole contacts between two space-times such that at least one of them is ME. MEs would accompany to both gravitational radiation and other forms or radiation classically and serve as their correlates. For massless extremals the metric tensor is of form

    g= m+ a ε⊗ ε+ b k⊗ k + c(ε⊗ kv +k⊗ ε) ,

    where m is the metric of empty Minkowski space. The curvature tensor is necessarily quadrilinear in polarization vector ε and light-like wave vector k (light-like ifor both M4 and ME metric) and from the general expression of Weyl tensor C in terms of R and g it is equal to curvature tensor: C=R.

    Hence the interpretation as graviton solution conforms with the GRT interpretation. Now however the energy momentum tensor for the induced Kähler form is non-vanishing and bilinear in velocity vector k and the interpretational problem is avoided.

What is interesting that also at GRT limit cosmological constant saves gravitons from reducing to vacuum solutions. The deviation of the energy density given by cosmological term from that for Minkowski metric is identifiable as gravitonic energy density. The mysterious cosmological constant would be necessary for making gravitons non-vacuum solutions. The value of graviton amplitude would be determined by the continuity conditions for Einstein's equations with cosmological term. The p-adic evolution of cosmological term predicted by TGD is however difficult to understand in GRT framework.

See the article Does GRT really allow gravitational radiation?. For background see the chapter Classical TGD.

Three reasons for the localization of induced spinor fields at string world sheets

There are now three good reasons for the modes of the induced spinor fields to be localized to 2-D string world sheets and partonic 2-surfaces - in fact, to the boundaries of string world sheets at them defining fermionic world lines. I list these three good reasons in the same order as I became aware of them.

  1. The first good reason is that this condition allows spinor modes to have well-defined electromagnetic charges - the induced classical W boson fields and perhaps also Z field vanish at string world sheets so that only em field and possibly Z field remain and one can have eigenstates of em charge.
  2. Second good reason actually a set of closely related good reasons. First, strong form of holography implied by the strong form of general coordinate invariance demands the ocalization: string world sheets and partonic 2-surfaces are "space-time genes". Also twistorial picture follows naturally if the locus for the restriction of spinor modes at the light-like orbits of partonic 2-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian is 1-D fermion world line. Thanks to holography fermions behave like point like particles, which are massless in 8-D sense. Thirdly, conformal invariance in the fermionic sector demands the localization.
  3. The third good reason emerges from the mathematical problem of field theories involving fermions: also in the models of condensed matter systems this problem is also encountered - in particular, in the models of high Tc superconductivity. For instance, AdS/CFT correspondence involving 10-D blackholes has been proposed as a solution - the reader can decide whether to take this seriously.

    Fermionic path integral is the source of problems. It can be formally reduced to the analog of partition function but the Boltzman weights (analogous to probabilities) are not necessary positive in the general case and this spoils the stability of the numerical computation. One gets rid of the sign problem if one can diagonalize the Hamiltonian, but this problem is believed to be NP-hard in the generic case. A further reason to worry in QFT context is that one must perform Wick rotation to transform action to Hamiltonian and this is a trick. It seems that the problem is much more than a numerical problem: QFT approach is somehow sick.

    The crucial observation giving the third good reason is that this problem is encountered only in dimensions D≥3 - not in dimensions D=1,2! No sign problem in TGD where second quantized fundamental fermions are at string world sheets!

A couple of comments are in order.
  1. Although the assumption about localization 2-D surfaces might have looked first a desperate attempt to save em charge, it now seems that it is something very profound. In TGD approach standard model and GRT emerge as an approximate description obtained by lumping the sheets of the many-sheeted space-time together to form a slightly curved region of Minkowski space and by identifying gauge potentials and gravitational field identified as sums of those associated with the sheets lumped together. The more fundamental description would not be plagued by the mathematical problem of QFT approach .
  2. Although fundamental fermions as second quantized induced spinor fields are 2-D character, it is the modes of the classical imbedding space spinor fields - eigenstates of four-momentum and standard model quantum numbers - that define the ground states of the super-conformal representations. It is these modes that correspond to the 4-D spinor modes of QFT limit. What goes wrong in QFT is that one assigns fermionic oscillator operators to these modes although second quantization should be carried out at deeper level and for the 2-D modes of the induced spinor fields: 2-D conformal symmetry actually makes the construction of these modes trivial.
To conclude, the condition that the theory is computable would pose a powerful condition on the theory. As a matter fact, this is not a new finding. The mathematical existence of Kähler geometry of "world of classical worlds" fixes its geometry more or less uniquely and therefore also the physics: one obtains a union of symmetric spaces labelled by zero modes of the metric and for symmetric space all points (now 3-surfaces) are geometrically equivalent meaning a gigantic simplification allowing to handle the infinite-dimensional case. Even for loop spaces the Kähler geometry is unique and has infinite-dimensional isometry group (Kac-Moody symmetries).

For details see the chapter Why TGD and What TGD is?.

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