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Towards M-Matrix

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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 About twistor lift of 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?.

Delicacies of the induced spinor structure and SUSY mystery

The discussion of induced spinor structure leads to a modification of an earlier idea (one of the many) about how SUSY could be realized in TGD in such a manner that experiments at LHC energies could not discover it and one should perform experiments at the other end of energy spectrum at energies which correspond to the thermal energy about .025 eV at room temperature. I have the feeling that this observation could be of crucial importance for understanding of SUSY.

The notion of induced spinor field deserves a more detailed discussion. Consider first induced spinor structures.

  1. Induced spinor field are spinors of M4× CP2 for which modes are characterized by chirality (quark or lepton like) and em charge and weak isospin.
  2. Induced spinor spinor structure involves the projection of gamma matrices defining induced gamma matrices. This gives rise to superconformal symmetry if the action contains only volume term.

    When Kähler action is present, superconformal symmetry requires that the modified gamma matrices are contractions of canonical momentum currents with imbedding space gamma matrices. Modified gammas appear in the modified Dirac equation and action, whose solution at string world sheets trivializes by super-conformal invariance to same procedure as in the case of string models.

  3. Induced spinor fields correspond to two chiralities carrying quark number and lepton number. Quark chirality does not carry color as spin-like quantum number but it corresponds to a color partial wave in CP2 degrees of freedom: color is analogous to angular momentum. This reduces to spinor harmonics of CP2 describing the ground states of the representations of super-symplectic algebra.

    The harmonics do not satisfy correct correlation between color and electroweak quantum numbers although the triality t=0 for leptonic waves and t=1 for quark waves. There are two manners to solve the problem.

    1. Super-symplectic generators applied to the ground state to get vanishing ground states weight instead of the tachyonic one carry color and would give for the physical states correct correlation: leptons/quarks correspond to the same triality zero(one partial wave irrespective of charge state. This option is assumed in p-adic mass calculations.
    2. Since in TGD elementary particles correspond to pairs of wormhole contacts with weak isospin vanishing for the entire pair, one must have pair of left and right-handed neutrinos at the second wormhole throat. It is possible that the anomalous color quantum numbers for the entire state vanish and one obtains the experimental correlation between color and weak quantum numbers. This option is less plausible since the cancellation of anomalous color is not local as assume in p-adic mass calculations.
The understanding of the details of the fermionic and actually also geometric dynamics has taken a long time. Super-conformal symmetry assigning to the geometric action of an object with given dimension an analog of Dirac action allows however to fix the dynamics uniquely and there is indeed dimensional hierarchy resembling brane hierarchy.
  1. The basic observation was following. The condition that the spinor modes have well-defined em charge implies that they are localized to 2-D string world sheets with vanishing W boson gauge fields which would mix different charge states. At string boundaries classical induced W boson gauge potentials guarantee this. Super-conformal symmetry requires that this 2-surface gives rise to 2-D action which is area term plus topological term defined by the flux of Kähler form.
  2. The most plausible assumption is that induced spinor fields have also interior component but that the contribution from these 2-surfaces gives additional delta function like contribution: this would be analogous to the situation for branes. Fermionic action would be accompanied by an area term by supersymmetry fixing modified Dirac action completely once the bosonic actions for geometric object is known. This is nothing but super-conformal symmetry.

    One would actually have the analog of brane-hierarchy consisting of surfaces with dimension D= 4,3,2,1 carrying induced spinor fields which can be regarded as independent dynamical variables and characterized by geometric action which is D-dimensional analog of the action for Kähler charged point particle. This fermionic hierarchy would accompany the hierarchy of geometric objects with these dimensions and the modified Dirac action would be uniquely determined by the corresponding geometric action principle (Kähler charged point like particle, string world sheet with area term plus Kähler flux, light-like 3-surface with Chern-Simons term, 4-D space-time surface with Kähler action).

  3. This hierarchy of dynamics is consistent with SH only if the dynamics for higher dimensional objects is induced from that for lower dimensional objects - string world sheets or maybe even their boundaries orbits of point like fermions. Number theoretic vision suggests that this induction relies algebraic continuation for preferred extremals. Note that quaternion analyticity means that quaternion analytic function is determined by its values at 1-D curves.
  4. Quantum-classical correspondences (QCI) requires that the classical Noether charges are equal to the eigenvalues of the fermionic charges for surfaces of dimension D=0,1,2,3 at the ends of the CDs. These charges would not be separately conserved. Charges could flow between objects of dimension D+1 and D - from interior to boundary and vice versa. Four-momenta and also other charges would be complex as in twistor approach: could complex values relate somehow to the finite life-time of the state?

    If quantum theory is square root of thermodynamics as ZEO suggests, the idea that particle state would carry information also about its life-time or the time scale of CD to which is associated could make sense. For complex values of αK there would be also flow of canonical and super-canonical momentum currents between Euclidian and Minkowskian regions crucial for understand gravitational interaction as momentum exchange at imbedding space level.

  5. What could be the physical interpretation of the bosonic and fermionic charges associated with objects of given dimension? Condensed matter physicists assign routinely physical states to objects of various dimensions: is this assignment much more than a practical approximation or could condensed matter physics already be probing many-sheeted physics?

From this one ends up to the possibility of identifying the counterpart of SUSY in TGD framework. There are several options to consider.

  1. The analog of brane hierarchy is realized also in TGD. Geometric action has parts assignable to 4-surface, 3-D light like regions between Minkowskian and Euclidian regions, 2-D string world sheets, and their 1-D boundaries. They are fixed uniquely. Also their fermionic counterparts - analogs of Dirac action - are fixed by super-conformal symmetry. Elementary particles reduce so composites consisting of point-like fermions at boundaries of wormhole throats of a pair of wormhole contacts.

    This forces to consider 3 kinds of SUSYs! The SUSYs associated with string world sheets and space-time interiors would be broken since there is a mixing between M4 chiralities in the modified Dirac action. The mass scale of the broken SUSY would correspond to the length scale of these geometric objects and one might argue that decoupling between the degrees of freedom considered occurs at high energies and explains why no evidence for SUSY has been observed at LHC. Also the fact that the addition of massive fermions at these dimensions can be interpreted differently. 3-D light-like 3-surfaces would be however an exception.

  2. For 3-D light-like surfaces the modified Dirac action associated with the Chern-Simons term does not mix M4 chiralities (signature of massivation) at all since modified gamma matrices have only CP2 part in this case. All fermions can have well-defined chirality. Even more: the modified gamma matrices have no M4 part in this case so that these modes carry no four-momentum - only electroweak quantum numbers and spin. Obviously, the excitation of these fermionic modes would be an ideal manner to create spartners of ordinary particles consting of fermion at the fermion lines. SUSY would be present if the spin of these excitations couples - to various interactions and would be exact in absence of coupling to interior spinor fields.

    What would be these excitations? Chern-Simons action and its fermionic counterpart are non-vanishing only if the CP2 projection is 3-D so that one can use CP2 coordinates. This strongly suggests that the modified Dirac equation demands that the spinor modes are covariantly constant and correspond to covariantly constant right-handed neutrino providing only spin.

    If the spin of the right-handed neutrino adds to the spin of the particle and the net spin couples to dynamics, N=2 SUSY is in question. One would have just action with unbroken SUSY at QFT limit? But why also right-handed neutrino spin would couple to dynamics if only CP2 gamma matrices appear in Chern-Simons-Dirac action? It would seem that it is independent degree of freedom having no electroweak and color nor even gravitational couplings by its covariant constancy. I have ended up with just the same SUSY-or-no-SUSY that I have had earlier.

  3. Can the geometric action for light-like 3-surfaces contain Chern-Simons term?
    1. Since the volume term vanishes identically in this case, one could indeed argue that also the counterpart of Kähler action is excluded. Moreover, for so called massless extremals of Kähler action reduces to Chern-Simons terms in Minkowskian regions and this could happen quite generally: TGD with only Kähler action would be almost topological QFT as I have proposed. Volume term however changes the situation via the cosmological constant. Kähler-Dirac action in the interior does not reduce to its Chern-Simons analog at light-like 3-surface.
    2. The problem is that the Chern-Simons term at the two sides of the light-like 3-surface differs by factor (-1)1/2 coming from the ratio of (g4)1/2 factors which themselves approach to zero: one would have the analog of dipole layer. This strongly suggests that one should not include Chern-Simons term at all.

      Suppose however that Chern-Simons terms are present at the two sides and αK is real so that nothing goes through the horizon forming the analog of dipole layer. Both bosonic and fermionic degrees of freedom for Euclidian and Minkowskian regions would decouple completely but currents would flow to the analog of dipole layer. This is not physically attractive.

      The canonical momentum current and its super counterpart would give fermionic source term ΓnΨint,+/- in the modified Dirac equation defined by Chern-Simons term at given side +/-: +/- refers to Minkowskian/Euclidian part of the interior. The source term is proportional to ΓnΨint,+/- and Γn is in principle mixture of M4 and CP2 gamma matrices and therefore induces mixing of M4 chiralities and therefore also 3-D SUSY breaking. It must be however emphasized that Γn is singular and one must be consider the limit carefully also in the case that one has only continuity conditions. The limit is not completely understood.

    3. If αK is complex, there is coupling between the two regions and the simplest assumption has been that there is no Chern-Simons term as action and one has just continuity conditions for canonical momentum current and hits super counterpart.
    The cautious conclusion is that 3-D Chern-Simons term and its fermionic counterpart are absent.
  4. What about the addition of fermions at string world sheets and interior of space-time surface (D=2 and D=4). For instance, in the case of hadrons D=2 excitations could correspond to addition of quark in the interior of hadronic string implying additional states besides the states obtained assuming only quarks at string ends. Let us consider the interior (D=4). The smallness of cosmological constant implies that the contribution to the four-momentum from interior should be rather small so that an interpretation in terms of broken SUSY might make sense. There would be mass m∼ .03 eV per volume with size defined by the Compton scale hbar/m. Note however that cosmological constant has spectrum coming as inverse powers of prime so that also higher mass scales are possible.

    This interpretation might allow to understand the failure to find SUSY at LHC. Sparticles could be obtained by adding interior right-handed neutrinos and antineutrinos to the particle state. They could be also associated with the magnetic body of the particle. Since they do not have color and weak interactions, SUSY is not badly broken. If the mass difference between particle and sparticle is of order m=.03 eV characterizing ρvac, particle and sparticle could not be distinguished in higher energy physics at LHC since it probes much shorter scales and sees only the particle. I have already earlier proposed a variant of this mechanism but without SUSY breaking.

    To discover SUSY one should do very low energy physics in the energy range m∼ .03 eV having same order of magnitude as thermal energy kT= 2.6× 10-2 eV at room temperature 25 oC. One should be able to demonstrate experimentally the existence of sparticle with mass differing by about m∼ .03 eV from the mass of the particle (one cannot of course exclude higher mass values if Λ has spectrum). An interesting question is whether the sfermions associated with standard fermions could give rise to Bose-Einstein condensates whose existence in the length scale of large neutron is strongly suggested by TGD view about living matter.

See the chapter Does the QFT Limit of TGD Have Space-Time Super-Symmetry?.

About minimal surface extremals of Kähler action

If the spectrum for the critical value of Kähler coupling strength is complex - say given by the complex zeros of zeta - the preferred extremals of Kähler action are minimal surfaces. This means that they satisfy simultaneously the field equations associated with two variational principles.

Conservation laws for the minimal surface extremals of Kähler action

Consider first the basic conservation laws.

  1. Complex value of αK means that conserved quantities are complex: this brings strongly in mind twistor approach. The value of cosmological constant is assumed to be real. There are two separate local conservations laws associated with the volume term and Kähler action respectively in both Minkowskian and Euclidian regions. This need not mean separate global conservation laws in Minkowskian and Euclidian regions. If there is non canonical momentum current between Minkowskian (M) and Euclidian (E) space-time regions the real and imaginary parts of conserved quantum numbers correspond schematically to the sums

    l Re(Q)= Re(1/αK)QK(E) + Im(1/αK)QK(M) +ρvac QV(M)

    Im(Q)=Im(1/αK)QK(E) + Re(1/αK)QK(M) .

    Here the subscripts V and K refer to the volume term and Kähler action respectively.

  2. If the canonical momentum current vanishes there both real and imaginary parts decompose to two separately conserved parts.

    Re(Q1)= Re(1/αK)QK(E) ,

    Re(Q2)= Im(1/αK)QK(M) +ρvac QV(M) ,

    Im(Q1)= Im(1/αK)QK(E) ,

    Im(Q2)= Re(1/αK)QK(M) .

    This looks strange and the natural assumption is that canonical momentum currents can flow between the Euclidian and Minkowskian regions and boundary conditions equate the components of normal currents at both sides.

Are minimal surface extremals of Kähler action holomorphic surfaces in some sense?

I have considered several ansätze for the general solutions of the field equations for the preferred extremals. One proposal is that preferred extremals as 4-surfaces of imbedding space with octonionic tangent space structure have quaternionic tangent space or normal space (so called M8-H duality). Second proposal is that preferred extremals can be seen as quaternion analytic surfaces. Third proposal relies on a fusion of complex and hyper-complex structures to what I call Hamilton-Jacobi structure. In Euclidian regions this would correspond to complex structure. Twistor approach suggests that the condition that the twistor lift of the space-time surface to a 6-D surface in the product of twistor spaces of M4 and CP2 equals to the twistor space of CP2. This proposal is highly interesting since twistor lift works only for M4× CP2. The intuitive picture is that the field equations are integrable and all these views might be consistent.

Preferred extremals of Kähler action as minimal surfaces would be a further proposal. Can one make conclusions about general form of solutions assuming that one has minimal surface extremals of Kähler action?

In D=2 case minimal surfaces are holomorphic surfaces or they hyper-complex variants and the imbedding space coordinates can be expressed as complex-analytic functions of complex coordinate or a hypercomplex analog of this. Field equations stating the vanishing of the trace gαβHkαβ if the second fundamental form Hkαβ== Dα&partial;βhk are satisfied because the metric is tensor of type (1,1) and second fundamental form of type (2,0) ⊕ (2,0). Field equations reduce to an algebraic identity and functions involved are otherwise arbitrary functions. The constraint comes from the condition that metric is of form (1,1) as holomorphic tensor.

This raises the question whether this finding generalizes to the level of 4-D space-time surfaces and perhaps allows to solve the field equations exactly in coordinates generalizing the hypercomplex coordinates for string world sheet and complex coordinates for the partonic 2-surface.

The known non-vacuum extremals of Kähler action are actually minimal surfaces. The common feature suggested already earlier to be common for all preferred extremals is the existence of generalization of complex structure.

  1. For Minkowskian regions this structure would correspond to what I have called Hamilton-Jacobi structure. The tangent space of the space-time surface X4 decomposes to local direct sum T(X4)= T(X2)⊕ T(Y2), where the 2-D tangent places T(X2) and T(Y2) define an integrable distribution integrating to a decomposition X4=X2× Y2. The complex structure is generalized to a direct some of hyper-complex structure in X2 meaning that there is a local light-like direction defining light-like coordinate u and its dual v. Y2 has complex complex coordinate (w,wbar). Minkowski space M4 has similar structure. It is still an open question whether metric decomposes to a direct sum of orthogonal metrics assignable to X2 and Y2 or is the most general analog of complex metric in question. guv and gwbar are certainly non-vanishing components of the induced metric. Metric could allow as non-vanishing components also guw and gvbarw. This slicing by pairs of surfaces would correspond to decomposition to a product of string world sheet and partonic 2-surface everywhere.

    In Euclidian regions ne would have 4-D complex structure with two complex coordinates (z,w) and their conjugates and completely analogous decompositions. In CP2 one has similar complex structure and actually Kähler structure extending to quaternionic structure. I have actually proposed that quaternion analyticity could provide the general solution of field equations.

  2. Assuming minimal surface property the field equations for Kähler action reduce to the vanishing of a sum of two terms. The first term comes from the variation with respect to the induced metric and is proportional to the contraction

    A=Jαγ JγβHkαβ .

    Second term comes from the variation with respect to induced Kähler form and is proportional to

    B=jα PksJslαhl .

    Here Pkl is projector to the normal space of space-time surface and jα= Dβ Jαβ is the conserved Kähler current.

    For the known extremals j vanishes or is light-like (for massless extremals) in which case A and B vanish separately.

  3. An attractive manner to satisfy field equations would be by assuming that the situation for 2-D minimal surface generalizes so that minimal surface equations are identically satisfied. Extremal property for Kähler action could be achieved by requiring that energy momentum tensor also for Kähler action is of type (1,1) so that one would have A=0. This implies jααsk=0. This can be true if j vanishes or is light-like as it is for the known extremals and if sk depend only on the light-like coordinate. In Euclidian regions one would have j=0.
  4. The proposed generalization is especially interesting in the case of cosmic string extremals of form X2× Y2, where X2⊂ M4 is minimal surface (string world sheet) and Y2 is complex homologically non-trivial sub-manifold of CP2 carrying Kähler magnetic charge. The generalization would be that the two transversal coordinates (w,wbar) in the plane orthogonal to the string world sheet defining polarization plane depend holomorphically on the complex coordinates of complex surface of CP2. This would transform cosmic string to flux tube.
  5. There are also solutions of form X2× Y2, where Y2 is Lagrangian sub-manifold of CP2 with vanishing Kähler magnetic charge and their deformations with (w,barw) depending on the complex coordinates of Y2 (see the slides of "On Lagrangian minimal surfaces on the complex projective plane" ). In this case Y2 is not complex sub-manifold of CP2 with arbitrary genus and induced Kähler form vanishes. The simplest choice for Y2 would be as homologically trivial geodesic sphere. Because of its 2-dimensionality Y2 has a complex structure defined by its induced metric so that solution ansatz makes sense also now.
See the new chapter About twistor lift of TGD or the article with the same title.

Does cosmological term in twistor action makes Kähler coupling genuine coupling parameter also classically?

The addition of the volume term to Kähler action has very nice interpretation as a generalization of equations of motion for a world-line extended to a 4-D space-time surface. The field equations generalize in the same manner for 3-D light-like surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian, for 2-D string world sheets, and for their 1-D boundaries defining world lines at the light-like 3-surfaces. For 3-D light-like surfaces the volume term is absent. Either light-like 3-surface is freely choosable in which case one would have Kac-Moody symmetry as gauge symmetry or that the extremal property for Chern-Simons term fixes the gauge.

The known non-vacuum extremals are minimal surface extremals of Kähler action and it might well be that the preferred extremal property realizing SH quite generally demands this. The addition of the volume term could however make Kähler coupling strength a manifest coupling parameter also classically when the phases of Λ and αK are same. Therefore quantum criticality for Λ and αK would have a precise local meaning also classically in the interior of space-time surface. The equations of motion for a world line of U(1) charged particle would generalize to field equations for a "world line" of 3-D extended particle.

This is an attractive idea consistent with standard wisdom but one can invent strong objections against it in TGD framework.

  1. All known non-vacuum extremals of Kähler action are minimal surfaces and the minimal surface vacuum extremals of Kähler action become non-vacuum extremals. This suggest that preferred extremals are minimal surface extremals of Kähler action so that the two dynamics apparently decouple. Minimal surface extremals are analogs for geodesics in the case of point-like particles: one might say that one has only gravitational interaction. This conforms with SH stating that gauge interactions at boundaries (orbits of partonic 2-surfaces and 2-surfaces at the ends of CD) correspond classically to the gravitational dynamics in the space-time interior.

    Note that at the boundaries of the string world sheets at light-like 3-surfaces the situation is different: one has equations of motion for geodesic line coupled to induce Kähler gauge potential and gauge coupling indeed appears classically as one might expect! For string world sheets one has only the topological magnetic flux term and minimal surface equation in string world sheet. Magnetic flux term gives the Kähler coupling at the boundary.

  2. Decoupling would allow to realize number theoretical universality since the field equations would not depend on coupling parameters at all. It is very difficult to imagine how the solutions could be expressible in terms of rational functions with coefficients in algebraic extension of rationals unless αK and Λ have very special relationship. If they have different phases, minimal surface extremals of Kähler action are automatically implied. If the values of αK correspond to complex zeros of Riemann ζ, also Λ should have same complex phase, in order to have genuine classical coupling. This looks somewhat un-natural but cannot be excluded.

    The most natural option is that Λ is real and αK corresponds to zeros of zeta. For trivial zeros the phases are different and decoupling occurs. For trivial zeros Λ and αK differ by imaginary unit so that again decoupling occurs.

  3. One can argue that the decoupling makes it impossible to understand coupling constant evolution. This is not the case. The point is that the classical charges assignable to super-symplectic algebra are sums over contributions from Kähler action and volume term and therefore depend on the coupling parameters. Their vanishing conditions for sub-algebra and its commutator with entire algebra give boundary conditions on preferred extremals so that coupling constant evolution creeps in classically!

    The condition that the eigenvalues of fermionic charge operators are equal to the classical charges brings in the dependence of quantum charges on coupling parameters. Since the elements of scattering matrix are expected to involve as building bricks the matrix elements of super-symplectic algebra and Kac-Moody algebra of isometry charges, one expectes that discrete coupling constant evolution creeps in also quantally via the boundary conditions for preferred extremals.

Although the above arguments seem to kill the idea that the dynamics of Kähler action and volume term could couple in space-time interior, one can compare this view (Option II) with the view based on complete decoupling (Option I).
  1. For Option I the coupling between the two dynamics could be induced just by the condition that the space-time surface becomes an analog of geodesic line by arranging its interior so that the U(1) force vanishes! This would generalize Chladni mechanism! The interaction would be present but be based on going to the nodal surfaces! Also the dynamics of string world sheets is similar: if the string sheets carry vanishing W boson classical fields, em charge is well-defined and conserved. One would also avoid the problems produced by large coupling constant between the two-dynamics present already at the classical level. At quantum level the fixed point property of quantum critical couplings would be the counterparts for decoupling.
  2. For Option II the coupling is of conventional form. When cosmological constant is small as in the scale of the known Universe, the dynamics of Kähler action is perturbed only very slightly by the volume term. The alternative view is that minimal surface equation has a very large perturbation proportional to the inverse of Λ so that the dynamics of Kähler action could serve as a controller of the dynamics defined by the volume term providing a small push or pull now and then. Could this sensitivity relate to quantum criticality and to the view about morphogenesis relying on Chladni mechanism in which field patterns control the dynamics with charged flux tubes ending up to the nodal surfaces of (Kähler) electric field (see this)? Magnetic flux tubes containing dark matter would in turn control and serve as template for the dynamics of ordinary matter.
Could the possible coupling of the two dynamics suggest any ideas about the values of αK and Λ at quantum criticality besides the expectation that cosmological constant is proportional to an inverse of p-adic prime?
  1. Number theoretic vision suggests the existence of preferred extremals represented by rational functions with rational or algebraic coefficients in preferred coordinates. For Option I one has preferred extremals of Kähler action which are minimal surfaces so that there is no coupling and no constraints on the ratio of couplings emerges: even better, both dynamics are independent of the coupling. All known non-vacuum extremals of Kähler action are indeed also minimal surfaces. For Option II the ratio of the coefficients Λ/8π G and 1/4παK should be rational or at most algebraic number. One must be however very cautious here: the minimal option allowed by strong form of holography is that the rational functions of proposed kind emerge only at the level of partonic 2-surfaces and string world sheets.
  2. I have proposed that that the inverse of Kähler coupling strength has spectrum coming as zeros of zeta or their imaginary parts (see this). The phases of complexified 1/αK and Λ/2G must be same in order to avoid the decoupling of Kähler action and minimal surface term implying minimal surface extremals of Kähler action.

    This conjecture is consistent with the rational function property only if αK and vacuum energy density ρvac appearing as the coefficient of volume term are proportional to the same possibly transcendental number with proportionality coefficient being an algebraic or rational number.

    If the phases are not identical (say Λ is real and one allows complex zeros) one has Option I and effective decoupling occurs. The coupling (Option2)) can occur for the trivial zeros of zeta if the volume term has coefficient iΛ/8πG rather than Λ/8π G to guarantee same phase as for 1/4παK. The coefficient iΛ/8πG would give in Minkowskian regions large real exponent of volume and this looks strange. In this case also number theoretical universality might make sense but SH would be broken in the sense that the space-time surfaces would not be analogous to geodesic lines.

  3. At quantum level number theoretical universality requires that the exponent of the total action defining vacuum functional reduces to the product of roots of unity and exponent of integer existing in finite-dimensional extension of p-adic numbers. This would suggest that total action reduces to a number of form q1+iq2π, qi rational number, so that its exponent is of the required form. Whether this can conform with the properties of zeros of zeta and properties of extremals is not clear.
ZEO suggests deep connections with the basic phenomenology of particle physics, quantum consciousness theory, and quantum biology and one can look the situation for both these options.
  1. Option I: Decoupling of the dynamics of Kähler action and volume term in space-time interior for all values of coupling parameters.
  2. Option II: Coupling of dynamics for trivial zeros of zeta and Λ→ iΛ.
Particle physics perspective

Consider a typical particle physics experiment. There are incoming and outgoing free particles moving along geodesics, these particles interact, and emanate as free particles from the interaction volume. This phenomenological picture does not follow from quantum field theory but is put in by hand, in particular the idea about interaction couplings becoming non-zero is involved. Also the role of the observer remains poorly understood.

The motion of incoming and outgoing particles is analogous to free motion along geodesic lines with particles generalized to 3-D extended objects. For both options these would correspond to the preferred extremals in the complement of CD within larger CD representing observer or measurement instrument. Decoupling would take place. In the interaction volume interactions are "coupled on" and particles interact inside the volume characterized by causal diamond (CD). What could be the TGD view translation of this picture?

  1. For Option I one would still have decoupling and the interpretation would be in terms of twistor picture in which one always has also in the internal lines on mass shell particles but with complex four-momenta. In TGD framework the momenta would be always complex due to the contribution of Euclidian regions defining the lines of generalized scattering diagrams. As explained coupling constant evolution can be understood also in this case and also classical dynamics depends on coupling parameters via the boundary conditions. The transitory period (control action) leading to the decoupled situation would be replaced by state function reduction, possibly to the opposite boundary.
  2. For Option II the transitory period would correspond to the coupling between the two classical dynamics and would take place inside CD after a phase transition identifiable as "big state function reduction" to time reversed mode. The problem is that in the interacting phase αK would not have a value approximately equal to the U(1) coupling strength of weak interactions (see this) so that the physical picture breaks down.
Quantum measurement theory in ZEO.
  1. For Option I state preparation and state function reduction would be in symmetric role. Also now there would be inherent asymmetry between zero energy states and their time reversals. With respect to observer the time reversed period would be invisible.
  2. For Option II state preparation for CD would correspond to a phase transition to a time reversed phase labelled by a trivial zero of zeta and Λ→ iΛ. In state function reduction to the original boundary of CD a phase transition to a phase labelled by non-trivial zero of zeta would occur and final state of free particles would emerge. The phase transitions would thus mean hopping from the critical line of zeta to the real axis and back and change the values of αK and possibly Λ. There would be strong breaking in time reversal symmetry.

    One cannot of course take this large asymmetry as an adhoc assumption: it should be induced by the presence of larger CD, which could also affect quite generally the values of αK and Λ (having also a spectrum of values).

TGD inspired theory of consciousness

What happens within sub-CD could be fundamental for the understanding of directed attention and sensory-motor cycle.

  1. The target of directed attention would correspond to the volume of CD - call it c - within larger CD - call it C representing the observer - attendee having c as part of its perceptive field. c would serve as a target of directed attention of C and thus define part of the perceptive field of c. c would correspond also to sub-self giving rise to a mental image of C. This would also allow to understand why the attention is directed rather than being completely symmetric with respect to C and c. For both options directed attention would correspond to sub-self c interpreted as mental image. There would be no difference.
  2. Quite generally, the self and time-reversed self could be seen as sensory input and motor response (Libet's findings). Directed attention would define the sensory input and sub-self could react to it by dying and re-incarnating as time-reversed subself. The two selves would correspond to sensory input and motor action following it as a reaction. Motor reaction would be sensory mental image in reversed time direction experienced by time reversed self. Only the description for the reaction would differ for the two options.

    The motor action would be time-reversed sensory perception for Option I. For Option II motor action would correspond to a different phase in which Kähler action and volume term couple classically.

TGD inspired quantum biology

The free geodesic line dynamics with vanishing U(1) Kähler force indeed brings in mind the proposed generalization of Chladni mechanism generating nodal surfaces at which charged magnetic flux tubes are driven (see this).

  1. For Option I the interiors of all space-time surfaces would be analogous to nodal surfaces and state function reductions would correspond to transition periods between different nodal surfaces. The decoupling would be dynamics of avoidance and could highly analogous to Chladni mechanism.
  2. For Option II the phase labelled by trivial zeros of zeta would correspond to period during which nodal surfaces are formed. This view about state function reduction and preparation as phase transitions in ZEO would provide classical description for the transition to the phase without direct interactions.
To sum up, it seems that the complete decoupling of the two dynamics (Option I) is favored by both SH, realization of preferred extremal property (perhaps as minimal surface extremals of Kähler action, number theoretical universality, discrete coupling constant evolution, and generalization of Chladni mechanism to a dynamics of avoidance.

For background see the new chapter About twistor lift of TGD or the article with the same title.

About twistor lift of TGD

The twistor lift of classical TGD is attractive physically but it is still unclear whether it satisfies all constraints. The basic implication of twistor lift would be the understanding of gravitational and cosmological constants. Cosmological constant removes the infinite vacuum degeneracy of Kähler action but because of the extreme smallness of cosmological constant Λ playing the role of inverse of gauge coupling strength, the situation for nearly vacuum extremals of Kähler action in the recent cosmology is non-perturbative. Cosmological constant and thus twistor lift make sense only in zero energy ontology (ZEO) involving causal diamonds (CDs) in an essential manner.

One motivation for introducing the hierarchy of Planck constants was that the phase transition increasing Planck constant makes possible perturbation theory in strongly interacting system. Nature itself would take care about the converge of the perturbation theory by scaling Kähler coupling strength αK to αK/n, n=heff/h. This hierarchy might allow to construct gravitational perturbation theory as has been proposed already earlier. This would for gravitation to be quantum coherent in astrophysical and even cosmological scales.

In this chapter this picture is studied in detail. The first interesting finding is that allowing Kähler coupling strength αK to correspond to zeros of zeta implies that for complex zeros the preferred extremals are mimimal surface extremals of Kähler action so that the values of coupling constants do not matter. The dynamics of Kägler action and volume term couple only for real zeros. This leads to an interpretation with profound implications for the views about what happens in particle physics experiment and in quantum measurement, for consciousness theory and for quantum biology. Second observation is that a fundamental length scale of biology - size scale of neuron and axon - would correspond to the p-adic length scale assignable to vacuum energy density assignable to cosmological constant and be therefore a fundamental physics length scale.

See the new chapter About twistor lift of TGD? or the article with the same title.

Still about induced spinor fields and TGD counterpart for Higgs

The understanding of the modified Dirac equation and of the possible classical counterpart of Higgs field in TGD framework is not completely satisfactory. The emergence of twistor lift of Kähler action inspired a fresh approach to the problem and it turned out that a very nice understanding of the situation emerges.

More precise formulation of the Dirac equation for the induced spinor fields is the first challenge. The well-definedness of em charge has turned out to be very powerful guideline in the understanding of the details of fermionic dynamics. Although induced spinor fields have also a part assignable space-time interior, the spinor modes at string world sheets determine the fermionic dynamics in accordance with strong form of holography (SH).

The well-definedness of em charged is guaranteed if induced spinors are associated with 2-D string world sheets with vanishing classical W boson fields. It turned out that an alternative manner to satisfy the condition is to assume that induced spinors at the boundaries of string world sheets are neutrino-like and that these string world sheets carry only classical W fields. Dirac action contains 4-D interior term and 2-D term assignable to string world sheets. Strong form of holography (SH) allows to interpret 4-D spinor modes as continuations of those assignable to string world sheets so that spinors at 2-D string world sheets determine quantum dynamics.

Twistor lift combined with this picture allows to formulate the Dirac action in more detail. Well-definedness of em charge implies that charged particles are associated with string world sheets assignable to the magnetic flux tubes assignable to homologically non-trivial geodesic sphere and neutrinos with those associated with homologically trivial geodesic sphere. This explains why neutrinos are so light and why dark energy density corresponds to neutrino mass scale, and provides also a new insight about color confinement.

A further important result is that the formalism works only for imbedding space dimension D=8. This is due the fact that the number of vector components is the same as the number of spinor components of fixed chirality for D=8 and corresponds directly to the octonionic triality.

p-Adic thermodynamics predicts elementary particle masses in excellent accuracy without Higgs vacuum expectation: the problem is to understand fermionic Higgs couplings. The observation that CP2 part of the modified gamma matrices gives rise to a term mixing M4 chiralities contain derivative allows to understand the mass-proportionality of the Higgs-fermion couplings at QFT limit.

See the chapter Higgs or something else?.

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 From Principles to Diagrams.

What happens to the extremals of Kähler action when volume term is introduced?

What happens to the extremals of Kähler action when volume term is introduced?

  1. The known non-vacuum extremals such as massless extremals (topological light rays) and cosmic strings are minimal surfaces so that they remain extremals and only the classical Noether charges receive an additional volume term. In particular, string tension is modified by the volume term. Homologically non-trivial cosmic strings are of form X2× Y2, where X2⊂ M4 is minimal surface and Y2⊂ CP2 is complex 2-surface and therefore also minimal surface.
  2. Vacuum degeneracy is in general lifted and only those vacuum extremals, which are minimal surfaces survive as extremals.
For CP2 type vacuum extremals the roles of M4 and CP2 are changed. M4 projection is light-like curve, and can be expressed as mk=fk(s) with light-likeness conditions reducing to Virasoro conditions. These surfaces are isometric to CP2 and have same Kähler and symplectic structures as CP2 itself. What is new as compared to GRT is that the induced metric has Euclidian signature. The interpretation is as lines of generalized scattering diagrams. The addition of the volume term forces the random light-like curve to be light-like geodesic and the action becomes the volume of CP2 in the normalization provided by cosmological constant. What looks strange is that the volume of any CP2 type vacuum extremals equals to CP2 volume but only the extremal with light-like geodesic as M4 projection is extremal of volume term.

Consider next vacuum extremals, which have vanishing induced Kähler form and are thus have CP2 projection belonging to at most 2-D Lagrangian manifold of CP2.

  1. Vacuum extremals with 2-D projections to CP2 and M4 are possible and are of form X2× Y2, X2 arbitrary 2-surface and Y2 a Lagrangian manifold. Volume term forces X2 to be a minimal surface and Y2 is Lagrangian minimal surface unless the minimal surface property destroys the Lagrangian character.

    If the Lagrangian sub-manifold is homologically trivial geodesic sphere, one obtains string like objects with string tension determined by the cosmological constant alone.

    Do more general 2-D Lagrangian minimal surfaces than geodesic sphere exist? For general Kähler manifold there are obstructions but for Kähler-Einstein manifolds such as CP2, these obstructions vanish (see this ). The case of CP2 is also discussed in the slides "On Lagrangian minimal surfaces on the complex projective plane" (see this). The discussion is very technical and demonstrates that Lagrangian minimal surfaces with all genera exist. In some cases these surfaces can be also lifted to twistor space of CP2.

  2. More general vacuum extremals have 4-D M4 projection. Could the minimal surface condition for 4-D M4 projection force a deformation spoiling the Lagrangian property? The physically motivated expectation is that string like objects give as deformations magnetic flux tubes for which string is thicknened so that it has a 2-D cross section. This would suggest that the deformations of string like objects X2× Y2, where Y2 is Lagrangian minimal surface, give rise to homologically trivial magnetic flux tubes. In this case Kähler magnetic field would vanish but the spinor connection of CP2 would give rise to induced magnetic field reducing to some U(1) subgroup of U(2). In particular, electromagnetic magnetic field could be present.
  3. p-Adically Λ behaves like 1/p as also string tension. Could hadronic string tension be understood also in terms of cosmological constant in hadronic p-adic length scale for strings if one assumes that cosmological constant for given space-time sheet is determined by its p-adic length scale?
The so called Maxwell phase which would correspond to small perturbations of M4 is also possible for 4-D Kähler action. For the twistor lift the volume term makes this phase possible. Maxwell phase is highly interesting since it corresponds to the intuitive view about what QFT limit of TGD could be.
  1. The field equations are a generalization of massless field equations for fields identifiable as CP2 coordinates and with a coupling to the deviation of the induced metric from M4 metric. It representes very weak perturbation. Hence the linearized field equations are expected to be an excellent approximation. The general challenge would be however the construction of exact solutions. One should also understand the conditions defining preferred extremals and stating that most of symplectic Noether charges vanish at the ends of space-time surface about boundaries of CD.
  2. Maxwell phase is the TGD analog for the perturbative phase of gauge theories. The smallness of the cosmological constant in cosmic length scales would make the perturbative approach useless in the path integral formulation. In TGD approach the path integral is replaced by functional integral involving also a phase but also now the small value of cosmological constant is a problem in long length scales. As proposed, the hierarchy of Planck constants would provide the solution to the problem.
  3. The value of cosmological constant behaving like Λ ∝ 1/p as the function of p-adic prime could be in short p-adic length scales large enough to allow a converging perturbative expansion in Maxwellian phase. This would conform with the idea that Planck constant has its ordinary value in short p-adic length scales.
  4. Does Maxwell phase allow extremals for which the CP2 projection is 2-D Lagrangian manifold - say a perturbation of a minimal Lagrangian manifold? This perturbation could be seen also as an alternative view about thickened minimal Lagrangian string allowing also M4 coordinates as local coordinates. If the projection is homologically trivial geodesic sphere this is the case. Note that solutions representable as maps M4→ CP2 are also possible for homologically non-trivial geodesic sphere and involve now also the induced Kähler form.
  5. The simplest deformations of canonically imbedded M4 are of form Φ= k• m, where Φ is an angle coordinate of geodesic sphere. The induced metric in M4 coordinates reads as gkl= mkl-R2kkkl and is flat and in suitably scaled space-time coordinates reduces to Minkowski metric or its Euclidian counterpart. kk is proportional to classical four-momentum assignable to the dark energy. The four-momentum is given by

    Pk = A× hbar kk ,

    A=[Vol(X3)/L4Λ] × (1+2x/1+x) ,

    x= R2k2 .

    Here kk is dimensionless since the the coordinates mk are regarded as dimensionless.

  6. There are interesting questions related to the singularities forced by the compactness of CP2. Eguchi-Hanson coordinates (r,θ,Φ,Ψ) (see this) allow to get grasp about what could happen.

    For the cyclic coordinates Ψ and Φ periodicity conditions allow to get rid of singularities. One can however have n-fold coverings of M4 also now.

    (r,θ) correspond to canonical momentum type canonical coordinates. Both of them correspond to angle variables (r/(1+r2)1/2 is essentially sine function). It is convenient to express the solution in terms of trigonometric functions of these angle variables. The value of the trigonometric function can go out of its range [-1,1] at certain 3-surface so that the solution ceases to be well-defined. The intersections of these surfaces for r and θ are 2-D surfaces. Many-sheeted space-time suggests a possible manner to circumvent the problem by gluing two solutions along the 3-D surfaces at which the singularities for either variable appear. These surfaces could also correspond to the ends of the space-time surface at the boundaries of CD or to the light-like orbits of the partonic 2-surfaces.

    Could string world sheets and partonic 2-surfaces correspond to the singular 2-surfaces at which both angle variables go out of their allowed range. If so, 2-D singularities would code for data as assumed in strong form of holography (SH). SH brings strongly in mind analytic functions for which also singularities code for the data. Quaternionic analyticity which makes sense would indeed suggest that co-dimension 2 singularities code for the functions in absence of 3-D counterpart of cuts (light-like 3-surfaces?)

  7. A more general picture might look like follows. Basic objects come in two classes. Surfaces X2× Y2, for which Y2 is either homologically non-trivial complex minimal 2-surface of CP2 of Lagrangian minimal surface. The perturbations of these two surfaces would also produce preferred extremals, which look locally like perturbations of M4. Quaternionic analyticity might be shared by both solution types. Singularities force many-sheetedness and strong form of holography.
Cosmological constant is expected to obey p-adic evolution and in very early cosmology the volume term becomes large. What are the implications for the vacuum extremals representing Robertson-Walker metrics having arbitrary 1-D CP2 projection?
  1. The TGD inspired cosmology involves primordial phase during a gas of cosmic strings in M4 with 2-D M4 projection dominates. The value of cosmological constant at that period could be fixed from the condition that homologically trivial and non-trivial cosmic strings have the same value of string tension. After this period follows the analog of inflationary period when cosmic strings condense are the emerging 4-D space-time surfaces with 4-D M4 projection and the M4 projections of cosmic strings are thickened. A fractal structure with cosmic strings topologically condensed at thicker cosmic strings suggests itself.
  2. GRT cosmology is obtained as an approximation of the many-sheeted cosmology as the sheets of the many-sheeted space-time are replaced with region of M4, whose metric is replaced with Minkowski metric plus the sum of deformations from Minkowski metric for the sheet. The vacuum extremals with 4-D M4 projection and arbitrary 1-D projection could serve as an approximation for this GRT cosmology. Note however that this representability is not required by basic principles.
  3. For cosmological solutions with 1-D CP2 projection minimal surface property forces the CP2 projection to belong to a geodesic circle S1. Denote the angle coordinate of S1 by Φ and its radius by R. For the future directed light-cone M4+ use the Robertson-Walker coordinates (a=(m02-rM2)1/2, r=arM, θ, φ), where (m0, rM, θ, φ) are spherical Minkowski coordinates. The metric of M4+ is that of empty cosmology and given by ds2 = da2-a22, where Ω2 denotes the line element of hyperbolic 3-space identifiable as the surface a=constant.

    One can can write the ansatz as a map from M4+ to S1 given by Φ= f(a) . One has gaa=1→ gaa= 1-R2(df/da)2. The field equations are minimal surface equations and the only non-trivial equation is associated with Φ and reads d2f/da2=0 giving Φ= ω a, where ω is analogous to angular velocity. The metric corresponds to a cosmology for which mass density goes as 1/a2 and the gravitational mass of comoving volume (in GRT sense) behaves is proportional to a and vanishes at the limit of Big Bang smoothed to "Silent whisper amplified to rather big bang for the critical cosmology for which the 3-curvature vanishes. This cosmology is proposed to results at the limit when the cosmic temperature approaches Hagedorn temperature.

  4. The TGD counterpart for inflationary cosmology corresponds to a cosmology for which CP2 projection is homologically trivial geodesic sphere S2 (presumably also more general Lagrangian (minimal) manifolds are allowed). This cosmology is vacuum extremal of Kähler action. The metric is unique apart from a parameter defining the duration of this period serving as the TGD counterpart for inflationary period during which the gas of string like objects condensed at space-time surfaces with 4-D M4 projection. This cosmology could serve as an approximate representation for the corresponding GRT cosmology.

    The form of this solution is completely fixed from the condition that the induced metric of a=constant section is transformed from hyperbolic metric to Euclidian metric. It should be easy to check whether this condition is consistent with the minimal surface property.

See the chapter From Principles to diagrams of "Towards M-Matrix" or the article About twistor lift of TGD.

Eigenstates of Yangian co-algebra generators as a manner to generate maximal entanglement?

Negentropically entangled objects are key entities in TGD inspired theory of consciousness and in the construction of tensor networks and the challenge is to understand how these could be constructed and what their properties could be. These states are diametrically opposite to unentangled eigenstates of single particle operators, usually elements of Cartan algebra of symmetry group. The entangled states should result as eigenstates of poly-local operators. Yangian algebras involve a hierarchy of poly-local operators, and twistorial considerations inspire the conjecture that Yangian counterparts of super-symplectic and other algebras made poly-local with respect to partonic 2-surfaces or end-points of boundaries of string world sheet at them are symmetries of quantum TGD. Could Yangians allow to understand maximal entanglement in terms of symmetries?

  1. In this respect the construction of maximally entangled states using bi-local operator Qz=Jx⊗ Jy - Jy⊗ Jx is highly interesting since entangled states would result by state function. Single particle operator like Jz would generate un-entangled states. The states obtained as eigenstates of this operator have permutation symmetries. The operator can be expressed as Qz=fzijJi⊗ Jj, where fABC are structure constants of SU(2) and could be interpreted as co-product associated with the Lie algebra generator Jz. Thus it would seem that unentangled states correspond to eigenstates of Jz and the maximally entangled state to eigenstates of co-generator Qz. Kind of duality would be in question.
  2. Could one generalize this construction to n-fold tensor products? What about other representations of SU(2)? Could one generalize from SU(2) to arbitrary Lie algebra by replacing Cartan generators with suitably defined co-generators and spin 1/2 representation with fundamental representation? The optimistic guess would be that the resulting states are maximally entangled and excellent candidates for states for which negentropic entanglement is maximized by NMP.
  3. Co-product is needed and there exists a rich spectrum of algebras with co-product (quantum groups, bialgebras, Hopf algebras, Yangian algebras). In particular, Yangians of Lie algebras are generated by ordinary Lie algebra generators and their co-generators subject to constraints. The outcome is an infinite-dimensional algebra analogous to one half of Kac-Moody algebra with the analog of conformal weight N counting the number of tensor factors. Witten gives a nice concrete explanation of Yangian for which co-generators of TA are given as QA= ∑i<j fABC TBi ⊗ TCj, where the summation is over discrete ordered points, which could now label partonic 2-surfaces or points of them or points of string like object. For a practically totally incomprehensible description of Yangian one can look at the Wikipedia article .
  4. This would suggest that the eigenstates of Cartan algebra co-generators of Yangian could define an eigen basis of Yangian algebra dual to the basis defined by the totally unentangled eigenstates of generators and that the quantum measurement of poly-local observables defined by co-generators creates entangled and perhaps even maximally entangled states. A duality between totally unentangled and completely entangled situations is suggestive and analogous to that encountered in twistor Grassmann approach where conformal symmetry and its dual are involved. A beautiful connection between generalization of Lie algebras, quantum measurement theory and quantum information theory would emerge.

For details see the chapter From Principles to Diagrams or the article with the same title.

Could N=2 super-conformal algebra be relevant for TGD?

The concrete realization of the super-conformal symmetry (SCS) in TGD framework has remained poorly understood. In particular, the question how SCS relates to super-conformal field theories (SCFTs) has remained an open question. The most general super-conformal algebra assignable to string world sheets by strong form of holography has N equal to the number of 4+4 =8 spin states of leptonic and quark type fundamental spinors but the space-time SUSY is badly broken for it. Covariant constancy of the generating spinor modes is replaced with holomorphy - kind of "half covariant constancy". I have considered earlier a proposal that N=4 SCA could be realized in TGD framework but given up this idea. Right-handed neutrino and antineutrino are excellent candidates for generating N=2 SCS with a minimal breaking of the corresponding space-time SUSY. Covariant constant neutrino is an excellent candidate for the generator of N=2 SCS. The possibility of this SCS in TGD framework will be considered in the sequel.

1. Questions about SCS in TGD framework

This work was inspired by questions not related to N=2 SCS, and it is good to consider first these questions.

1. 1 Could the super-conformal generators have conformal weights given by poles of fermionic zeta?

The conjecture (see this) is that the conformal weights for the generators super-symplectic representation correspond to the negatives of h= -ksk of the poles sk fermionic partition function ζF(ks)=ζ(ks)/ζ(2ks) defining fermionic partition function. Here k is constant, whose value must be fixed from the condition that the spectrum is physical. ζ(ks) defines bosonic partition function for particles whos energies are given by log(p), p prime. These partition functions require complex temperature but is completely sensible in Zero Energy Ontology (ZEO), where thermodynamics is replaced with its complex square root.

For non-trivial zeros 2ks=1/2+iy of ζ(2ks) s would correspond pole s= (1/2+iy)/2k of zF(ks). The corresponding conformal weights would be h=(-1/2-iy)/2k. For trivial zeros 2ks=-2n, n=1,2,.. s=-n/k would correspond to conformal weights h=n/k>0. Conformal confinement is assumed meaning that the sum of imaginary parts of of generators creating the state vanishes.

What can one say about the value of k? The pole of ζ(ks) at s=1/k would correspond to pole and conformal weight h=-1/k. For k=1 the trivial conformal weights would be positive integers h=1,2,...: this certainly makes sense. This gives for the real part for non-trivial conformal weights h=-1/4. By conformal confinement both pole and its conjugate belong to the state so that this contribution to conformal weight is negative half integers: this is consistent with the facts about super-conformal representations. For the ground state of super-conformal representation the conformal weight for conformally confined state would be h=- K/2. In p-adic mass calculations one would have K=6 (see this) .

The negative ground state conformal weights of particles look strange but p-adic mass calculations require that the ground state conformal weights of particles are negative: h=-3 is required.

1.2 What could be the origin of negative ground state conformal weights?

Super-symplectic conformal symmetries are realized at light-cone boundary and various Hamiltonians defined analogs of Kac-Moody generators are proportional functions f(rM)HJ,m HA, where HJ,m correspond to spherical harmonics at the 2-sphere RM=constant and HA is color partial wave in CP2, f(rM) is a partial wave in radial light-like coordinate which is eigenstate of scaling operator L0=rMd/dRM and has the form (rM/r0)-h, where h is conformal weight which must be of form h=-1/2+iy.

To get plane wave normalization for the amplitudes

(rM/r0) h=(rM/r0)-1/2exp(iyx) ,

x=log(rM/r0) ,

one must assume h=-1/2+iy. Together with the invariant integration measure drM this gives for the inner product of two conformal plane waves exp(iyix), x=log(rM/r0) the desired expression ∫ exp[iy1-y2)x] dx= δ(y1-y2), where dx= drM/rM is scaling invariance integration measure. This is just the usual inner product of plane waves labelled by momenta yi.

If rM/r0 can be identified as a coordinate along fermionic string (this need not be always the case) one can interpret it as real or imaginary part of a hypercomplex coordinate at string world sheet and continue these wave functions to the entire string world sheets. This would be very elegant realization of conformal invariance.

1.3. How to relate degenerate representations with h>0 to the massless states constructed from tachyonic ground states with negative conformal weight?

This realization would however suggest that there must be also an interpretation in which ground states with negative conformal weight hvac=-k/2 are replaced with ground states having vanishing conformal weights hvac=0 as in minimal SCAs and what is regarded as massless states have conformal weights h= -hvac>0 of the lowest physical state in minimal SCAs.

One could indeed start directly from the scaling invariant measure drM/rM rather than allowing it to emerge from drM. This would require in the case of p-adic mass calculations that has representations satisfying Virasoro conditions for weight h=-hvac>0. p-Adic mass squared would be now shifted downwards and proportional to L0+hvac. There seems to be no fundamental reason preventing this interpretation. One can also modify scaling generator L0 by an additive constant term and this does not affect the value of c. This operation corresponds to replacing basis {zn} with basis {zn+1/2}.

What makes this interpretation worth of discussing is that the entire machinery of conformal field theories with non-vanishing central charge and non-vanishing but positive ground state conformal weight becomes accessible allowing to determine not only the spectrum for these theories but also to determine the partition functions and even to construct n-point functions in turn serving as basic building bricks of S-matrix elements (see this) .

ADE classification of these CFTs in turn suggests at connection with the inclusions of hyperfinite factors and hierarchy of Planck constants. The fractal hierarchy of broken conformal symmetries with sub-algebra defining gauge algebra isomorphic to entire algebra would give rise to dynamic symmetries and inclusions for HFFs suggest that ADE groups define Kac-Moody type symmetry algebras for the non-gauge part of the symmetry algebra.

2. Questions about N=2 SCS

N=2 SCFTs has some inherent problems. For instance, it has been claimed that they reduce to topological QFTs. Whether N=2 can be applied in TGD framework is questionable: they have critical space-time dimension D=4 but since the required metric signature of space-time is wrong.

2.1 Inherent problems of N=2 SCS

N=2 SCS has some severe inherent problems.

  1. N=2 SCS has critical space-time dimension D=4, which is extremely nice. On the other, N=2 requires that space-time should have complex structure and thus metric signature (4,0), (0,4) or (2,2) rather than Minkowski signature. Similar problem is encountered in twistorialization and TGD proposal is Hamilton-Jacobi structure (se the appendix of (see this), which is hybrid of hypercomplex structure and Kähler structure. There is also an old proposal by Pope et al (see this) that one can obtain by a procedure analogous to dimensional reduction N=2 SCS from a 6-D theory with signature (3,3). The lifting of Kähler action to twistor space level allows the twistor space of M4 to have this signature and the degrees of freedom of the sphere S2 are indeed frozen.
  2. There is also an argument by Eguchi that N=2 SCFTs reduce under some conditions to mere topological QFTs (see this). This looks bad but there is a more refined argument that N=2 SCFT transforms to a topological CFT only by a suitable twist (see this). This is a highly attractive feature since TGD can be indeed regarded as almost topological QFT. For instance, Kähler action in Minkowskian regions could reduce to Chern-Simons term for a very general solution ansatz. Only the volume term having interpretation in terms of cosmological constant (see this) (extremely small in recent cosmology) would not allow this kind of reduction. The topological description of particle reactions based on generalized Feynman diagrams identifiable in terms of space-time regions with Euclidian signature of the induced metric would allow to build n-point functions in the fermionic sector as those of a free field theory. Topological QFT in bosonic degrees of freedom would correspond naturally to the braiding of fermion lines.

2.2 Can one really apply N=2 SCFTs to TGD?

TGD version of SCA is gigantic as compared to the ordinary SCA. This SCA involves super-symplectic algebra associated with metrically 2-dimensional light-cone boundary (light-like boundaries of causal diamonds) and the corresponding extended conformal algebra (light-like boundary is metrically sphere S2). Both these algebras have conformal structure with respect to the light-like radial coordinate rM and conformal algebra also with respect to the complex coordinate of S2. Symplectic algebra replaces finite-dimensional Lie algebra as the analog of Kac-Moody algebra. Also light-like orbits of partonic 2-surfaces possess this SCA but now Kac-Moody algebra is defined by isometries of imbedding space. String world sheets possess an ordinary SCA assignable to isometries of the imbedding space. An attractive interpretation is that rM at light-cone boundary corresponds to a coordinate along fermionic string extendable to a hypercomplex coordinate at string world sheet.

N=8 SCS seems to be the most natural candidate for SCS behind TGD: all fermion spin states would correspond to generators of this symmetry. Since the modes generating the symmetry are however only half-covariantly constant (holomorphic) this SUSY is badly broken at space-time level and the minimal breaking occurs for N=2 SCS generated by right-handed neutrino and antineutrino.

The key motivation for the application of minimal N=2 SCFTs to TGD is that SCAs for them have a non-vanishing central charge c and vacuum weight h≥ 0 and the degenerate character of ground state allows to deduce differential equations for n-point functions so that these theories are exactly solvable. It would be extremely nice is scattering amplitudes were basically determined by n-point functions for minimal SCFTs.

A further motivation comes from the following insight. ADE classification of N=2 SCFTs is extremely powerful result and there is connection with the hierarchy of inclusions of hyperfinite factors of type II1, which is central for quantum TGD. The hierarchy of Planck constants assignable to the hierarchy of isomorphic sub-algebras of the super-symplectic and related algebras suggest interpretation in terms of ADE hierarchy a rather detailed view about a hierarchy of conformal field theories and even the identification of primary fields in terms of critical deformations.

The application N=2 SCFTs in TGD framework can be however challenged. The problem caused by the negative value of vacuum conformal weight has been already discussed but there are also other problems.

  1. One can argue that covariantly constant right-handed neutrino - call it νR - defines a pure gauge super-symmetry and it has taken along time to decide whether this is the case or not. Taking at face value the lacking evidence for space-time SUSY from LHC would be easy but too light-hearted manner to get rid of the problem.

    Could it be that at space-time level covariantly constant right-handed neutrino (νR) and its antiparticle (ν*R) generates pure gauge symmetry so that the resulting sfermions correspond to zero norm states? The oscillator operators for νR at imbedding space level have commutator proportional to pkγk vanishing at the limit of vanishing massless four-momentum. This would imply that they generate sfermions as zero norm states. This argument is however formulated at the level of imbedding space: induced spinor modes reside at string world sheets and covariant constancy is replaced by holomorphy!

    At the level of induced spinor modes located at string world sheets the situation is indeed different. The anti-commutators are not proportional to pkγk but in Zero Energy Ontology (ZEO) can be taken to be proportional to nkγk where nk is light-like vector dual to the light-like radial vector of the point of the light-like boundary of causal diamond CD (part of light-one boundary) considered. Therefore also constant νR and ν*R are allowed as non-zero norm states and the 3 sfermions are physical particles. Both ZEO and strong form of holography (SH) would play crucial role in making the SCS dynamical symmetry.

  2. Second objection is that LHC has failed to detect sparticles. In TGD framework this objection cannot be taken seriously. The breaking of N=2 SUSY would be most naturally realized as different p-adic length scales for particle and sparticle. The mass formula would be the same apart from different p-adic mass scale. Sparticles could emerge at short p-adic length scale than those studied at LHC (labelled by Mersenne primes M89 and MG,79= (1+i)79).

    One the other hand, one could argue that since covariantly constant right-handed neutrino has no electroweak-, color- nor gravitational interactions, its addition to the state should not change its mass. Again the point is however that one considers only neutrinos at string world sheet so that covariant constancy is replaced with holomorphy and all modes of right-handed neutrino are involved. Kähler Dirac equation brings in mixing of left and right-handed neutrinos serving as signature for massivation in turn leading to SUSY breaking. One can of course ask whether the p-adic mass scales could be identical after all. Could the sparticles be dark having non-standard value of Planck constant heff=n× h and be created only at quantum criticality (see this).

This is a brief overall view about the most obvious problems and proposed solution of them in TGD framework and in the following I will discuss the details. I am of course not a SCFT professional. I however dare to trust my physical intuition since experience has taught to me that it is better to concentrate on physics rather than get drowned in poorly understood mathematical technicalities.

For details see the new chapter Could N=2 Super-Conformal Algebra Be Relevant For TGD? or the article with the same title.

Tensor Networks and S-matrices

The concrete construction of scattering amplitudes has been the toughest challenge of TGD and the slow progress has occurred by identification of general principles with many side tracks. One of the key problems has been unitarity. The intuitive expectation is that unitarity should reduce to a local notion somewhat like classical field equations reduce the time evolution to a local variational principle. The presence of propagators have been however the the obstacle for locally realized unitarity in which each vertex would correspond to unitary map in some sense.

TGD suggests two approaches to the construction of S-matrix.

  1. The first approach is generalization of twistor program (this). What is new is that one does not sum over diagrams but there is a large number of equivalent diagrams giving the same outcome. The complexity of the scattering amplitude is characterized by the minimal diagram. Diagrams correspond to space-time surfaces so that several space-time surfaces give rise to the same scattering amplitude. This would correspond to the fact that the dynamics breaks classical determinism. Also quantum criticality is expected to be accompanied by quantum critical fluctuations breaking classical determinism. The strong form of holography would not be unique: there would be several space-time surfaces assignable as preferred extremals to given string world sheets and partonic 2-surfaces defining "space-time genes".
  2. Second approach relies on the number theoretic vision and interprets scattering amplitudes as representations for computations with each 3-vertex identifiable as a basic algebraic operation (this). There is an infinite number of equivalent computations connecting the set of initial algebraic objects to the set of final algebraic objects. There is a huge symmetry involved: one can eliminate all loops moving the end of line so that it transforms to a vacuum tadpole and can be snipped away. A braided tree diagram is left with braiding meaning that the fermion lines inside the line defined by light-like orbit are braided. This kind of braiding can occur also for space-like fermion lines inside magnetic flux tubes and defining correlate for entanglement. Braiding is the TGD counterpart for the problematic non-planarity in twistor approach.
Third approach involving local unitary as an additional key element is suggested by tensor networks relying on the notion of perfect entanglement discussed by Preskill et al (see this and this). A detailed representation can be found in the article of Preskill et al ).


It is certainly clear from the beginning that the possibly existing description of S-matrix in terms of tensor networks cannot correspond to the perturbative QFT description in terms of Feynman diagrams.

  1. Tensor network description relates interior and boundary degrees in holography by a isometry. Now however unitary matrix has quite different role. It could correspond to U-matrix relating zero energy states to each other or to the S-matrix relating to each other the states at boundary of CD and at the shifted boundary obtained by scaling. These scalings shifting the second boundary of CD and increasing the distance between the tips of CD define the analog of unitary time evolution in ZEO. The U-matrix for transitions associated with the state function reductions at fixed boundary of CD effectively reduces to S-matrix since the other boundary of CD is not affected.

    The only manner one could see this as holography type description would be in terms of ZEO in which zero energy states are at boundaries of CD and U-matrix is a representation for them in terms of holography involving the interior states representing scattering diagram in generalized sense.

  2. The appearance of small gauge coupling constant tells that the entanglement between "states" in state spaces whose coordinates formally correspond to quantum fields is weak and just opposite to that defined by a perfect tensor. Quite generally, coupling constant might be the fatal aspect of the vertices preventing the formulation in terms of perfect entanglement.

    One should understand how coupling constant emerges from this kind of description - or disappears from standard QFT description. One can think of including the coupling constant to the definition of gauge potentails: in TGD framework this is indeed true for induced gauge fields. There is no sensical manner to bring in the classical coupling constants in the classical framework and the inverse of Kähler coupling strength appears only as multiplier of the Kähler action analogous to critical temperature.

    More concretely, there are WCW spin degrees of freedom (fermionic degrees of freedom) and WCW orbital degrees of freedom involving functional integral over WCW. Fermionic contribution would not involve coupling constants whereas the functional integral over WCW involving exponential of vacuum functional could give rise to the coupling constants assignable to the vertices in the minimal tree diagram.

  3. The decomposition S= 1+iT of unitary S-matrix giving unitarity as the condition -i(T-T) +TT=0 reflects the perturbative thinking. If one has only isometry instead of unitary transformation, this decomposition becomes problematic since T and T whose some appears in the formula act in different spaces. One should have the generalization of Id as a "trivial" isometry. Alternatively, one should be able to extend the state space Hin by adding a tensor factor mapped trivially in isometry.
  4. There are 3- and 4-vertices rather than only -say, 3-vertices as in tensor networks. For non-Abelian Chern-Simons term for simple Lie group one would have besides kinetic term only 3-vertex Tr(A∧ A ∧ A) defining the analog of perfect tensor entanglement when interpreted as co-product involving 3-D permutation symbol and structure constants of Lie algebra. Note also that for twistor Grassmannian approach the fundamental vertices are 3-vertices. It must be however emphasized that QFT description emerges from TGD only at the limit when one identifies gauge potentials as sums of induced gauge potentials assignable to the space-time sheets, which are replaced with single piece of Minkowski space.
  5. Tensor network description does not contain propagators since the contractions are between perfect tensors. It is to make sense propagators must be eliminated. The twistorial factorization of massless fermion propagator suggest that this might be possible by absorbing the twistors to the vertices.
These reasons make it clear that the proposed idea is just a speculative question. Perhaps the best strategy is to look this crazy idea from different view points: the overly optimistic view developing big picture and the approach trying to debunk the idea.

The overly optimistic vision

With these prerequisites one can follow the optimistic strategy and ask how tensor networks could allow to generalize the notion of unitary S-matrix in TGD framework.

  1. Tensor networks provide an elegant representation of holography mapping interior states isometrically (in Hilbert space sense) to boundary states or vice versa for selected subsets of states defining the code subspace for holographic quantum error correcting code. Again the tensor net is highly non-unique but there is some minimal tensor net characterizing the complexity of the entangled boundary state.
  2. Tensor networks have two key properties, which might be abstracted and applied to the construction of S-matrix in zero energy ontology (ZEO): perfect tensors define isometry for any subspace defined by the index subset of perfect tensor to its complement and the non-unique graph representing the network. As far as the construction of Hilbert space isometry between local interior states and highly non-local entangled boundary states is considered, these properties are enough.
One cannot avoid the idea that these three constructions are different aspects of one and same construction and that tensor net construction with perfect tensors representing vertices could provide and additional strong constraint to the long sought for explicit recipe for the construction of scattering amplitudes. How tensor networks could the allow to generalize the notion of unitary S-matrix in TGD framework?
  1. Tensor networks suggests the replacement of unitary correspondence with the more general notion of Hilbert space isometry. This generalization is very natural in TGD since one must allow phase transitions increasing the state space and it is quite possible that S-matrix represents only isometry: this would mean that SS=Idin holds true but SS=Idout does not even make sense. This conforms with the idea that state function reduction sequences at fixed boundary of causal diamonds defining conscious entities give rise evolution implying that the size of the state space increases gradually as the system becomes more complex. Note that this gives rise to irreversibility understandandable in terms of NMP (this). It might be even impossible to formally restore unitary by introducing formal additional tensor factor to the space of incoming states if the isometric map of the incoming state space to outgoing state space is inclusion of hyperfinite factors.
  2. If the huge generalization of the duality of old fashioned string models makes sense, the minimal diagram represesenting scattering is expected to be a tree diagram with braiding and should allow a representation as a tensor network. The generalization of the tensor network concept to include braiding is trivial in principle: assign to the legs connecting the nodes defined by perfect tensors unitary matrices representing the braiding - here topological QFT allows realization of the unitary matrix. Besides fermionic degrees of freedom having interpretation as spin degrees of freedom at the level of "World of Classical Worlds" (WCW) there are also WCW orbital degrees of freedom. These two degrees of freedom factorize in the generalized unitarity conditions and the description seems much simpler in WCW orbital degrees of freedom than in WCW spin degrees of freedom.
  3. Concerning the concrete construction there are two levels involved, which are analogous to descriptions in terms of boundary and interior degrees of freedom in holography. The level of fundamental fermions assignable to string world sheets and their boundaries and the level of physical particles with particles assigned to sets of partonic 2-surface connected by magnetic flux tubes and associated fermionic strings. One could also see the ends of causal diamonds as analogous to boundary degrees of freedom and the space-time surface as interior degrees of freedom.
The description at the level of fundamental fermions corresponds to conformal field theory at string world sheets.
  1. The construction of the analogs of boundary states reduces to the construction of N-point functions for fundamental fermions assignable to the boundaries of string world sheets. These boundaries reside at 3-surfaces at the space-like space-time ends at CDs and at light-like 3-surfaces at which the signature of the induced space-time metric changes.
  2. In accordance with holography, the fermionic N-point functions with points at partonic 2-surfaces at the ends of CD are those assignable to a conformal field theory associated with the union of string world sheets involved. The perfect tensor is assignable to the fundamental 4-fermion scattering which defines the microscopy for the geometric 3-particle vertices having twistorial interpretation and also interpretation as algebraic operation.

    What is important is that fundamental fermion modes at string world sheets are labelled by conformal weights and standard model quantum numbers. No four-momenta nor color quantum numbers are involved at this level. Instead of propagator one has just unitary matrix describing the braiding.

  3. Note that four-momenta emerging in somewhat mysterious manner to stringy scattering amplitudes and mean the possibility to interpret the amplitudes at the particle level.
Twistorial and number theoretic constructions should correspond to particle level construction and also now tensor network description might work.
  1. The 3-surfaces are labelled by four-momenta besides other standard model quantum numbers but the possibility of reducing diagram to that involving only 3-vertices means that momentum degrees of freedom effectively disappear. In ordinary twistor approach this would mean allowance of only forward scattering unless one allows massless but complex virtual momenta in twistor diagrams. Also vertices with larger number of legs are possible by organizing large blocks of vertices to single effective vertex and would allow descriptions analogous to effective QFTs.
  2. It is highly non-trivial that the crucial factorization to perfect tensors at 3-vertices with unitary braiding matrices associated with legs connecting them occurs also now. It allows to split the inverses of fermion propagators into sum of products of two parts and absorb the halves to the perfect tensors at the ends of the line. The reason is that the inverse of massless fermion propagator (also when masslessness is understood in 8-D sense allowing M4 mass to be non-vanishing) to be express as bilinear of the bi-spinors defining the twistor representing the four-momentum. It seems that this is absolutely crucial property and fails for massive (in 8-D sense) fermions.

For the details see the new chapter From Principles to Diagrams or the article with the same title.

Twistor googly problem transforms from a curse to blessing in TGD framework

There was a nice story with title "Michael Atiyah’s Imaginative State of Mind" about mathematician Michael Atyiah in Quanta Magazine. The works of Atyiah have contributed a lot to the development of theoretical physics. What was pleasant to hear that Atyiah belongs to those scientists who do not care what others think. As he tells, he can afford this since he has got all possible prices. This is consoling and encouraging even for those who have not cared what others think and for this reason have not earned any prizes. Nor even a single coin from what they have been busily doing their whole lifetime!

In the beginning of the story "twistor googly problem" was mentioned. I had to refresh my understanding about googly problem. In twistorial description the modes of massless fields (rather than entire massless fields) in space-time are lifted to the modes in its 6-D twistor-space and dynamics reduces to holomorphy. The analog of this takes place also in string models by conformal invariance and in TGD by its extension.

One however encounters googly problem: one can have twistorial description for circular polarizations with well-defined helicity +1/-1 but not for general polarization states - say linear polarizations, which are superposition of circular polarizations. This reflects itself in the construction of twistorial amplitudes in twistor Grassmann program for gauge fields but rather implicitly: the amplitudes are constructed only for fixed helicity states of scattered particles. For gravitons the situation gets really bad because of non-linearity.

Mathematically the most elegant solution would be to have only +1 or -1 helicity but not their superpositions implying very strong parity breaking and chirality selection. Parity parity breaking occurs in physics but is very small and linear polarizations are certainly possible! The discusion of Penrose with Atyiah has inspired a possible solution to the problem known as "palatial twistor theory". Unfortunately, the article is behind paywall too high for me so that I cannot say anything about it.

What happens to the googly problem in TGD framework? There is twistorialization at space-time level and imbedding space level.

  1. One replaces space-time with 4-surface in H=M4×CP2 and lifts this 4-surface to its 6-D twistor space represented as a 6-surface in 12-D twistor space T(H)=T(M4)×T(CP2). The twistor space has Kähler structure only for M4 and CP2 so that TGD is unique. This Kähler structure is needed to lift the dynamics of Kähler action to twistor context and the lift leads to the a dramatic increase in the understanding of TGD: in particular, Planck length and cosmological constant with correct sign emerge automatically as dimensional constants besides CP2 size.
  2. Twistorialization at imbedding space level means that spinor modes in H representing ground states of super-symplectic representations are lifted to spinor modes in T(H). M4 chirality is in TGD framework replaced with H-chirality, and the two chiralities correspond to quarks and leptons. But one cannot superpose quarks and leptons! "Googly problem" is just what the superselection rule preventing superposition of quarks and leptons requires in TGD!
One can look this in more detail.
  1. Chiral invariance makes possible for the modes of massless fields to have definite chirality: these modes correspond to holomorphic or antiholomorphic amplitudes in twistor space and holomorphy (antiholomorphy is holomorphy with respect to conjugates of complex coordinates) does not allow their superposition so that massless bosons should have well-defined helicities in conflict with experimental facts. Second basic problem of conformally invariant field theories and of twistor approach relates to the fact that physical particles are massive in 4-D sense. Masslessness in 4-D sense also implies infrared divergences for the scattering amplitudes. Physically natural cutoff is required but would break conformal symmetry.
  2. The solution of problems is masslessness in 8-D sense allowing particles to be massive in 4-D sense. Fermions have a well-defined 8-D chirality - they are either quarks or leptons depending on the sign of chirality. 8-D spinors are constructible as superpositions of tensor products of M4 spinors and of CP2 spinors with both having well-defined chirality so that tensor product has chiralities (ε1, ε2), εi=+/- 1, i=1,2. H-chirality equals to ε=ε1ε2. For quarks one has ε= 1 (a convention) and for leptons ε=-1. For quark states massless in M4 sense one has either (ε12) = (1,1) or (ε12) = (-1,-1) and for massive states superposition of these. For leptons one has either (ε1, ε2) = (1,-1) or (ε1, ε2) = (-1,1) in massless case and superposition of these in massive case.
  3. The twistorial lift to T(M4)× T(CP2) of the ground states of super-symplectic representations represented in terms of tensor products formed from H-spinor modes involves only quark and lepton type spinor modes with well-defined H-chirality. Superpositions of amplitudes in which different M4 helicities appear but M4 chirality is always paired with completely correlating CP2 chirality to give either ε=1 or ε=-1. One has never a superposition of of different chiralities in either M4 or CP2 tensor factor. I see no reason forbidding this kind of mixing of holomorphicities and this is enough to avoid googly problem. Linear polarizations and massive states represent states with entanglement between M4 and CP2 degrees of freedom. For massless and circularly polarized states the entanglement is absent.
  4. This has interesting implications for the massivation. Higgs field cannot be scalar in 8-D sense since this would make particles massive in 8-D sense and separate conservation of B and L would be lost. Theory would also contain a dimensional coupling. TGD counterpart of Higgs boson is actually CP2 vector, and one can say that gauge bosons and Higgs combine to form 8-D vector. This correctly predicts the quantum numbers of Higgs. Ordinary massivation by constant vacuum expectation value of vector Higgs is not an attractive idea since no covariantly constant CP2 vector field exists so that Higgsy massivation is not promising except at QFT limit of TGD formulated in M4. p-Adic thermodynamics gives rise to 4-D massivation but keeps particles massless in 8-D sense. It also leads to powerful and correct predictions in terms of p-adic length scale hypothesis.
Addition: Anonymous reader gave me a link to the paper of Penrose and this inspired further more detailed considerations of googly problem.
  1. After the first reading I must say that I could not understand how the proposed elimination of conjugate twistor by quantization of twistors solves the googly problem, which means that both helicities are present (twistor Z and its conjugate) in linearly polarized classical modes so that holomorphy is broken classically.
  2. I am also very skeptic about quantizing of either space-time coordinates or twistor space coordinates. To me quantization is natural only for linear objects like spinors. For bosonic objects one must go to higher abstraction level and replace superpositions in space-time with superpositions in field space. Construction of "World of Classical Worlds" (WCW) in TGD means just this.
  3. One could however think that circular polarizations are fundamental and quantal linear combination of the states carrying circularly polarized modes give rise to linear and elliptic polarizations. Linear combination would be possible only at the level of field space (WCW in TGD), not for classical fields in space-time. If so, then the elimination of conjugate Z by quantization suggested by Penrose would work.
  4. Unfortunately, Maxwell's equations allow classically linear polarisations! In order to achieve classical-quantum consistency, one should modify classical Maxwell's equations somehow so that linear polarizations are not possible. Googly problem is still there!
What about TGD?
  1. Massless extremals representing massless modes are very "quantal": they cannot be superposed classically unless both momentum and polarisation directions for them (they can depend space-time point) are exactly parallel. Optimist would guess that the classical local classical polarisations are circular. No, they are linear! Superposition of classical linear polarizations at level of WCW can give rise to local linear but not local circular polarization! Something more is needed.
  2. The only sensible conclusion is that only gauge boson quanta (not classical modes) represented as pairs of fundamental fermion and antifermion in TGD framework can have circular polarization! And indeed, massless bosons - in fact, all elementary particles- are constructed from fundamental fermions and they allow only two M4, CP 2 and M4× CP2 helicities/-chiralities analogous to circular polarisations. B and L conservation would transform googly problem to a superselection rule as already described.
To sum up, both the extreme non-linearity of Kähler action, the representability of all elementary particles in terms of fundamental fermions and antifermions, and the generalization of conserved M4 chirality to conservation of H-chirality would be essential for solving the googly problem in TGD framework.

For background see the chapter From Principles to giagrams or the article From Principles to Diagrams.

Could M4 Kähler form introduce new gravitational physics?

The introduction of M4 Kähler form strongly suggested by the twistor formulation of TGD could bring in new gravitational physics.

  1. As found, the twistorial formulation of TGD assigns to M4 a self dual Kähler form whose square gives Minkowski metric. It can (but need not if M4 twistor space is trivial as bundle) contribute to the 6-D twistor counterpart of Kähler action inducing M4 term to 4-D Kähler action vanishing for canonically imbedded M4.
  2. Self-dual Kähler form in empty Minkowski space satisfies automatically Maxwell equations and has by Minkowskian signature and self-duality a vanishing action density. Energy momentum tensor is proportional to the metric so that Einstein Maxwell equations are satisfied for a non-vanishing cosmological constant! M4 indeed allows a large number of self dual Kähler fields (I have christened them as Hamilton-Jacobi structures). These are probably the simplest solutions of Einstein-Maxwell equations that one can imagine!
  3. There however exist quite a many Hamilton-Jacobi structures. However, if this structure is to be assigned with a causal diamond (CD) it must satisfy additional conditions, say SO(3) symmetry and invariance under time translations assignable to CD. Alternatively, covariant constancy and SO(2)⊂ SO(3) symmetry might be required.
    1. In the case of causal diamond (CD) a spherically symmetric self-dual monopole Kähler form with non-vanishing components Jtr= εtrθφJθφ, Jθφ=cos(θ) carrying radial electric and magnetic fields with identical gravitational charges looks rather natural option. The time-like line connecting the tips of CD would carry a genuine self-dual monopole so that Dirac monopole would not be in question. The potential associated with J could be chosen to be Aμ ↔ (1/r,0,0,sin(θ). I have considered this kind of possibility earlier in context of TGD inspired model of anyons but gave up the idea.

      The moduli space for CDs with second tip fixed would be hyperbolic space H3=SO(3,1)/SO(3) or a space obtained by identifying points at the orbits of some discrete subgroup of SO(3,1) as suggested by number theoretic considerations. This induced Kähler field could make the blackholes with center at this line to behave like M4 magnetic monopoles if the M4 part of Kähler form is induced into the 6-D lift of Kähler action with extremely small coefficients of order of magnitude of cosmological constant. Cosmological constant and the possibility of CD monopoles would thus relate to each other.

    2. Covariant constancy is an alternative option. This would leave only the fields Jtz =Jxy=1 unique apart from Lorentz transformation: it would be attractive to assign this Kähler with given CD to define a preferred plane M2 required also by the number theoretic vision. Now however rotational invariance is broken to SO(1,1)× SO(2). SO(3,1)/SO(1,1)× SO(2) would define moduli for CDs. Magnetic and electric parts of Kähler form would be in z-direction and flux tubes would tend to be in this direction. One would have clearly a preferred direction and it is difficult to imagine how the gravitational field of blackhole could correlate with these fluxes unless one assigns to each flux tube its own CD.
  4. A further interpretational problem is that the classical coupling of M4 Kähler gauge potential to induced spinors is not small. Can one really tolerate this kind of coupling equivalent to a coupling to a self dual monopole field carrying electric and magnetic charges? One could of course consider the condition that the string world sheets carrying spinor modes are such that the induced M4 Kähler form vanishes and gauge potential become pure gauge. M4 projection would be 2-D Lagrange manifold whereas CP2 projection would carry vanishing induce W and possibly also Z0 field in order that em charge is well defined for the modes. These conditions would fix the string world sheets to a very high degree in terms of maps between this kind of 2-D sub-manifolds of M4 and CP2. Spinor dynamics would be determined by the avoidance of interaction!

    It must be emphasized that the imbedding space spinor modes characterizing the ground states of super-symplectic representations would not couple to the monopole field so that at this level Poincare invariance is not broken. The coupling would be only at the space-time level and force spinor modes to Lagrangian sub-manifolds.

  5. At the static limit of GRT and for gij≈ δij implying SO(3) symmetry there is very close analogy with Maxwell's equations and one can speak of gravi-electricity and gravi-magnetism with 4-D vector potential given by the components of g. The genuine U(1) gauge potential does not however relate to the gravimagnetism in GRT sense. Situation would be analogous to that for CP2, where one must add to the spinor connection U(1) term to obtain respectable spinor structure. Now the U(1) term would be added to trivial spinor connection of flat M4: its presence would be justified by twistor space Kähler structure. If the induced M4 Kähler form is present as a classical physical field it means genuinely new contribution to gravitational interaction and assignable to cosmological constant.

    I have talked much about gravitational flux or gravitons are carried along Kähler magnetic monopole flux tubes. This is quite respectable hypothesis. One can however ask whether the gravitational interaction could be mediated along flux tubes of M4 Kähler magnetic field carrying monopole flux. For the proposed SO(3) symmetric option the flux tubes would emanate radially from the origin and one could assign to each gravitating object CD. It is of course quite possible that Kähler magnetic flux tubes and gravitational flux tubes are one and same thing in astrophysical systems. Note however that Kähler magnetic monopole fluxes do not involve genuine monopole like M4 Kähler fluxes in SO(3) symmetric case.

For background see the chapter From Principles to giagrams or the article From Principles to Diagrams.

Cosmic evolution of the radius of the fiber of the twistor space of space-time surface

I have continued the little calculations inspired by the surprising finding that twistorial lift of Kähler action based dynamics immediately leads to the identification of cosmological length scales as fundamental classical length scales appearing in 6-D Kähler action, whose dimensional reduction gives Kähler action plus small cosmological term with correct sign to explain together with magnetic flux tube tension accelerating cosmic expansion. Whether Planck length emerges classically from from quantum theory remains still an open question.

For a fleeting moment I thought that for the twistor space of Minkowski space the 2-D fiber could be hyperbolic sphere H2 (t2-x2-y2 =-RH2) rather than sphere S2 as it is for CP2 with Euclidian signature of metric. I however soon realized that the infinite area of H2 implies that 6-D Kähler action is infinite and that there are many other difficulties.

The correct manner to define Minkowskian variant of twistor space is by starting from the generalization of complex and Kähler structures for M4= M2+ E2 of local tangent space to longitudinal (defined by light-like vector) and to transversal directions (polarizations orthogonal to the light-like vector. The decomposition can depend on point but the distributions of two planes must integrated to 2-D surfaces. In E2 one has complex structure and in M2 its hyper-complex variant. In M2 has decomposition of replacing complex numbers by hyper-complex numbers so that complex coordinate x+iy is replaced with w=t+ie, i2=-1 and e2=-1.

It took time to realize I have actually carried out this generalization years ago with quite different motivations and called the resulting structure Hamilton-Jacobi structure! The twistor fiber is defined by projections of 4-D antisymmetric tensors (in particular induced Kähler form) to the orthogonal complement of unique time direction determed by the sum of light-like vector and its dual in M2. This part of tensor could be called magnetic. Th magnetic part of the tensor defines a direction and one has natural metric making the space of directions sphere S2 with metric having signature (-1,-1). This requires that twistor space has metric signature (-1,-1,1,-1,-1,-1) (I also considered seriously the signature (1,1,1,-1,-1,-1) so that there are three time-like coordinates) .

The radii of the spheres associated with M4 and CP2 define two fundamental scales and the scaling of 6-D Käler action brings in third fundamental length scale. On possibility is that the radii of the two spheres are actually identical and essentially equal to CP2 radius. Second option is that the radius of S2(M4) equals to Planck length, which would be therefore a fundamental length scale.

The radius RD of the 2-D fiber of twistor space assignable to space-time surfaces is dynamical. In Euclidian space-time regions the fiber is sphere: a good guess is that its order of magnitude is determined by the winding numbers of the maps from S2(X4)→ S2(M4) and S2(X4)→ S2(CP2). The winding numbers (1,0) and (0,1) represent the simplest options. The question is whether one could say something non-trivial about cosmic evolution of RD as function of cosmic time. This seems to be the case.

Before continuing it is good to recall how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.

  1. The first term is essentially the ordinary Kähler action multiplied by the area of S2(X4), which is compensated by the length scale, which can be taken to be the area 4π R2(M4) of S2(M4). This makes sense for winding numbers (w1,w2)=(n,0) meaning that S2(CP2) is effectively absent but S2(M4) is present.
  2. Second term is the analog of Kähler action assignable assignable to the projection of S2(M4) Kähler form. The corresponding Kähler coupling strength αK (M4) is huge - so huge that one has

    αK (M4)4π R2(M4)== L2 ,

    where 1/L2 is of the order of cosmological constant and thus of the order of the size of the recent Universe. αK(M4) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as αK(M4) ∝ p≈ 2k, p prime, k prime.

  3. The Kähler form assignable to M4 is not assumed to contribute to the action since it does not contribute to spinor connection of M4. One can of course ask whether it could be present. For canonically imbedded M4 self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small.Breaking of Lorentz invariance is however a possible problem. If αK(M4) is given by above expression, then this contribution is extremely small.
Hence one can consider the possibility that the action is just the sum of full 6-D Kähler actions assignable to T(M4) and T(CP2) but with different values of αK if one has (w1,w2)=(n,0). Also other w2≠ 0 is possible but corresponds to gigantic cosmological constant.

Given the parameter L2 as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive. One can actually get estimate for the evolution of RD as function of cosmic time if one accepts Friedman cosmology as an approximation of TGD cosmology. One can actually get estimate for the evolution of RD as function of cosmic time if one accepts Friedman cosmology as an approximation of TGD cosmology.

  1. Assume critical mass density so that one has

    ρcr= 3H2/8π G .

  2. Assume that the contribution of cosmological constant term to the mass mass density dominates. This gives ρ≈ ρvac=Λ/8π G. From ρcrvac one obtains

    Λ= 3H2 .

  3. From Friedman equations one has H2= ((da/dt)/a)2, where a corresponds to light-cone proper time and t to cosmic time defined as proper time along geodesic lines of space-time surface approximated as Friedmann cosmology. One has

    Λ= 3/gaaa2

    in Robertson-Walker cosmology with ds2= gaada2-a232.

  4. Combining this equations with the TGD based equation

    Λ= 8π2G/L2RD2

    one obtains

    2G/L2RD2= 3/gaaa2.

  5. Assume that quantum criticality applies so that L has spectrum given by p-adic length scale hypothesis so that one discrete p-adic length scale evolution for the values of L. There are two options to consider depending on whether p-adic length scales are assigned with light-cone proper time a or with cosmic time t

    T= a (Option I) , T=t (Option II).

    Both options give the same general formula for the p-adic evolution of L(k) but with different interpretation of T(k).

    L(k)/Lnow= T(k)/Tnow , T(k)= L(k) = 2(k-151)/2× L(151) , L(151)≈ 10 nm .

    Here T(k) is assumed to correspond to primary p-adic length scale. An alternative - less plausible - option is that T(k) corresponds to secondary p-adic length scale L2(k)=2k/2L(k) so that T(k) would correspond to the size scale of causal diamond. In any case one has L ∝ L(k). One has a discretized version of smooth evolution

    L(a) = Lnow × (T/Tnow) .

Consider now the predictions.
  1. Feeding into the formula following from two expressions for Λ one obtains an expression for RD(a)

    RD/lP= (8/3)1/2π× (a/L(a)× gaa1/2

    This equation tells that RD is indeed dynamical, and becomes very small at very early times since gaa becomes very small. As a matter of fact, in very early cosmic string dominated cosmology gaa would be extremely small constant (see this). In late cosmology gaa→ 1 holds true and one obtains at this limit

    RD(now)= (8/3)1/2π× (anow/Lnow) × lP ≈ 4.4 ×(anow/Lnow) × lP .

  2. For T= t option RD/lP remains constant during both matter dominated cosmology, radiation dominated cosmology, and string dominated cosmology since one has a∝ tn with n= 1/2 during radiation dominated era, n= 2/3 during matter dominated era, and n=1 during string dominated era (see this). This gives

    RD/lP=(8/3)1/2π× at (gaa1/2(t(end)/L(end)) = (8/3)1/2π×(1/n)(t(end)/L(end)) .

    Here "end"> refers the end of the string or radiation dominated period or to the recent time in the case of matter dominated era. The value of n would have evolved as RD/lP∝ (1/n)(tend/Lend), n∈ [1,3/2,2}. During radiation dominated cosmology RD ∝ a1/2 holds true. The value of RD would be very nearly equal to R(M4) and R(M4) would be of the same order of magnitude as Planck length. In matter dominated cosmology would would have RD ≈ 2.2 (t(now)/L(now)) × lP .

  3. For RD(now)=lP one would have

    Lnow/anow =(8/3)1/2π≈ 4.4 .

    In matter dominated cosmology gaa=1 gives tnow=(2/3)× anow so that predictions differ only by this factor for options I and II. The winding number for the map S2(X4)→ S2(CP2) must clearly vanish since otherwise the radius would be of order R.

  4. For RD(now)= R one would obtain

    anow/Lnow =(8/3)1/2π× (R/lP)≈ 2.1× 104 .

    One has Lnow=106 ly: this is roughly the average distance scale between galaxies. The size of Milky Way is in the range 1-1.8 × 105 ly and of an order of magnitude smaller.

  5. An interesting possibility is that RD(a) evolves from RD ≈ R(M4) ≈ lP to RD ≈ R. This could happen if the winding number pair (w1,w2)=(1,0) transforms to (w1,w2)=(0,1) during transition to from radiation (string) dominance to matter (radiation) dominance. RD/lP radiation dominated cosmology would be related by a factor

    RD(rad)/RD(mat)>= (3/4)(t(rad,end)/L(rad,end))× (L(now)/t(now))

    to that in matter dominated cosmology. Similar factor would relate the values of RD/lP in string dominated and radiation dominated cosmologies. The condition RD(rad)/RD(mat)=lP/R expressing the transformation of winding numbers would give

    L(now)/L(rad,end) =(4/3) (lP/R) (t(now)/t(rad,end)) .

    One has t(now)/t(rad,end)≈ .5× 106 and lP/R =2.5× 10-4 giving L(now)/L(rad,end)≈ 125, which happens to be near fine structure constant.

  6. For the twistorial lifts of space-time surfaces for which cosmological constant has a reasonable value , the winding numbers are equal to (w1,w2)=(n,0) so that RD=n1/2 R(S2(M4)) holds true in good approximation. This conforms with the observed constancy of RD during various cosmological eras, and would suggest that the ratio t(end)/L(end) characterizing these periods is same for all periods. This determines the evolution for the values of αK(M4).
R(M4)≈ lP seems rather plausible option so that Planck length would be fundamental classical length scale emerging naturally in twistor approach. Cosmological constant would be coupling constant like parameter with a spectrum of critical values given by p-adic length scales.

For background see the chapter From Principles to giagrams or the article From Principles to Diagrams.

Twistorial approach and connection with General Relativity

For year or two ago I ended up with a vision about how twistor approach could generalize to TGD framework. A more explicit realization of twistorialization as lifting of the preferred extremal X4 of Kähler action to corresponding 6-D twistor space X6 identified as surface in the 12-D product of twistor spaces of M4 and CP2 allowing Kähler structure suggests itself: this makes these spaces completely unique twistorially and seems more or less obvious that the Kähler structure must have profound physical meaning. It turned out that it has: the projection of Kähler form defines the representation of preferred quaternionic imaginary unit needed to assign twistor structure to space-time surface. Almost equally obvious idea is that the lifting of the dynamics for space-time surface to that for its twistor space in the product of twistor spaces of M4 and CP2 must be based on 6-D Kähler action.

Contrary to the original expectations, the twistorial approach is not mere reformulation but leads to a first principle identification of cosmological constant and perhaps also of gravitational constant and to a modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure.

There are some new results forcing a profound modification of the recent view about TGD but consistent with the general picture. A more explicit realization of twistorialization as lifting of the preferred extremal X4 of Kähler action to corresponding 6-D twistor space X6 identified as surface in the 12-D product of twistor spaces of M4 and CP2 allowing Kähler structure suggests itself.

The action principle in 6-D context is also Kähler action, which dimensionally reduces to Kähler action plus cosmological term. This brings in the radii of spheres S2 associated with the twistor space of CP2 presumably determined by CP2 radius and radius of S2 associated with M4 twistor space for which an attractive identification is as Planck length, which would be now purely classical parameter. The radius of S2 associated with space-time surface is determined by induced metric and is emergent length scale. The normalization of 6-D Kähler action by a scale factor with dimension which is inverse length squared brings in a further length scale closely related to cosmological constant which is also dynamical and has correct sign to explain accelerated expansion of the Universe.

The dimensionally reduced dynamics is a highly non-trivial modification of the dynamics of Kähler action however preserving the known extremals and basic properties of Kähler action and allowing to interpret induced Kähler form in terms of preferred imaginary unit defining twistor structure.

In the sequel I will discuss the recent understanding of twistorizalization, which is considerably improved from that in the earlier formulation. I formulate the dimensional reduction of 6-D Kähler action and consider the physical interpretation. After that I proceed to discuss the basic principles behind the recent view about scattering amplitudes as generalized Feynman diagrams.

1. Some mathematical background

First I will try to clarify the mathematical details related to the twistor spaces and how they emerge in the recent context. I do not regard myself as a mathematician in technical sense and I can only hope that the representation based on physical intuition does not contain serious mistakes.

1.1. Imbedding space is twistorially unique

It took roughly 36 years to learn that M4 and CP2 are twistorially unique. Space-times are surfaces in H=M4× CP2. M4 and CP2 are unique 4-manifolds in the sense that both allow twistor space with Kähler structure: Kähler structure is the crucial concept. Strictly speaking, M4 and its Euclidian variant E4 allow both twistor space and the twistor space of M4 is Minkowskian variant T(M4)= SU(2,2)/SU(2,1)× U(1) of 6-D twistor space CP3= SU(4)/SU(3)× U(1) of E4. The twistor space of CP2 is 6-D T(CP2)= SU(3)/U(1)× U(1), the space for the choices of quantization axes of color hypercharge and isospin.

This leads to a proposal - just a proposal - for the formulation of TGD in which space-time surfaces X4 in H are lifted to twistor spaces X6, which are sphere bundles over X4 and such that they are surfaces in 12-D product space T(M4)× T(CP2) such the twistor structure of X4 are in some sense induced from that of T(M4)× T(CP2). What is nice in this formulation is that one can use all the machinery of algebraic geometry so powerful in superstring theory (Calabi-Yau manifolds).

1.2 What does twistor structure in Minkowskian signature really mean?

What twistor structure in Minkowskian signature really means geometrically has remained a confusing question for me. The problems associated with the Minkowskian signature of the metric are encountered also in twistor Grassmann approach to scattering amplitudes but are circumvented by performing Wick rotation that is using E4 or S4 instead of M4 and applying algebraic continuation. Also complexification of Minkowksi space for momenta is used. These tricks do not apply now.

Let us try to collect thoughts about what is involved.

  1. Instead of M4 one considers the conformal compactification M4c of M4 identifiable as the boundary of light-cone boundary of 6-D Minkowski space with signature (1,1,-1,-1,-1), whose points differing by scaling are identified. One has a slicing by spheres of signature (-1,-1,-1) and varying radius ρ and these spheres are projectively identified so that one can "fix the gauge" by choosing ρ=ρ0. Since one has light-cone, the contribution dρ2 to the line element vanishes and one obtains ds2= ρ022- ρ20 ds2(S3). Conformal compactification means that the scale ρ0 of the metric is not unique. The scaling of the metric of the twistor space ρ02. Conformal invariance of the theory saves from problems.
  2. The Euclidian version of the twistor space of M4 corresponds to the twistor space of S4 identifiable as CP3= SU(4)/SU(3)× U(1) identifiable in terms of complex 2+2-spinors. The twistor space of M4c is SU(2,2)/SU(2,1)× U(1) (see this) and can be seen as a kind of algebraic continuation of CP3=SU(4)/SU(3)× U(1). This space is complex manifold but it is not completely clear to me whether this really guarantees the existence of Kähler structure consistent with the complex structure.
  3. If the Minkowskian variant of the twistor space (rather than only that associated with S4) is to have complex structure in the ordinary sense of the word, its metric must have even signature. M4c has signature (1,-1-1,-1) so that the signature of the analog of S2 fiber should have signature which is (1,-1) to give even signature (1,-1,1-1,-1,-1) for the twistor space. The sphere S2 would be replaced with its non-compact hyperbolic counterpart SO(2,1)/SO(1,1) and has metric signature (1,-1). One cannot assign to it finite size in the usual sense. However, since this space corresponds to hyperboloid t2-x2-y2=-R2 (3-D mass shell), one can assign to it finite hyperbolic radius RH. There are however problems.
    1. One cannot assign to H2 finite size in the usual sense. However, since this space corresponds to hyperboloid t2-x2-y2=-R2 (3-D mass shell), one could assign to it finite hyperbolic radius RH. In dimensional reduction of 6-D Kähler action however the integral over H2 gives its area if the restriction of J to H2 has square equal to metric as is extremely natural to assume. The area is RH2 times an infinite number and 4-D dimensionally reduced action would have infinite value. At the limit RH=0 (2-D light-cone boundary) the area vanishes as also the dimensionally reduced action.
    2. For even signature of twistor space the determinant of the induced 6-metric would be real in both Euclidian and Minkowskian space-time regions. Both Euclidian Minkowskian regions contribute to Kähler function (as was the original wrong assumption using |det(g)1/2| in volume element). The exponent of Kähler action in Minkowskian regions would not define phase as QFT picture demands.
    3. This picture is in conflict with the vision about the fiber as space S2 of directions defined by antisymmetric forms. The hidden assumption is that one has field of preferred time-like directions n and one considers induced Kähler form at space-like 3-surface with metric signature (-1,-1,-1) with n as time-like normal field.

      Could one imagine fixing of space-like direction field defining normals for a slicing by 3-surfaces with metric signature (1,-1,-1)? If so, one would end up with SO(2,1)/SO(1,1) as the fiber characterizing directions of projections of J to this subspace. The slicing by 3-surfaces parallel to the light-like 3-surfaces at the boundaries of Minkowskian and Euclidian space-time regions could indeed do the job. The light-likeness of these 3-surfaces also fits nicely with conformal invariance. The above problems are however enough to guarantee that the lifetime of H2 option was rather precisely 24 hours.

  4. The only alternative, which comes in mind is a hypercomplex generalization of the Kähler structure. This requires that the metric of the Minkowskian twistor space has signature (-1,-1,1,-1,-1,-1). This would give 1 time-like direction and the hypercomplex coordinate would correspond to a sub-space with signature (1,-1). Hypercomplex coordinate can be represented as h=t+iez, i2=-1,e2=-1. Kähler form representing imaginary unit would be replaced with eJ. One would consider sub-spaces of complexified quaternions spanned by real unit and units eIk, k=1,2,3 as representation of the tangent space of space-time surfaces in Minkowskian regions. This is familiar already from M8 duality (see this).

    One could regard Minkowskian twistor space as a kind of Wick rotation of the Euclidian twistor space. Hyper-complex numbers do not define number field since for light-like hypercomplex numbers t+iez, t=+/- z do not have finite inverse. Hypercomplex numbers allow a generalization of analytic functions used routinely in physics. Fiber would be sphere S2 with metric signature (-1,-1). Cosmological term would be finite and the sign of the cosmological term in the dimensionally reduced action would be positive as required. Also metric determinant would be imaginary as required. At this moment I cannot invent any killer objection against this option.

1.3 What the induction of twistor structure could mean?

To proceed one must make explicit the definition of twistor space. The 2-D fiber S2 consists of antisymmetric tensors of X4 which can be taken to be self-dual or anti-self-dual by taking any antisymmetric form and by adding to its plus/minus its dual. Each tensor of this kind defines a direction - point of S2. These points can be also regarded as quaternionic imaginary units. One has a natural metric in S2 defined by the X4 inner product for antisymmetric tensors: this inner product depends on space-time metric. Kähler action density is example of a norm defined by this inner product in the special case that the antisymmetric tensor is induced Kähler form. Induced Kähler form defines a preferred imaginary unit and is needed to define the imaginary part ω(X,Y)= ig(X,-JY) of hermitian form h= h+iω.

Consider now what the induction of twistor structure could mean.

  1. The induction procedure for Kähler structure of 12-D twistor space T requires that the induced metric and Kähler form of the base space X4 of X6 obtained from T is the same as that obtained by inducing from H=M4× CP2. Since the Kähler structure and metric of T is lift from H this seems obvious. Projection would compensate the lift.
  2. This is not yet enough. The Kähler structure and metric of F projected from T must be same as those lifted from X4. The connection between metric and ω implies that this condition for Kähler form is enough. The antisymmetric Kähler forms in fiber obtained in these two manners co-incide. Since Kähler form has only one component in 2-D case, one obtains single constraint condition giving a commutative diagram stating that the direct projection to F equals with the projection to the base followed by a lift to fiber. The resulting induced Kähler form is not covariantly constant but in fiber F one has J2=-g.

    As a matter of fact, this condition might be trivially satisfied as a consequence of the bundle structure of twistor space. The Kähler form from S2× S2 can be projected to S2 associated with X4 and by bundle projection to a two-form in X4. The intuitive guess - which might be of course wrong - is that this 2-form must be same as that obtained by projecting the Kähler form of CP2 to X4. If so then the bundle structure would be essential but what does it really mean?

  3. Intuitively it seems clear that X6 must decompose locally to a product X4× S2 in some sense. This is true if the metric and Kähler form reduce to direct sums of contributions from the tangent spaces of X4 and S2. This guarantees that 6-D Kähler action decomposes to a sum of 4-D Kähler action and Kähler action for S2.

    This could be however too strong a condition. Dimensional reduction occurs in Kaluza-Klein theories and in this case the metric can have also components between tangent spaces of the fiber and base being interpreted as gauge potentials. This suggests that one should formulate the condition in terms of the matrix T↔ gαμgβν-gανgβμ defining the norm of the induced Kähler form giving rise to Kähler action. T maps Kähler form J↔ Jαβ to a contravariant tensor Jc↔ Jαβ and should have the property that Jc(X4) (Jc( S2)) does not depend on J( S2) (J(X4)).

    One should take into account also the self-duality of the form defining the imaginary unit. In X4 the form S=J+/- *J is self-dual/anti-self dual and would define twistorial imaginary unit since its square equals to -g representing the negative of the real unit. This would suggest that 4-D Kähler action is effectively replaced with (J+/- *J)∧(J+/- *J)/2 =J*∧J +/- J∧J, where *J is the Hodge dual defined in terms of 4-D permutation tensor ε. The second term is topological term (Abelian instanton term) and does not contribute to field equations. This in turn would mean that it is the tensor T+/- ε for which one can demand that Sc(X4) (Sc(S2)) does not depend on S(S2) (S(X4)).

2. Surprise: twistorial dynamics does not reduce to a trivial reformulation of the dynamics of Kähler action

I have thought that twistorialization classically means only an alternative formulation of TGD. This is definitely not the case as the explicit study demonstrated. Twistor formulation of TGD is in terms of of 6-D twistor spaces T(X4) of space-time surfaces X4⊂ M4× CP2 in 12-dimensional product T=T(M4)× T(CP2) of 6-D twistor spaces of T(M4) of M4 and T(CP2) of CP2. The induced Kähler form in X4 defines the quaternionic imaginary unit defining twistor structure: how stupid that I realized it only now! I experienced during single night many other "How stupid I have been" experiences.

Classical dynamics is determined by 6-D variant of Kähler action with coefficient 1/L2 having dimensions of inverse length squared. Since twistor space is bundle, a dimensional reduction of 6-D Kähler action to 4-D Kähler action plus a term analogous to cosmological term - space-time volume - takes place so that dynamics reduces to 4-D dynamics also now. Here one must be careful: this happens provided the radius of F associated with X4 does not depend on point of X4. The emergence of cosmological term was however completely unexpected: again "How stupid I have been" experience. The scales of the spheres and the condition that the 6-D action is dimensionless bring in 3 fundamental length scales!

2.1 New scales emerge

The twistorial dynamics gives to several new scales with rather obvious interpretation. The new fundamental constants that emerge are the radius of hyperbolic sphere associated with T(M4) and of sphere associated with T(CP2). The radius of the fiber associated with X4 is not a fundamental constant but determined by the induced metric. By above argument the fiber is sphere for Euclidian signature and hyperbolic sphere for Minkowskian signature.

  1. For CP2 twistor space the radius of S2 must be apart from numerical constant equal to CP2 radius R. For M4 the simplest assumption is that also now the radius for S2(M4 equals to R(M4=R so that Planck length would not emerge from fundamental theory classically. Second option is that it does and one has R(M4=lP.
  2. If the signature of S2(M4) is (-1,-1) both Minkowskian and Euclidian space-time regions have S2(X4) with the same signature (-1,-1). The radius RD of S2(X4) is dynamically determined.
Recall first how the cosmological constant emerges from TGD framework. The key point is that the 6-D Kähler action contains two terms.
  1. The first term is essentially the ordinary Kähler action multiplied by the area of S2(X4), which is compensated by the length scale, which can be taken to be the area 4π R2(M4) of S2(M4). This makes sense for winding numbers (w1,w2)=(n,0) meaning that S2(CP2) is effectively absent but S2(M4) is present.
  2. Second term is the analog of Kähler action assignable assignable to the projection of S2(M4) Kähler form. The corresponding Kähler coupling strength αK (M4) is huge - so huge that one has

    αK (M4)4π R2(M4)== L2 ,

    where 1/L2 is of the order of cosmological constant and thus of the order of the size of the recent Universe. αK(M4) is also analogous to critical temperature and the earlier hypothesis that the values of L correspond to p-adic length scales implies that the values of come as αK(M4) ∝ p≈ 2k, p prime, k prime.

  3. The Kähler form assignable to M4 is not assumed to contribute to the action since it does not contribute to spinor connection of M4. One can of course ask whether it could be present. For canonically imbedded M4 self-duality implies that this contribution vanishes and for vacuum extremals of ordinary Kähler action this contribution is small.Breaking of Lorentz invariance is however a possible problem. If αK(M4) is given by above expression, then this contribution is extremely small.
Hence one can consider the possibility that the action is just the sum of full 6-D Kähler actions assignable to T(M4) and T(CP2) but with different values of αK if one has (w1,w2)=(n,0). Also other w2≠ 0 is possible but corresponds to gigantic cosmological constant.

Given the parameter L2 as it is defined above, one can deduce an expression for cosmological constant Λ and show that it is positive.

  1. 6-D Kähler action has dimensions of length squared and one must scale it by a dimensional constant: call it 1/L2. L is a fundamental scale and in dimensional reduction it gives rise to cosmological constant. Cosmological constant Λ is defined in terms of vacuum energy density as Λ =8π Gρvac can have two interpretations. Λ can correspond to a modification of Einstein-Hilbert action or - as now - to an additional term in the action for matter. In the latter case positive Λ means negative pressure explaining the observed accelerating expansion. It is actually easy to deduce the sign of Λ.

    1/L2 multiplies both Kähler action - FijFij (∝ E2-B2 in Minkowskian signature). The energy density is positive. For Kähler action the sign of the multiplier must be positive so that 1/L2 is positive. The volume term is fiber space part of action having same form as Kähler action. It gives a positive contribution to the energy density and negative contribution to the pressure.

    In Λ= 8π Gρvac one would have ρvac=π/L2RD2 as integral of the -FijFij over S2 given the π/RD2 (no guarantee about correctness of numerical constants). This gives Λ= 8π2G/L2RD2. Λ is positive and the sign is same as as required by accelerated cosmic expansion. Note that super string models predict wrong sign for Λ. Λ is also dynamical since it depends on RD, which is dynamical. One has 1/L2 =kΛ, k=8π2G/RD2 apart from numerical factors.

    The value of L of deduced from Euclidian and Minkowskian regions in this formal manner need not be same. Since the GRT limit of TGD describes space-time sheets with Minkowskian signature, the formula seems to be applicable only in Minkowskian regions. Again one can argue that one cannot exclude Euclidian space-time sheets of even macroscopic size and blackholes and even ordinary concept matter would represent this kind of structures.

  2. L is not size scale of any fundamental geometric object. This suggests that L is analogous to αK and has value spectrum dictated by p-adic length scale hypothesis. In fact, one can introduce the ratio of ε=R2/L2 as a dimensionless parameter analogous to coupling strength what it indeed is in field equations. If so, L could have different values in Minkowskian and Euclidian regions.
  3. I have earlier proposed that RU==(1/Λ)1/2 is essentially the p-adic length scale Lp ∝ p1/2= 2k/2, p≈ 2k, k prime, characterizing the cosmology at given time and satisfies RU∝ a meaning that vacuum energy density is piecewise constant but on the average decreases as 1/a2, a cosmic time defined by light-cone proper time. A more natural hypothesis is that L satisfies this condition and in turn implies similar behavior or RU. p-Adic length scales would be the critical values of L so that also p-adic length scale hypothesis would emerge from quantum critical dynamics! This conforms with the hypothesis about the value spectrum of αK labelled in the same manner (see this).
  4. At GRT limit the magnetic energy of the flux tubes gives rise to an average contribution to energy momentum tensor, which effectively corresponds to negative pressure for which the expansion of the Universe accelerates. It would seem that both contributions could explain accelerating expansion. If the dynamics for Kähler action and volume term are coupled, one would expect same orders of magnitude for negative pressure and energy density - kind of equipartition of energy.
Consider first the scales emerging from GRT picture. RU ≈ (1/Λ1/2≈ 1026 m = 10 Gly is not far from the recent size of the Universe defined as c× t ≈ 13.8 Gly. The derived size scale L1==(RU× lP)1/2 is of the order of L1=.5× 10-4 meters, the size of neuron. Perhaps this is not an accident. To make life of the reader easier I have collected the basic numbers to the following table.

m(CP2)≈ 5.7× 1014 GeV mP=2.435 × 1018 GeV R(CP2)/lP≈ 4.1× 103
RU= 10 Gy t= 13.8 Gy L1= (lPRU)1/2=.5 × 10-4

Let us consider now some quantitative estimates. R(X4) depends on homotopy equivalence classes of the maps from S2(X4)→ S2(M4) and S2(X4)→ S2(CP2) - that is winding numbers wi , i=0,2 for these maps. The simplest situations correspond to the winding numbers (w1,w2)=(1,0) and (w1,w2)=(0,1) . For (w1,w2)=(1,0) M4 contribution to the metric of S2(X4) dominates and one has R(X4)≈ R(M4) . For R(M4)=lP so Planck length would define a fundamental length and Planck mass and Newton's constant would be quantal parameters. For (w1,w2)=(0,1) the radius of sphere would satisfy RD≈ R ( CP2 size): now also Planck length would be quantal parameter.

Consider next additional scales emerging from TGD picture.

  1. One has L = ( 23/2π lP/RD)× RU. In Minkowskian regions with RH=lP this would give L = 8.9× RU: there is no obvious interpretation for this number. If one takes the formula seriously in Euclidian regions one obtains the estimate L=29 Mly. The size scale of large voids varies from about 36 Mly to 450 Mly (see this).
  2. Consider next the derived size scale L2=(L× lP)1/2 = [L/RU]1/2 × L1 = [23/2π lP/RD]1/2× L1. For RD=lP one has L2 ≈ 3L1. For RD=R making sense in Euclidian regions, this is of the order of size of neutrino Compton length: 3 μm, the size of cellular nucleus and rather near to the p-adic length scale L(167)= 2.6 m, corresponds to the largest miracle Gaussian Mersennes associated with k=151,157,163,167 defining length scales in the range between cell membrane thickness and the size of cellular nucleus. Perhaps these are co-incidences are not accidental. Biology is something so fundamental that fundamental length scale of biology should appear in the fundamental physics.
The formulas and predictions for different options are summarized by the following table.

Option L=[23/2π lP/RD]× RU L2=(LlP)1/2 = [23/2π lP/RD]1/2× L1
RD= R 29 Mly ≈ 3 μ m
RD=lP 8.9RU ≈ 3L1=1.5× 10-4 m

In the case of M4 the radius of S2 cannot be fixed it remains unclear whether Planck length scale is fundamental constant or whether it emerges.

2.2 What about extremals of the dimensionally reduced 6-D Kähler action?

It seems that the basic wisdom about extremals of Kähler action remains unaffected and the motivations for WCW are not lost. What is new is that the removal of vacuum degeneracy is forced by twistorial action.

  1. All extremals, which are either vacuum extremals or minimal surfaces remain extremals. In fact, all extremals that I know. For minimal surfaces the dynamics of the volume term and 4-D Kähler action separate and field equations for them are separately satisfied. The vacuum degeneracy motivating the introduction of WCW is preserved. The induced Kähler form vanishes for vacuum extremals and the imaginary unit of twistor space is ill-defined. Hence vacuum extremals cannot belong to WCW. This correspond to the vanishing of WCW metric for vacuum extremals.
  2. For non-minimal surfaces Kähler coupling strength does not disappear from the field equations and appears as a genuine coupling very much like in classical field theories. Minimal surface equations are a generalization of wave equation and Kähler action would define analogs of source terms. Field equations would state that the total isometry currents are conserved. It is not clear whether other than minimal surfaces are possible, I have even conjectured that all preferred extremals are always minimal surfaces having the property that being holomorphic they are almost universal extremals for general coordinate invariant actions.
  3. Thermodynamical analogy might help in the attempts to interpret. Quantum TGD in zero energy ontology (ZEO) corresponds formally to a complex square root of thermodynamics. Kähler action can be identified as a complexified analog of free energy. Complexification follows both from the fact that g1/2 is real/imaginary in Euclidian/Minkowskian space-time regions. Complex values are also implied by the proposed identification of the values of Kähler coupling strength in terms of zeros and pole of Riemann zeta in turn identifiable as poles of the so called fermionic zeta defining number theoretic partition function for fermions (see this). The thermodynamical for Kähler action with volume term is Gibbs free energy G= F-TS= E-TS+PV playing key role in chemistry.
  4. The boundary conditions at the ends of space-time surfaces at boundaries of CD generalize appropriately and symmetries of WCW remain as such. At light-like boundaries between Minkowskian and Euclidian regions boundary conditions must be generalized. In Minkowkian regions volume can be very large but only the Euclidian regions contribute to Kähler function so that vacuum functional can be non-vanishing for arbitrarily large space-time surfaces since exponent of Minkowskian Kähler action is a phase factor.
  5. One can worry about almost topological QFT property. Although Kähler action from Minkowskian regions at least would reduce to Chern-Simons terms with rather general assumptions about preferred extremals, the extremely small cosmological term does not. Could one say that cosmological constant term is responsible for "almost"?

    It is interesting that the volume of manifold serves in algebraic geometry as topological invariant for hyperbolic manifolds, which look locally like hyperbolic spaces Hn=SO(n,1)/SO(n). See also the article Volumes of hyperbolic manifolds and mixed Tate motives. Now one would have n=4. It is probably too much to hope that space-time surfaces would be hyperbolic manifolds. In any case, by the extreme uniqueness of the preferred extremal property expressed by strong form of holography the volume of space-time surface could also now serve as topological invariant in some sense as I have earlier proposed. What is intriguing is that AdSn appearing in AdS/CFT correspondence is Lorentzian analogue Hn.

To sum up, the twistor lift of the dynamics of Kähler action allows to understand the origin of Planck length and cosmological constant. Here the earlier picture has been incomplete. Also the size scale of large voids and two fundamental biological length scales appear. p-Adic length scale hypothesis is realized in terms of the scaling factor of the 6-D Käler action defining giving rise to a dimensionless coupling constant. What is most remarkable that since only M4 and CP2 allow twistor space with Kähler structure, TGD is completely unique in twistor formulation.

For background see the chapter From Principles to Diagrams or the article From Principles to Diagrams.

From Principles to Diagrams

The generalization of twistor diagrams to TGD framework has been very inspiring (and also frightening) mission impossible and allowed to gain deep insights about what TGD diagrams could be mathematically. I of course cannot provide explicit formulas but the general structure for the construction of twistorial amplitudes in N=4 SUSY suggests an analogous construction in TGD thanks to huge symmetries of TGD and unique twistorial properties of M4× CP2.

I try to summarize the big vision. Several guiding principles are involved and have gradually evolved to a coherent whole. The generalization of twistor diagrams to TGD framework has been very inspiring (and also frightening) mission impossible and allowed to gain deep insights about what TGD diagrams could be mathematically. I of course cannot provide explicit formulas but the general structure for the construction of twistorial amplitudes in N=4 SUSY suggests an analogous construction in TGD thanks to huge symmetries of TGD and unique twistorial properties of M4× CP2.

I try to summarize the big vision. Several guiding principles are involved and have gradually evolved to a coherent whole.

Imbedding space is twistorially unique

It took roughly 36 years to learn that M4 and CP2 are twistorially unique.

  1. Space-times are surfaces in M4× CP2. M4 and CP2 are unique 4-manifolds in the sense that both allow twistor space with Kähler structure: Kähler structure is the crucial concept. Strictly speaking, M4 and its Euclidian variant E4 allow both twistor space and the twistor space of M4 is Minkowskian variant T(M4)= SU(2,2)/SU(2,1)× U(1) of 6-D twistor space CP3= SU(4)/SU(3)× U(1) of E4. The twistor space of CP2 is 6-D T(CP2)= SU(3)/U(1)× U(1), the space for the choices of quantization axes of color hypercharge and isospin.
  2. This leads to a proposal for the formulation of TGD in which space-time surfaces X4 in H are lifted to twistor spaces X6, which are sphere bundles over X4 and such that they are surfaces in 12-D product space T(M4)× T(CP2) such the twistor structure of X4 are in some sense induced from that of T(M4)× T(CP2). What is nice in this formulation is that one can use all the machinery of algebraic geometry so powerful in superstring theory (Calabi-Yau manifolds). It was a complete surprise that a clear examination of this ideas leads to a profound understanding of the relationship between TGD and GRT (this will be discussed in later blog posting). Planck length emerges whereas fundamental constant as also cosmological constant emerges dynamically from the length scale parameter appearing in 6-D Kähler action. One can say, that twistor extension is absolutely essential for really understanding the gravitational interactions although the modification of Kähler action is extremely small due to the huge value of length scale defined by cosmological constant.
  3. Masslessness (masslessness in complex sense for virtual particles in twistorialization) is essential condition for twistorialization. In TGD massless is masslessness in 8-D sense for the representations of superconformal algebras. This suggests that 8-D variant of twistors makes sense. 8-dimensionality indeed allows octonionic structure in the tangent space of imbedding space. One can also define octonionic gamma matrices and this allows a possible generalization of 4-D twistors to 8-D ones using generalization of sigma matrices representing quaternionic units to to octonionic sigma "matrices" essential for the notion of twistors. These octonion units do not of course allow matrix representation unless one restricts to units in some quaternionic subspace of octonions. Space-time surfaces would be associative and thus have quaternionic tangent space at each point satisfying some additional conditions.

Strong form of holography

Strong form of holography (SH) following from general coordinate invariance (GCI) for space-times as surfaces states that the data assignable to string world sheets and partonic 2-surfaces allows to code for scattering amplitudes. The boundaries of string world sheets at the space-like 3-surfaces defining the ends of space-time surfaces at boundaries of causal diamonds (CDs) and the fermionic lines along light-like orbits of partonic 2-surfaces representing lines of generalized Feynman diagrams become the basic elements in the generalization of twistor diagrams (I will not use the attribute "Feynman" in precise sense, one could replace it with "twistor" or even drop away). One can assign fermionic lines massless in 8-D sense to flux tubes, which can also be braided.

One obtains a fractal hierarchy of braids with strands, which are braids themselves. At the lowest level one has braids for which fermionic lines are braided. This fractal hierarchy is unavoidable and means generalization of the ordinary Feynman diagram. I have considered some implications of this hierarchy (see this).

The existence of WCW demands maximal symmetries

Quantum TGD reduces to the construction of Kähler geometry of infinite-D "world of classical worlds" (WCW), of associated spinor structure, and of modes of WCW spinor fields which are purely classical entities and quantum jump remains the only genuinely quantal element of quantum TGD. Quantization without quantization, would Wheeler say.

By its infinite-dimensionality, the mere mathematical existence of the Kähler geometry of WCW requires maximal isometries. Physics is completely fixed by the mere condition that its mathematical description exists.

Super-symplectic and other symmetries of WCW are in decisive role. These symmetry algebras have conformal structure and generalize and extend the conformal symmetries of string models (Kac-Moody algebras in particular). These symmetries give also rise to the hierarchy of Planck constants. The super-symplectic symmetries extend to a Yangian algebra, whose generators are polylocal in the sense that they involve products of generators associated with different partonic surfaces. These symmetries leave scattering amplitudes invariant. This is an immensely powerful constraint, which remains to be understood.

Quantum criticality

Quantum criticality (QC) of TGD Universe is a further principle. QC implies that Kähler coupling strength is mathematically analogous to critical temperature and has a discrete spectrum. Coupling constant evolution is replaced with a discrete evolution as function of p-adic length scale: sequence of jumps from criticality to a more refined criticality or vice versa (in spin glass energy landscape you at bottom of well containing smaller wells and you go to the bottom of smaller well).

This implies that either all radiative corrections (loops) sum up to zero (QFT limit) or that diagrams containing loops correspond to the same scattering amplitude as tree diagrams so that loops can eliminated by transforming them to arbitrary small ones and snipping away moving the end points of internal lines along the lines of diagram (fundamental description).

Quantum criticality at the level of super-conformal symmetries leads to the hierarchy of Planck constants heff=n× h labelling a hierarchy of sub-algebras of super-symplectic and other conformal algebras isomorphic to the full algebra. Physical interpretation is in terms of dark matter hierarchy. One has conformal symmetry breaking without conformal symmetry breaking as Wheeler would put it.

Physics as generalized number theory, number theoretical universality

Physics as generalized number theory vision has important implications. Adelic physics is one of them. Adelic physics implied by number theoretic universality (NTU) requires that physics in real and various p-adic numbers fields and their extensions can be obtained from the physics in their intersection corresponding to an extension of rationals. This is also enormously powerful condition and the success of p-adic length scale hypothesis and p-adic mass calculations can be understood in the adelic context.

In TGD inspired theory of consciousness various p-adic physics serve as correlates of cognition and p-adic space-time sheets can be seen as cognitive representations, "thought bubbles". NTU is closely related to SH. String world sheets and partonic 2-surfaces with parameters (WCW coordinates) characterizing them in the intersection of rationals can be continued to space-time surfaces by preferred extremal property but not always. In p-adic context the fact that p-adic integration constants depend on finite number of pinary digits makes the continuation easy but in real context this need not be possible always. It is always possible to imagine something but not always actualize it!

Scattering diagrams as computations

Quantum criticality as possibility to eliminate loops has a number theoretic interpretation. Generalized Feynman diagram can be interpreted as a representation of a computation connecting given set X of algebraic objects to second set Y of them (initial and final states in scattering) (trivial example: X={3,4} → 3× 4 = 12 → 2× 6 → {2,6}=Y. The 3-vertices (a× b=c) and their time-reversals represent algebraic product and co-product.

There is a huge symmetry: all diagrams representing computation connecting given X and Y must produce the same amplitude and there must exist minimal computation. The task of finding this computation is like finding the simplest representation for the formula X=Y and the noble purpose of math teachers is that we should learn to find it during our school days. This generalizes the duality symmetry of old fashioned string models: one can transform any diagram to a tree diagram without loops. This corresponds to quantum criticality in TGD: coupling constants do not evolve. The evolution is actually there but discrete and corresponds to infinite number critical values for Kahler coupling strength analogous to temperature.

Reduction of diagrams with loops to braided tree-diagrams

  1. In TGD pointlike particles are replaced with 3-surfaces and by SH by partonic 2-surfaces. The important implication of 3-dimensionality is braiding. The fermionic lines inside light-like orbits of partonic 2-surfaces can be knotted and linked - that is braided (this is dynamical braiding analogous to dance). Also the fermionic strings connecting partonic 2-surfaces at space-like 3-surfaces at boundaries of causal diamonds (CDs) are braided (space-like braiding).

    Therefore ordinary Feynman diagrams are not enough and one must allow braiding for tree diagrams. One can also imagine of starting from braids and allowing 3-vertices for their strands (product and co-product above). It is difficult to imagine what this braiding could mean. It is better to imagine braid and allow the strands to fuse and split (annihilation and pair creation vertices).

  2. This braiding gives rise in the planar projection representation of braids to a generalization of non-planar Feynman diagrams. Non-planar diagrams are the basic unsolved problem of twistor approach and have prevented its development to a full theory allowing to construct exact expressions for the full scattering amplitudes (I remember however that Nima Arkani-Hamed et al have conjectured that non-planar amplitudes could be constructed by some procedure: they notice the role of permutation group and talk also about braidings (describable using covering groups of permutation groups)). In TGD framework the non-planar Feynman diagrams correspond to non-trivial braids for which the projection of braid to plane has crossing lines, say a and b, and one must decide whether the line a goes over b or vice versa.
  3. An interesting open question is whether one must sum over all braidings or whether one can choose only single braiding. Choice of single braiding might be possible and reflect the failure of string determinism for Kähler action and it would be favored by TGD as almost topological quantum field theory (TQFT) vision in which Kähler action for preferred extremal is topological invariant.

Scattering amplitudes as generalized braid invariants

The last big idea is the reduction of quantum TGD to generalized knot/braid theory (I have talked also about TGD as almost TQFT). The scattering amplitude can be identified as a generalized braid invariant and could be constructed by the generalization of the recursive procedure transforming in a step-by-step manner given braided tree diagram to a non-braided tree diagram: essentially what Alexander the Great did for Gordian knot but tying the pieces together after cutting. At each step one must express amplitude as superposition of amplitudes associated with the different outcomes of splitting followed by reconnection. This procedure transforms braided tree diagram to a non-braided tree diagrams and the outcome is the scattering amplitude!

For background see the chapter From Principles to Diagrams or the article From Principles to Diagrams.

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