Tesla inspires still

Tesla has served as a source of inspiration for free energy researchers decade after decade. The claims assigned with Tesla and free energy are not consistent with the prevailing belief systems, and it is very interesting to look whether they could make sense in TGD framework predicting a lot of new physics. I have even written a book studying these claims systematically (see this). The book is written about two decades ago and TGD has developed a lot after than and it is interesting to take a glimple at it with the recent understanding of TGD.

1. Some claims associated with Tesla and free energy research

In the following some claims assigned with Tesla and free energy research are summarized.

1. Tesla explained his observations with scalar waves. Maxwellian electrodynamics does not however allow massless photons with vanishing spin. Tesla also believed on aether but to my view Tesla was here a child of his time.
   

   My recent view is that scalar waves are not possible as single-sheeted structures in TGD framework. Many-sheeted space-time could however allow effective scalar waves as two-sheeted structures realized as a pair of massless extremals (MEs) representing waves of opposite polarization propagating in the same direction. From the point of view of test particles the effect of MEs is indeed like that of a scalar wave. This variant of scalar wave could explain the findings claimed by Tesla. Also the analogs of light waves propagating with arbitrary small velocity and even of standing waves make sense as pairs of MEs with opposite momentum directions.
   
   

2. The notion of free energy is different from the standard notion of free energy appearing in thermodynamics. Over-unity effects assigned also with Tesla are a typical claim. In strong form it would mean non-conservation of energy and academic community concludes that since energy is conserved, these claims are crackpot non-sense. In weaker form the claims would mean transformation of heat to work with efficiency larger than the upper bound predicted by Carnot law.
   

   Second law however kills hopes about perpetual motion machines based on this kind of over-unity effects. Second law assumes fixed arrow of time. There exists however strong empirical evidence for the possibility that the arrow of time can change - phase conjugate waves are the key example. This led Fantappie to propose the notion of syntropy as entropy in reversed time direction. Second law in reverse time direction could also allow an error correction mechanism in quantum computation: Nature itself would do it. Phase conjugate waves are indeed known to perform error correction.
   

   Quantum TGD relies on zero energy ontology (ZEO). ZEO allows both arrows of time and a temporary change of the arrow of time could make possible to break the standard laws of thermodynamics at least temporarily and in short enough scales. ZEO indeed plays a key role in TGD inspired quantum biology and quantum theory of consciousness.
   
   

3. Recall that Tesla reported strange findings in electronic circuits subjected to sudden pulses created by putting switches on or off. Could it be that these pulses were accompanied by macroscopic quantum jumps changing the arrow of time and inducing over-unity effects and breaking of second law in the standard sense? I considered this possibility
   for a couple of decades about (see this). Could one think of taking this possibility seriously and replicating the studies of Tesla?
   
   

In the following I will consider the issue of energy conservation in ZEO. Classically energy is well-defined and conserved in TGD Universe. But what about energy conservation in quantum sense? ZEO involves delocalization of states in time and this could allow energy conservation only in some resolution determined by the scale of the increment of time in given state function reduction inducing a shift of the active boundary of CD farther from the passive one.

2. Energy is conserved classically in TGD but what about conservation in quantum sense in ZEO?

ZEO guarantees classical conservation laws. What about the situation at quantum level? Could the energy associated with the positive energy part of zero energy state increase in quantum transitions and lead to over-unity effects? In principle, conservation laws do not prevent this quantally.

1. Recall that zero energy states are identified as superpositions of pairs (a,b) formed from states a and b having opposite total quantum numbers and being assigned with the opposite boundaries of causal diamond (CD). The states at the passive boundary B of CD are not affected whereas the states at the active boundary A are affected by a sequence of unitary time evolutions also shifting A farther away from B (in statistical sense at least).
   

   Each unitary evolution induces a de-localization of A in its moduli space and "small" SFR induces its localization (including time localization meaning time measurement). This sequence would approximately conserve the energies of the states in the superposition. This in the approximation that their energies are large in the energy scale Δ E =hbar_eff/Delta t defined by the time increment Δ t in single unitary time evolution. Large value of h_eff makes the conservation worse for a given Δ t. Unitarity together with the approximate energy conservation implies that the average energy is approximately conserved.
   
   

2. Negative energy signals sent from A to its geometric past and received at B in remote metabolism would correspond to "big" SFR. If the notion of remote metabolism giving effectively rise to over-unitary effect is to make sense, the approximate energy conservation should fail in "big" SFRs in quantal sense. For this to be the case, the first unitary evolution of B followed by "small" SFR energy conservation should be a bad approximation. This does not however seem plausible if one assumes energy conservation for the next state function reductions. What could be so special in the first state function reduction?
   
   
3. Why the energy conservation made approximate by the finite size of CD and finite duration of unitary evolution, should fail badly in some situations? According to the number theoretic vision, "small" SFRs preserve the extension of rationals defining the adele and therefore also hbar_eff/hbar₀=n identifiable as the dimension of the extension. hbar_eff/hbar₀=n can however change n_old→ n_new in "big" SFRs forced to occur when "small" SFRs preserving n_old are not anymore possible. If a large increase of h_eff occurs in the "big" SFR, the Δ E=hbar_eff/Delta t increases if Δ t is still of the same order of magnitude. The approximate energy conservation could fail badly enough to make possible remote metabolism.
   
   
4. In the subsequent SFRs energy conservation should however hold true in good approximation. The values of Δ t should be large in the subsequent "small" SFRs, and Δ t should scale as Δ t ∝ n to guarantee that Δ E remains the same. As a quantum scale Δ t analogous to Compton length is indeed proportional to n. In the first reduction one must have of n=n_old but in the subsequent reductions one must have n=n_new to guarantee energy conservation in the same approximation as before.
   

   To sum up: in the first "small" SFR one should have Δ E∝ n_new and Δ t∝ n_new. Can one really deduce this from the basic TGD?
   
   

5. ZEO suggests that evolution means a continual increase of the size of CD so that small CD could eventually grow to even cosmic size (whether this occurs always or whether zero energy state can become pure vacuum at both boundaries of CD remains an open problem). CD with a cosmic size should however have huge energy. This would not only require non-conservation of energy in quantal sense but also its increase in statistical sense at least. Why should the energy increase?
   

   The increase would relate directly to the basic defining property of ZEO. Preferred time direction means that the transfer of energy quantum numbers can take place only from the active boundary of CD to the passive boundary in "big" SFR. This allows interpretation as remote metabolism implying increase of the magnitude of energy.
   
   

3. Could Nature provide an error correction mechanism for quantum computation?

Error correction has turned out to be major problem in the attempts to construct quantum computers. It is believed to be necessary because quantum entanglement is extremely fragile for the standard value of Planck constant. In TGD the situation changes. Large values of h_eff increasing the time scale of entanglement are possible and reversed time evolutions in quantum sense imply second law in reversed time direction meaning spontaneous reduction of entropy in the standard time direction. Nature itself would provide the needed error correction mechanism perhaps applied routinely in living systems (for instance, to correct mutations of DNA and transcription and translation errors).

To sum up, this picture is extremely interesting from the point of view of future technologies. One can even challenge the cherised law of energy conservation at quantum level (classically it remains exact in TGD Universe). Could one consider the possibility that the energy of system could be increased by the evolution by "big"> state function reductions increasing the value of h_eff? Could one at least temporarily reduce entropy by inducing time evolutions in opposite time direction? TGD strongly suggests that these mechanisms are at work in biology. Maybe energy and iquantum nformation technologists could learn something from living matter?

See the article Tesla still inspires or the chapter Construction of Quantum Theory: More about Matrices of "Towards M-matrix".

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

Articles and other material related to TGD.

A connection of singularities of minimal surfaces with generation of Higgs vacuum expectation?

String world sheet appear as singularities of space-time surfaces as minimal surfaces. At string world sheets minimal surface equations fail and there is transfer of Noether charges associated with Kähler and volume degrees of freedom at string world. This has interpretation as analog for the interaction of charged particle with Maxwell field.

What about the physical interpretation of the singular divergences of the isometry currents J_A of the volume action located at string world sheet?

1. The divergences of J_A are proportional to the trace of the second fundamental form H formed by the covariant derivatives of gradients ∂_αh^k of H-coordinates in the interior and vanish. The singular contribution at string world sheets is determined by the discontinuity of the isometry current J_A and involves only the first derivatives ∂_αh^k.
   
   
2. One of the first questions after ending up with TGD for 41 years ago was whether the trace of H in the case of CP₂ coordinates could serve as something analogous to Higgs vacuum expectation value. The length squared for the trace has dimensions of mass squared. The discontinuity of the isometry currents for SU(3) parts in h=u(2) and its complement t, whose complex coordinates define u(2) doublet. u(2) is in correspondence with electroweak algebra and t with complex Higgs doublet. Could an interpretation as Higgs or even its vacuum expectation make sense?
   
   
3. p-Adic thermodynamics explains fermion masses elegantly (understanding of boson masses is not in so good shape) in terms of thermal mixing with excitations having CP₂ mass scale and assignable to short string associated with wormhole contacts. There is also a contribution from long strings connecting wormhole contacts and this could be important for the understanding of weak gauge boson masses. Could the discontinuity of isometry currents determine this contribution to mass. Edges/folds would carry mass.
   
   
4. The non-singular part of the divergence multiplying 2-D delta function has dimension 1/length squared and the square of this vector in CP₂ metric has dimension of mass squared. Could the interpretation of the discontinuity as Higgs expectation make sense? If so, Higgs expectation would vanish in the space-time interior.
   

   Could the interior modes of the induced spinor field - or at least the interior mode of right-handed neutrino ν_R having no couplings to weak or color fields - be massless in 8-D or even 4-D sense? Could ν_R and νbar_R generate an unbroken N=2 SUSY in interior whereas inside string world sheets right-handed neutrino and antineutrino would be eaten in neutrino massivation and the generators of N=2 SUSY would be lost somewhat like charged components of Higgs!
   

   If so, particle physicists would be trying to find SUSY from wrong place. Space-time interior would be the correct place. Would the search of SUSY be condensed matter physics rather than particle physics?
   

   
   Remark: There is an interesting delicacy involved. Consider an edge at 3-surface. The divergence from the discontinuity contains contributions from two normal coordinates proportional to a delta function for the normal coordinate and coming from the discontinuity. The discontinuity must be however localized to the string rather than 2-surface. There must be present also a delta function for the second normal coordinate. Hence the value of normal discontinuity must be infinite along the string. One would have infinitely sharp edge. A concrete example is provided by function y= |x|^α, α<1. This kind of situation is encountered in Thom's catastrophe theory for the projection of the catastrophe: in this case one has α=1/2. This argument generalizes to 3-D case but visualization is possible only as a motion of infinitely sharp edge of 3-surface.
   

   Kähler form and metric are second degree monomials of partial derivatives so that an attractive assumption is that g_αβ, J_αβ and therefore also the components of volume and Kähler energy momentum tensor are continuous. This would allow ∂_nih^k to become infinite and change sign at the discontinuity as the idea about infinitely sharp edge suggests. This would reduce the continuity conditions for canonical momentum currents to rather simple form
   

   T^n_in_jΔ ∂_njh^k=0 .
   

   which in turn would give
   

   T^n_in_j=0
   

   stating that canonical momentum is conserved but transferred between Kähler and volume degrees of freedom. One would have a condition for a continuous quantity conforming with the intuitive view about boundary conditions due to conservation laws. The condition would state that energy momentum tensor reduces to that for string world sheet at the singularity so that the system becomes effectively 2-D. I have already earlier proposed this condition.
   

   The reduction of 4-D locally to effectively 2-D system raises the question whether any separate action is needed for string world sheets (and their boundaries)? The generated 2-D action would be similar to the proposed 2-D action. By super-conformal symmetry similar generation of 2-D action would take place also in the fermionic degrees of freedom. I have proposed also this option already earlier.
   

See the chapter The Recent View about Twistorialization in TGD Framework or the article More about the construction of scattering amplitudes in TGD framework.

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

Articles and other material related to TGD.

Idea before its time: space-time surfaces as Kähler calibrated surfaces

When ideas stop flowing, it is best to stay calm and do something practical. Updating of books or homepage is not rocket science but gives a feeling that one is doing something useful. I realized that 7 books have grown so that they have about thousand pages and decided to divide them to two pieces: the result is that the number of books grew to the magic number 24.

This led to the updating of the introductions of books. I have the habit of writing introductions so that they reflect the latest overall view - books themselves contain older archeological layers and inconsistencies are unavoidable. Also at this time I experienced several not merely pleasant surprises.

A pleasant surprise was that the discrete coupling contant evolution predicted by TGD implying the vanishing of loop corrections, simplifying twistorial scattering amplitudes and their recursion formulas dramatically, and also implying that scattering amplitudes reduce to sums of resonance contributions. I realized that this is nothing but the Veneziano duality, which served as starting point of dual resonance models leading to string picture and later to super string theories.

This suggests a new insight possibly allowing to get out of the dead end of super string models. What would be the really deep thing would be the sum over resonances picture. The continuous cuts are obtained only approximately at the limit when the density of poles becomes large enough.

In string model picture this is not possible since one cannot obtain anything resembling gauge theories. In TGD framework ot seems however possible to circumvent all the objections that I have managed to discover. The first crucial element is that in TGD also classical conserved quantities can be complex (finite width for resonances needed for unitarity). Second crucial element is that string tension has discrete spectrum reducing to that for cosmological constant.

A surprise that looked unpleasant at first was the finding that I had talked about so called calibrations of sub-manifolds as something potentially important for TGD and later forgotten the whole idea! A closer examination however demonstrated that I had ended up with the analog of this notion completely independently later as the idea that preferred extremals are minimal surfaces apart form 2-D singular surfaces, where there would be exchange of Noether charges between Kähler and volume degrees of freedom.

1. The original idea that I forgot too soon was that the notion of calibration (see this) generalizes and could be relevant for TGD. A calibration in Riemann manifold M means the existence of a k-form φ in M such that for any orientable k-D sub-manifold the integral of φ over M equals to its k-volume in the induced metric. One can say that metric k-volume reduces to homological k-volume.
   

   Calibrated k-manifolds are minimal surfaces in their homology class since the variation of the integral of φ is identically vanishing. Kähler calibration is induced by the k^th power of Kähler form and defines calibrated sub-manifold of real dimension 2k. Calibrated sub-manifolds are in this case precisely the complex sub-manifolds. In the case of CP₂ they would be complex curves (2-surfaces) as has become clear.
   
   

2. By the Minkowskian signature of M⁴ metric, the generalization of calibrated sub-manifold so that it would apply in M⁴× CP₂ is non-trivial. Twistor lift of TGD however forces to introduce the generalization of Kähler form in M⁴ (responsible for CP breaking and matter antimatter asymmetry) and calibrated manifolds in this case would be naturally analogs of string world sheets and partonic 2-surfaces as minimal surfaces. Cosmic strings are Cartesian products of string world sheets and complex curves of CP₂. Calibrated manifolds, which do not reduce to Cartesian products of string world sheets and complex surfaces of CP₂ should also exist and are minimal surfaces.
   

   One can also have 2-D calibrated surfaces and they could correspond to string world sheets and partonic 2-surfaces which also play key role in TGD. Even discrete points assignable to partonic 2-surfaces and representing fundamental fermions play a key role and would trivially correspond to calibrated surfaces.
   
   

3. Much later I ended up with the identification of preferred extremals as minimal surfaces by totally different route without realizing the possible connection with the generalized calibrations. Twistor lift and the notion of quantum criticality led to the proposal that preferred extremals for the twistor lift of Kähler action containing also volume term are minimal surfaces. Preferred extremals would be separately minimal surfaces and extrema of Kähler action and generalization of complex structure to what I called Hamilton-Jacobi structure would be an essential element. Quantum criticality outside singular surfaces would be realized as decoupling of the two parts of the action. May be all preferred extremals be regarded as calibrated in generalized sense.
   

   If so, the dynamics of preferred extremals would define a homology theory in the sense that each homology class would contain single preferred extremal. TGD would define a generalized topological quantum field theory with conserved∈dexNoether charge Noether charges (in particular rest energy) serving as generalized topological invariants having extremum in the set of topologically equivalent 3-surfaces.
   
   

4. The experience with CP₂ would suggest that the Kähler structure of M⁴ defining the counterpart of form φ is unique. There is however infinite number of different closed self-dual Kähler forms of M⁴ defining what I have called Hamilton-Jacobi structures. These forms can have subgroups of Poincare group as symmetries. For instance, magnetic flux tubes correspond to given cylindrically symmetry Kähler form. The problem disappears as one realizes that Kähler structures characterize families of preferred extremals rather than M⁴.
   
   

If the notion of calibration indeed generalizes, one ends up with the same outcome - preferred extremals as minimal surfaces with 2-D string world sheets and partonic 2-surfaces as singularities - from many different directions.

1. Quantum criticality requires that dynamics does not depend on coupling parameters so that extremals must be separately extremals of both volume term and Kähler action and therefore minimal surfaces for which these degrees of freedom decouple except at singular 2-surfaces where the necessary transfer of Noether charges between two degrees of freedom takes place at these. One ends up with string picture but strings alone are of course not enough. For instance, the dynamical string tension is determined by the dynamics for the twistor lift.
   
   
2. Almost topological QFT picture implies the same outcome: topological QFT property fails only at the string world sheets.
   
   
3. Discrete coupling constant evolution, vanishing of loop corrections, and number theoretical condition that scattering amplitudes make sense also in p-adic number fields, requires a representation of scattering amplitudes as sum over resonances realized in terms of string world sheets.
   
   
4. In the standard QFT picture about scattering incoming states are solutions of free massless field equations and interaction regions the fields have currents as sources. This picture is realized by the twistor lift of TGD in which the volume action corresponds to geodesic length and Kähler action to Maxwell action and coupling corresponds to a transfer of Noether charges between volume and Kähler degrees of freedom. Massless modes are represented by minimal surfaces arriving inside causal diamond (CD) and minimal surface property fails in the scattering region consisting of string world sheets.
   
   
5. Twistor lift forces M⁴ to have generalize Kähler form and this in turn strongly suggests a generalization of the notion of calibration. At physics side the implication is the understanding of CP breaking and matter anti-matter asymmetry.
   
   
6. M⁸-H duality requires that the dynamics of space-time surfaces in H is equivalent with the algebraic dynamics in M⁸. The effective reduction to almost topological dynamics implied by the minimal surface property implies this. String world sheets (partonic 2-surfaces) in H would be images of complex (co-complex sub-manifolds) of X⁴⊂ M⁸ in H. This should allows to understand why the partial derivatives of imbedding space coordinates can be discontinuous at these edges/folds but there is no flow between interior and singular surface implying that string world sheets are minimal surfaces (so that one has conformal invariance).
   
   
   

See the chapter The Recent View about Twistorialization in TGD Framework or the article More about the construction of scattering amplitudes in TGD framework.

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

Articles and other material related to TGD.

Twistors in TGD and unexpected connection with Veneziano duality

The twistorialization of TGD has two aspects. The attempt to generalize twistor Grassmannian approach emerged first. It was however followed by the realization that also the twistor lift of TGD at classical space-time level is needed. It turned out that that the progress in the understanding of the classical twistor lift has been much faster - probably this is due to my rather limited technical QFT skills.

Twistor lift at space-time level

8-dimensional generalization of ordinary twistors is highly attractive approach to TGD. The reason is that M⁴ and CP₂ are completely exceptional in the sense that they are the only 4-D manifolds allowing twistor space with Kähler structure. The twistor space of M⁴× CP₂ is Cartesian product of those of M⁴ and CP₂. The obvious idea is that space-time surfaces allowing twistor structure if they are orientable are representable as surfaces in H such that the properly induced twistor structure co-incides with the twistor structure defined by the induced metric.

In fact, it is enough to generalize the induction of spinor structure to that of twistor structure so that the induced twistor structure need not be identical with the ordinary twistor structure possibly assignable to the space-time surface. The induction procedure reduces to a dimensional reduction of 6-D Kähler action giving rise to 6-D surfaces having bundle structure with twistor sphere as fiber and space-time as base. The twistor sphere of this bundle is imbedded as sphere in the product of twistor spheres of twistor spaces of M⁴ and CP₂.

This condition would define the dynamics, and the original conjecture was that this dynamics is equivalent with the identification of space-time surfaces as preferred extremals of Kähler action. The dynamics of space-time surfaces would be lifted to the dynamics of twistor spaces, which are sphere bundles over space-time surfaces. What is remarkable that the powerful machinery of complex analysis becomes available.

It however turned out that twistor lift of TGD is much more than a mere technical tool. First of all, the dimensionally reduction of 6-D Kähler action contained besides 4-D Kähler action also a volume term having interpretation in terms of cosmological constant. This need not bring anything new, since all known extremals of Kähler action with non-vanishing induced Kähler form are minimal surfaces. There is however a large number of imbeddings of twistor sphere of space-time surface to the product of twistor spheres. Cosmological constant has spectrum and depends on length scale, and the proposal is that coupling constant evolution reduces to that for cosmological constant playing the role of cutoff length. That cosmological constant could transform from a mere nuisance to a key element of fundamental physics was something totally new and unexpected.

1. The twistor lift of TGD at space-time level forces to replace 4-D Kähler action with 6-D dimensionally reduced Kähler action for 6-D surface in the 12-D Cartesian product of 6-D twistor spaces of M⁴ and CP₂. The 6-D surface has bundle structure with twistor sphere as fiber and space-time surface as base.
   

   Twistor structure is obtained by inducing the twistor structure of 12-D twistor space using dimensional reduction. The dimensionally reduced 6-D Kähler action is sum of 4-D Kähler action and volume term having interpretation in terms of a dynamical cosmological constant depending on the size scale of space-time surface (or of causal diamond CD in zero energy ontology (ZEO)) and determined by the representation of twistor sphere of space-time surface in the Cartesian product of the twistor spheres of M⁴ and CP₂.
   
   

2. The preferred extremal property as a representation of quantum criticality would naturally correspond to minimal surface property meaning that the space-time surface is separately an extremal of both Kähler action and volume term almost everywhere so that there is no coupling between them. This is the case for all known extremals of Kähler action with non-vanishing induced Kähler form.
   

   Minimal surface property could however fail at 2-D string world sheets, their boundaries and perhaps also at partonic 2-surfaces. The failure is realized in minimal sense if the 3-surface has 1-D edges/folds (strings) and 4-surface 2-D edges/folds (string world sheets) at which some partial derivatives of the imbedding space coordinates are discontinuous but canonical momentum densities for the entire action are continuous.
   

   There would be no flow of canonical momentum between interior and string world sheet and minimal surface equations would be satisfied for the string world sheet, whose 4-D counterpart in twistor bundle is determined by the analog of 4-D Kähler action. These conditions allow the transfer of canonical momenta between Kähler- and volume degrees of freedom at string world sheets. These no-flow conditions could hold true at least asymptotically (near the boundaries of CD).
   

   M⁸-H duality suggests that string world sheets (partonic 2-surfaces) correspond to images of complex 2-sub-manifolds of M⁸ (having tangent (normal) space which is complex 2-plane of octonionic M⁸).
   
   

3. Cosmological constant would depend on p-adic length scales and one ends up to a concrete model for the evolution of cosmological constant as a function of p-adic length scale and other number theoretic parameters (such as Planck constant as the order of Galois group): this conforms with the earlier picture.
   

   Inflation is replaced with its TGD counterpart in which the thickening of cosmic strings to flux tubes leads to a transformation of Kähler magnetic energy to ordinary and dark matter. Since the increase of volume increases volume energy, this leads rapidly to energy minimum at some flux tube thickness. The reduction of cosmological constant by a phase transition however leads to a new expansion phase. These jerks would replace smooth cosmic expansion of GRT. The discrete coupling constant evolution predicted by the number theoretical vision could be understood as being induced by that of cosmological constant taking the role of cutoff parameter in QFT picture.
   
   

Twistor lift at the level of scattering amplitudes and connection with Veneziano duality

The classical part of twistor lift of TGD is rather well-understood. Concerning the twistorialization at the level of scattering amplitudes the situation is much more difficult conceptually - I already mentioned my limited QFT skills.

1. From the classical picture described above it is clear that one should construct the 8-D twistorial counterpart of theory involving space-time surfaces, string world sheets and their boundaries, plus partonic 2-surfaces and that this should lead to concrete expressions for the scattering amplitudes.
   

   The light-like boundaries of string world sheets as carriers of fermion numbers would correspond to twistors as they appear in twistor Grassmann approach and define the analog for the massless sector of string theories. The attempts to understand twistorialization have been restricted to this sector.
   
   

2. The beautiful basic prediction would be that particles massless in 8-D sense can be massive in 4-D sense. Also the infrared cutoff problematic in twistor approach emerges naturally and reduces basically to the dynamical cosmological constant provided by classical twistor lift.
   

   One can assign 4-momentum both to the spinor harmonics of the imbedding space representing ground states of super-conformal representations and to light-like boundaries of string world sheets at the orbits of partonic 2-surfaces. The two four-momenta should be identical by quantum classical correspondence: this could be seen as a concretization of Equivalence Principle. Also a connection with string model emerges.
   
   

3. As far as symmetries are considered, the picture looks rather clear. Ordinary twistor Grassmannian approach boils down to the construction of scattering amplitudes in terms of Yangian invariants for conformal group of M⁴. Therefore a generalization of super-symplectic symmetries to their Yangian counterpart seems necessary. These symmetries would be gigantic but how to deduce their implications?
   
   
4. The notion of positive Grassmannian is central in the twistor approach to the scattering amplitudes in N=4 SUSYs. TGD provides a possible generalization and number theoretic interpretation of this notion. TGD generalizes the observation that scattering amplitudes in twistor Grassmann approach correspond to representations for permutations. Since 2-vertex is the only fermionic vertex in TGD, OZI rules for fermions generalizes, and scattering amplitudes are representations for braidings.
   

   Braid interpretation encourages the conjecture that non-planar diagrams can be reduced to ordinary ones by a procedure analogous to the construction of braid (knot) invariants by gradual un-braiding (un-knotting).
   
   

This is however not the only vision about a solution of non-planarity. Quantum criticality provides different view leading to a totally unexpected connection with string models, actually with the Veneziano duality, which was the starting point of dual resonance model in turn leading via dual resonance models to super string models.

1. Quantum criticality in TGD framework means that coupling constant evolution is discrete in the sense that coupling constants are piecewise constant functions of length scale replaced by dynamical cosmological constant. Loop corrections would vanish identically and the recursion formulas for the scattering amplitudes (allowing only planar diagrams) deduced in twistor Grassmann would involve no loop corrections. In particular, cuts would be replaced by sequences of poles mimicking them like sequences of point charge mimic line charges. In momentum discretization this picture follows automatically.
   
   
2. This would make sense in finite measurement resolution realized in number theoretical vision by number-theoretic discretization of the space-time surface (cognitive representation) as points with coordinates in the extension of rationals defining the adele. Similar discretization would take place for momenta. Loops would vanish at the level of discretization but what would happen at the possibly existing continuum limit: does the sequence of poles integrate to cuts? Or is representation as sum of resonances something much deeper?
   
   
3. Maybe it is! The basic idea of behind the original Veneziano amplitudes (see this) was Veneziano duality. This 4-particle amplitude was generalized by Yoshiro Nambu, Holber-Beck Nielsen, and Leonard Susskind to N-particle amplitude (see this) based on string picture, and the resulting model was called dual resonance model. The model was forgotten as QCD emerged. Later came superstring models and led to M-theory. Now it has become clear that something went wrong, and it seems that one must return to the roots. Could the return to the roots mean a careful reconsideration of the dual resonance model?
   
   
4. Recall that Veneziano duality (1968) was deduced by assuming that scattering amplitude can be described as sum over s-channel resonances or t-channel Regge exchanges and Veneziano duality stated that hadronic scattering amplitudes have representation as sums over s- or t-channel resonance poles identified as excitations of strings. The sum over exchanges defined by t-channel resonances indeed reduces at larger values of s to Regge form.
   

   The resonances had zero width, which was not consistent with unitarity. Further, there were no counterparts for the sum of s-, t-, and u-channel diagrams with continuous cuts in the kinematical regions encountered in QFT approach. What puts bells ringing is the u-channel diagrams would be non-planar and non-planarity is the problem of twistor Grassmann approach.
   
   

5. Veneziano duality is true only for s- and t- channels but not been s- and u-channel. Stringy description makes t-channel and s-channel pictures equivalent. Could it be that in fundamental description u-channels diagrams cannot be distinguished from s-channel diagrams or t-channel diagrams? Could the stringy representation of the scattering diagrams make u-channel twist somehow trivial if handles of string world sheet representing stringy loops in turn representing the analog of non-planarity of Feynman diagrams are absent? The permutation of external momenta for tree diagram in absence of loops in planar representation would be a twist of π in the representation of planar diagram as string world sheet and would not change the topology of the string world sheet and would not involve non-trivial world sheet topology.
   

   For string world sheets loops would correspond to handles. The presence of handle would give an edge with a loop at the level of 3-surface (self energy correction in QFT). Handles are not allowed if the induced metric for the string world sheet has Minkowskian signature. If the stringy counterparts of loops are absent, also the loops in scattering amplitudes should be absent.
   

   This argument applies only inside the Minkowskian space-time regions. If string world sheets are present also in Euclidian regions, they might have handles and loop corrections could emerge in this manner. In TGD framework strings (string world sheets) are identified to 1-D edges/folds of 3-surface at which minimal surface property and topological QFT property fails (minimal surfaces as calibrations). Could the interpretation of edge/fold as discontinuity of some partial derivatives exclude loopy edges: perhaps the branching points would be too singular?
   
   

A reduction to a sum over s-channel resonances is what the vanishing of loops would suggest. Could the presence of string world sheets make possible the vanishing of continuous cuts even at the continuum limit so that continuum cuts would emerge only in the approximation as the density of resonances is high enough?

The replacement of continuous cut with a sum of infinitely narrow resonances is certainly an approximation. Could it be that the stringy representation as a sum of resonances with finite width is an essential aspect of quantum physics allowing to get rid of infinities necessarily accompanying loops? Consider now the arguments against this idea.

1. How to get rid of the problems with unitarity caused by the zero width of resonances? Could finite resonance widths make unitarity possible? Ordinary twistor Grassmannian approach predicts that the virtual momenta are light-like but complex: obviously, the imaginary part of the energy in rest frame would have interpretation as resonance with.
   

   In TGD framework this generalizes for 8-D momenta. By quantum-classical correspondence (QCC) the classical Noether charges are equal to the eigenvalues of the fermionic charges in Cartan algebrable (maximal set of mutually commuting observables) and classical TGD indeed predicts complex momenta (Kähler coupling strength is naturally complex). QCC thus supports this proposal.
   
   

2. Sum over resonances/exchanges picture is in conflict with QFT picture about scattering of particles. Could finite resonance widths due to the complex momenta give rise to the QFT type scattering amplitudes as one develops the amplitudes in Taylor series with respect to the resonance width? Unitarity condition indeed gives the first estimate for the resonance width.
   

   QFT amplitudes should emerge in an approximation obtained by replacing the discrete set of finite width resonances with a cut as the distance between poles is shorter than the resolution for mass squared.
   

   In superstring models string tension has single very large value and one cannot obtain QFT type behavior at low energies (for instance, scattering amplitudes in hadronic string model are concentrated in forward direction). TGD however predicts an entire hierarchy of p-adic length scales with varying string tension. The hierarchy of mass scales corresponding roughly to the lengths and thickness of magnetic flux tubes as thickened cosmic strings and characterized by the value of cosmological constant predicted by twistor lift of TGD. Could this give rise to continuous QCT type cuts at the limit when measurement resolution cannot distinguish between resonances?
   
   

At this age one develops the habit of looking back to the days of youth. I remember that I had intention to make some kind of thesis (perhaps it was MsC) and went to Dr. Claus Montonen well-known from Montonen-Olive duality proposed in 1977, the same year that I discovered the basic idea of TGD. Claus Montonen proposed that I could work with the analytic formulas for scattering amplitudes in dual resonance models (these models were studied during period 1968-1973). I must have looked at the problem but have probably concluded that I am unable to do anything useful. More than four decades later I met these amplitudes again!

See the article More about the construction of scattering amplitudes in TGD framework or the chapter The Recent View about Twistorialization in TGD Framework of "Towards M-matrix".

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

Articles and other material related to TGD.

When we will start to make theoretical physics again?

The following is comment to FB discussion about what TGD as a TOE really means and also about the sad situation prevailing in the forefront of theoretical physics now.

TGD as TOE

Of course TGD is TOE but in much more general sense that usually. TGD is also a quantum theory of consciousness and life.

TGD started as a theory of gravitation but during the first two years it became clear that TGD is also a generalization of string models allowing to understand the origin of standard model symmetries. The basic problem of GRT (lost classical conservation laws) was the starting point of TGD - colleagues still fail to realize the existence of the problem!

The problems of string models (spontaneous compactification needed to get space-time) or those of GUTS (loss of separate conservation of baryon and lepton numbers and failure to find any evidence for this prediction plus fine tuning problems) could have also been starting points of TGD. It became also clear that TGD is a fusion of quantum gravity and standard model. Dark matter and energy characterized by cosmological constant could have been also the starting points: cosmic strings providing the solution to the problem of galactic dark mattter and of cosmological constant could have also lead to TGD.

Later the basic paradox of quantum measurement theory and problems of biology (the origin of macroscopic quantum coherence) led to zero energy ontology and adelic physics as a number theoretic generalization of physics predicting the hierarchy of Planck constants as explanation of dark matter. TGD became also a quantum theory of biology, cognition, and consciousness.

What about future?

I can safely say that my work is done now, and I can only hope and wait that colleagues become mature to realize the situation. There are some good signs: even many string theorists admit that superstrings were a failure. For instance Claus Montonen, famous finnish colleague, admitted this publicly some time ago in radio program.

The gurus however feel themselves forced to keep their public belief in strings. Since they are of my age and happy professors in a good health, their life expection is around one century. Since science indeed proceeds funeral by funeral, year 2050 is very optimistic guess for the year after which the change of the tide can happen: at that time I have been buried for decades so that I will not see the day of glory.

It is clear that academic theoretical physics will experience a long stagnation period inducing also a stagnation of experimental particle physics. Theoretical physics migh continue as a kind of net activity by laymen with minimal knowledge and understanding - somewhat like aether theories do. There is no point in building an accelerator with 50 billion dollar costs if it is clear from the beginning that it will only demonstrate that there is no evidence for the predicted dark particles or susy partners.

Half century is a short time in science. I feel myself like a stranger, like a representative of a collapsed civilization, among colleagues who could not be less interested in the idea of TOE, which inspired me and my generation so deeply. They could be equally well making money in stock market. I feel like a long distance runner who enters the goal and finds that audience has long ago lost interest in the competion and gone home.

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

Articles and other material related to TGD.

Some comments about classical conservation laws in Zero Energy Ontology

In Zero Energy Ontology (ZEO), the basic geometric structure is causal diamond (CD), which is a subset of M⁴× CP₂ identified as an intersection of future and past directed light cones of M⁴ with points replaced with CP₂. Poincare symmetries are isometries of M⁴× CP₂ but CD itself breaks Poincare symmetry.

Whether Poincare transformations can act as global symmetries in the "world of classical worlds" (WCW), the space of space-time surfaces - preferred extremals - connecting 3-surfaces at opposite boundaries of CD, is not quite clear since CD itself breaks Poincare symmetry. One can even argue that ZEO is not consistent with Poincare invariance. By holography one can either talk about WCW as pairs of 3-surfaces or about space of preferred extremals connecting the members of the pair.

First some background.

1. Poincare transformations act symmetries of space-time surfaces representing extremals of the classical variational principle involved, and one can hope that this is true also for preferred extremals. Preferred extremal property is conjectured to be realized as a minimal surface property implied by appropriately generalized holomorphy property meaning that field equations are separately satisfied for Kähler action and volume action except at 2-D string world sheets and their boundaries. Twistor lift of TGD allows to assign also to string world sheets the analog of Kähler action.
   
   
2. String world sheets and their light-like boundaries carry elementary particle quantum numbers identified as conserved Noether charges assigned with second quantized induced spinors solving modified Dirac equation determined by the action principle determining the preferred extremals - this gives rise to super-conformal symmetry for fermions.
   
   
   
3. The ground states of super-symplectic and super-Kac-Moody representations correspond to spinor harmonics with well-defined Poincare quantum numbers. Excited states are obtained using generators of symplectic algebra and have well-defined four-momenta identifiable also as classical momenta. Quantum classical correspondence (QCC) states that classical charges are equal to the eigenvalues of Poincare generators in the Cartan algebra of Poincare algebra. This would hold quite generally.
   
   
4. In ZEO one assigns opposite total quantum numbers to the boundaries of CD: this codes for the conservation laws. The action of Poincare transformations can be non-trivial at second (active) boundary of CD only and one has two kinds of realizations of Poincare algebra leaving either boundary of CD invariant. Since Poincare symmetries extend to Kac-Moody symmetries analogous to local gauge symmetries, it should be possible to achieve trivial action at the passive boundary of CD so that the Cartan algebra of symmetries act non-trivially only at the active boundary of CD. Physical intuition suggests that Poincare transformations on the entire CD treating it as a rigid body correspond to trivial center of mass quantum numbers.
   
   

How do Poincare transformations act on 3-surfaces at the active boundary of CD?

1. Zero energy states are superpositions of 4-D preferred extremals connecting 3-D surfaces at boundaries of CD, the ends of space-time. One should be able to construct the analogs of plane waves as superpositions of space-time surfaces obtained by translating the active boundary of CD and 3-surfaces at it so that the size of CD increases or decreases. The translate of a preferred extremal is a preferred extremal associated with the new pair of 3-surfaces and has size and thus also shape different from those of the original. Clearly, classical theory becomes an essential part of quantum theory.
   
   
2. Four-momentum eigenstate is an analog of plane wave which is superposition of the translates of a preferred extremal. In practice it is enough to have wave packets so that in given resolution one has a cutoff for the size of translations in various directions. As noticed, QCC requires that the eigenvalues of Cartan algebra generators such as momentum components are equal to the classical charges.
   
   

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

Articles and other material related to TGD.

About gauge bosons and their decay vertices in TGD framework

The attempt to understand how unitarity of scattering amplitudes emerges led as side track to a more detailed view about gauge bosons as flux tubes carrying monopole flux and consisting of two long portions with Minkowskian signature and two short portions represented by wormhole contacts. Also a more detailed view about decay vertices emerged.

There question is how elementary particles and their basic interaction vertices could be realized in this framework.

1. In TGD framework particle would correspond to pair of wormhole contact associated with closed magnetic flux tube carrying monopole flux. Strongly flattened rectangle with Minkowskian flux tubes as long edges with length given by weak scale and Euclidian wormhole contacts as short edges with CP₂ radius as lengths scale is a good visualization. 3-particle vertex corresponding to the replication of this kind of flux tube rectangle to two rectangles would replace 3-vertex of Feynman graph. There is analogy with DNA replication. Similar replication is expected to be possible also for the associated closed fermionic strings.
   
   
2. Denote the wormhole contacts by A and B and their opposite throats by A_i and B_i, i=1,2. For fermions A₁ can be assumed to carry the electroweak quantum numbers of fermion. For electroweak bosons A₁ and A₂ (for instance) could carry fermion and anti-fermion, whose quantum numbers sum up to those of ew gauge boson. These "corner fermions" can be called active.
   

   Also other distributions of quantum numbers must be considered. Fermion and anti-fermion could in principle reside at the same throat - say A₁. One can however assume that second wormhole contact, say A has quantum numbers of fermion or weak boson (or gluon) and second contact carries quantum numbers screening weak isospin.
   
   

3. The model assumes that the weak isospin is neutralized in length scales longer than the size of the flux tube structure given by electro-weak scale. The screening fermions can be called passive. If the weak isospin of W^+/- boson is neutralized in the scale of flux tube, 2 ν_Lνbar_R pairs are needed (lepton number for these pairs must vanish) for W^-. For Z νbar_Lν_R and ν_Lνbar_R are needed. The pairs of passive fermions could reside in the interior of flux tube, at string world sheet or at its corners just like active fermions. The first extreme is that the neutralizing neutrino-antineutrino pairs reside in interior at the opposite long edges of the rectangular flux tube. Second extreme is that they are at the corners of rectangular closed string.
   
   
4. Rectangular closed string containing active fermion at wormhole A (say) and with members of isospin neutralizing neutrino-antineutrino pair at the throats of B serves as basic units. In scales shorter than string length the end A would behave like fermion with weak isospin. At longer scales physical fermion would be hadron like entity with vanishing isospin and one could speak of confinement of weak isospin.
   

   From these physical fermions one can build gauge bosons as bound states. Weak bosons and also gluons would be pairs of this kind of fermionic closed strings connecting wormhole contacts A and B. Gauge bosons (and also gravitons) could be seen as composites of string like physical fermions with vanishing net isospin rather than those of point like fundamental fermions.
   
   

5. The decay of weak boson to fermion-antifermion pair would be flux tube replication in which closed strings representing physical fermion and anti-fermion continue along different copies of flux tube structure. The decay of boson to two bosons - say W→ WZ - by replication of flux tube would require creation of a pair of physical fermionic closed strings representing Z. This would correspond to a V-shaped vertex with the edge of V representing closed fermionic closed string turning backwards in time. In decays like Z→ W⁺W^- two closed fermion strings would be created in the replication of flux tube. Rectangular fermionic string would turns backwards in time in the replication vertex and the rectangular strings of Z would be shared between W⁺ and W^-.
   
   

This mesonlike picture about weak bosons as bound states of fermions sounds complex as compared with standard model picture. On the other hand only the spinor fields assignable to single fermion family are present. A couple of comments concerning this picture are in order.

1. M⁸ duality provides a different perspective. In M⁸ picture these vertices could correspond to analogs of local 3 particle vertices for octonionic superfield, which become nonlocal in the map taking M⁸=M⁴× CP₂ surfaces to surfaces in H=M⁴× CP₂. The reason is that M⁴ point is mapped to M⁴ point but the tangent space at E⁴ point is mapped to a point of CP₂. If the point in M⁸ corresponds to a self-intersection point the tangent space at the point is not unique and point is mapped to two distinct points. There local vertex in M⁸ would correspond to non-local vertex in H and fermion lines could just begin. This would mean that at H-level fermion line at moment of replication and V-shaped fermion line pair beginning at different point of throat could correspond to 3-vertex at M⁸ level.
   
   
2. The 3-vertex representing replication could have interpretation in terms of quantum criticality: in reversed direction of time two branches of solution of classical field equations would co-incide.
   
   

See the article More about the construction of scattering amplitudes in TGD framework or the chapter The Recent View about Twistorialization in TGD Framework.

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

Articles and other material related to TGD.

More about the construction of scattering amplitudes in TGD framework

During years I have considered several proposals for S-matrix in TGD framework - perhaps the most realistic proposal relies on the generalization of twistor Grassmann approach to TGD context. There are several questions waiting for an answer. How to achieve unitarity? What it is to be a particle in classical sense? Can one identify TGD analogs of quantum fields? Could scattering amplitudes have interpretation as Fourier transforms of n-point functions for the analogs of quantum fields?

Unitarity is certainly the issue number 1 and in the sequel almost trivial solution to the unitarity problem based on the existence of super-symplectic transformations acting as isometries of "world of classical worlds" implying infinite number of conserved Noether charges in turn guaranteeing unitarity. Also quantum classical correspondence and the role of string world sheets for strong form of holography are discussed. What is found that number theoretic view justifies the assignment of action to string world sheets.

See the article More about the construction of scattering amplitudes in TGD framework or the chapter The Recent View about Twistorialization in TGD Framework.

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

Articles and other material related to TGD.

About TGD counterparts of classical field configurations in Maxwell's theory

Classical physics is an exact part of TGD so that the study of extremals of dimensionally reduces 6-D Kähler action can provide a lot of intuition about quantum TGD and see how quantum-classical correspondence is realized.
In the following the goal is to develop further understanding about TGD counterparts of the simplest field configurations in Maxwell's theory.

About differences between Maxwell's ED and TGD

TGD differs from Maxwell's theory in several important aspects.

1. The TGD counterparts of classical electroweak gauge potentials are induced from component of spinor connection of CP₂. Classical color gauge potentials corresponds to the projections of Killing vector fields of color isometries.
   
   
2. Also M⁴ has Kähler potential, which is induced to space-time surface and gives rise to an additional U(1) force. The couplings of M⁴ gauge potential to quarks and leptons are of same sign whereas the couplings of CP₂ Kähler potential to B and L are of opposite sign so that the contributions to 6-D Kähler action reduce to separate terms without interference term.
   

   Coupling to induced M⁴ Kähler potential implies CP breaking. This could explain the small CP breaking in hadronic systems and also matter antimatter asymmetry in which there are opposite matter-antimatter asymmetries inside cosmic strings and their exteriors respectively. A priori it is however not obvious that the CP breaking is small.
   
   

3. General coordinate invariance implies that there are only 4 local field like degrees of freedom so that for extremals with 4-D M⁴ projection corresponding to GRT space-time both metric, electroweak and color gauge potentials can be expressed in terms four CP₂ coordinates and their gradients. Preferred extremal property realized as minimal surface condition means that field equations are satisfied separately for the 4-D Kähler and volume action reduces the degrees of freedom further.
   

   If the CP₂ part of Kähler form is non-vanishing, minimal surface conditions can be guaranteed by a generalization of holomorphy realizing quantum criticality (satisfied by known extremals). One can say that there is no dependence on coupling parameters. If CP₂ part of Kähler form vanishes identically, the minimal surface condition need not be guaranteed by holomorphy. It is not at all clear whether quantum criticality and preferred extremal property allow this kind of extremals.
   
   

4. Supersymplectic symmetries act as isometries of "world of classical worlds" (WCW). In a well-defined sense supersymplectic symmetry generalizes 2-D conformal invariance to 4-D context. The key observation here is that light-like 3-surfaces are metrically 2-D and therefore allow extended conformal invariance.
   

   Preferred extremal property realizing quantum criticality boils down to a condition that sub-algebra of SSA and its commutator with SSA annihilate physical states and that corresponding Noether charges vanish. These conditions could be equivalent with minimal surface property. This implies that the set of possible field patterns is extremely restricted and one might talk about "archetypal" field patterns analogous to partial waves or plane waves in Maxwell's theory.
   
   

5. Linear superposition of the archetypal field patterns is not possible. TGD however implies the notion of
   many-sheeted space-time and each sheet can carry its own field pattern. A test particle which is space-time surface itself touches all these sheets and experiences the sum of the effects caused by fields at various sheets. Effects are superposed rather than fields and this is enough. This means weakening of the superposition principle of Maxwell's theory and the linear superposition of fields at same space-time sheet is replaced with set theoretic union of space-time sheets carrying the field patterns whose effects superpose.
   

   This observation is also essential in the construction of QFT limit of TGD. The gauge potentials in standard model and gravitational field in general relativity are superpositions of those associated with space-time sheets idealized with slightly curved piece of Minkowski space M⁴.
   
   

6. An important implication is that each system has field identity - field body or magnetic body (MB). In Maxwell's theory superposition of fields coming from different sources leads to a loss of information since one does not anymore now which part of field came from particular source. In TGD this information loss does not happen and this is essential for TGD inspired quantum biology.
   

   Remark: An interesting algebraic analog is the notion of co-algebra. Co-product is analogous to reversal of product AB= C in the sense that it assigns to C and a linear combination of products ∑ A_i⊗ B_i such that A_iB_i=C. Quantum groups and co-algebras are indeed important in TGD and it might be that there is a relationship. In TGD inspired quantum biology magnetic body plays a key role as an intentional agent receiving sensory data from biological body and using it as motor instrument.
   
   

7. I have already earlier considered a space-time correlate for second quantization in terms of sheets of covering for h_eff=nh₀. I have proposed that n factorizes as n=n₁n₂ such that n₁ (n₂) is the number sheets for space-time surface as covering of CP₂ (M⁴). One could have quantum mechanical linear superposition of space-time sheets, each with a particular field pattern. This kind state would correspond to single particle state created by quantum field in QFT limit. For instance, one could have spherical harmonic for orientations of magnetic flux tube or electric flux tube.
   

   One could also have superposition of configurations containing several space-time sheets simultaneously as analogs of many-boson states. Many-sheeted space-time would correspond to this kind many-boson states. Second quantization in quantum field theory (QFT) could be seen as an algebraic description of many-sheetedness having no obvious classical correlate in classical QFT.
   
   

8. Flux tubes should be somehow different for gravitational fields, em fields, and also weak and color gauge fields. The value of n=n₁n₂ for gravitational flux tubes is very large by Nottale formula hbar_eff= hbar_gr= GMm/v₀. The value of n₂ for gravitational flux tubes is n₂∼ 10⁷ if one accepts the formula G= R²/n₂hbar. For em fields much smaller values of n and therefore of n₂ are suggestive. There the value of n measuring in adelic physics algebraic complexity and evolutionary level would distinguish between gravitational and em flux tubes.
   

   Large value of n would mean quantum coherence in long scales. For gravitation this makes sense since screening is absent unlike for gauge interactions. Note that the large value of h_eff=h_gr implies that α_em= e²/4πℏ_eff is extremely small for gravitational flux tubes so that they would indeed be gravitational in an excellent approximation.
   

   n would be the dimension of extension of rationals involved and n₂ would be the number space-time sheets as covering of M⁴. If this picture is correct, gravitation would correspond to much larger algebraic complexity and much larger value of Planck constant. This conforms with the intuition that gravitation plays essential role in the quantum physics of living matter.
   

   There are also other number theoretic characteristics such as ramified primes of the extension identifiable as preferred p-adic primes in turn characterizing elementary particle. Also flux tubes mediating weak and strong interactions should allow characterization in terms of number theoretic parameters. There are arguments that in atomic physics one has h=6h₀. Since the quantum coherence scale of hadrons is smaller than atomic scale, one can ask whether one could have h_eff<h.
   
   

In this article CP₂ type extremals will be considered from the point of view of quantum criticality and the view about string world sheets, their lightlike boundaries as carriers of fermion number, and the ends as point like particles as singularities acting as sources for minimal surfaces satisfying non-linear generalization of d'Alembert equation.

I will also discuss the delicacies associated with M⁴ Kähler structure and its connection with what I call Hamilton-Jacobi structure and with M⁸ approach based on classical number fields. I will argue that the breaking of CP symmetry associated with M⁴ Kähler structure is small without any additional assumptions: this is in contrast with the earlier view.

The difference between TGD and Maxwell's theory and consider the TGD counterparts of simple em field configurations will be also discussed. Topological field quantization provides a geometric view about formation of atoms as bound states based on flux tubes as correlates for binding, and allows to identify space-time correlates for second quantization. These considerations force to take seriously the possibility that preferred extremals besides being minimal surfaces also possess generalized holomorphy reducing field equations to purely algebraic conditions and that minimal surfaces without this property are not preferred extremals. If so, at microscopic level only CP₂ type extremals, massless extremals, and string like objects and their deformations would exist as preferred extremals and serve as building bricks for the counterparts of Maxwellian field configurations and the counterparts of Maxwellian field configurations such as Coulomb potential would emerge only at the QFT limit.

See the article About TGD counterparts of classical field configurations in Maxwell's theory or the chapter About the Identification of the Preferred Kähler Action.

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

Articles and other material related to TGD.

Generalized conformal symmetry, quantum criticality, catastrophe theory, and coupling constant evolution

The notion of quantum criticality allows two realizations: as stationarity of S² part of the twistor lift of Kähler action and in terms of zeros of zeta are key elements in the explicit proposal for discrete coupling constant evolution reducing to that for cosmological constant.

Quantum criticality from different perspectives

Quantum criticality is however much more general notion, and one must ask how this view relates to the earlier picture.

1. At the real number side continuous coupling constant evolution makes sense. What does this mean? Can one say that quantum criticality makes possible only adelic physics together with large h_eff/h₀=n as dimension for extension of rationals. This hierarchy is essential for life and cognition.
   

   Can one conclude that living systems correspond to quantum critical values of S(S²) and therefore α_K and in-animate systems correspond to other values of α_K? But wouldn't his mean that one gives up the original vision that α_K is analogous to critical temperature. The whole point was that this would make physics unique?
   

   From mathematical view point also continuous α_K can make sense. α_K can be continuous if it corresponds to a higher-dimensional critical manifold at which two or more preferred extremals associated with the same parameter values co-incide - roots of polynomial P(x,a,b) depending on parameters a,b serves as the canonical example. The degree of quantum criticality would vary and there would be a hierarchy of critical systems characterized by the dimension of the critical manifold. One would have a full analog of statistical physics. For mathematician this is the only convincing interpretation.
   

   2-D cusp catastrophe serves as a basic example helping to generalize. Cusp corresponds to the roots of dP₄/dx=0 of third order polynomial P₄(x,a,b), where (a,b) are control variables. The projection of region with 3 real roots to (a,b)-plane is bounded by critical lines forming a roughly V-shaped structure. d²P₄/dx² vanishes at the edges of V, where two roots co-incide and d³P₄/dx³ vanishes at the tip of V, where 3 roots co-incide.
   
   

2. A hierarchy of quantum criticalities has been actually assumed. The hierarchy of representations for super-symplectic algebra realizing 4-D analog of super-conformal symmetries allows an infinite hierarchy of representations for which infinite-D sub-algebra isomorphic to a full algebra and its commutator with the full algebra annihilate physical states. Also classical Noether charges vanish. What is new is that conformal weights are non-negative integers. The effective dimensions of these systems are finite - at least in the sense that one one has finite-D Lie algebra (or its quantum counterpart) or corresponding Kac-Moody algebra as symmetries. This realization of quantum criticality generalize the idea that conformal symmetry accompanies 2-D criticality.
   

   This picture conforms also with the vision about hierarchy of hyper-finite-factors with included hyper-finite factor defining measurement resolution. Hyper-finiteness indeed means finite-dimensionality in excellent approximation.
   
   

TGD as catastrophe theory and quantum criticality as prerequisite for the Euclidian signature of WCW metric

It is good to look more precisely how the catastrophe theoretic setting generalizes to TGD.

1. The value of the twistor lift of Kähler action defining Kähler function very probably corresponds to a maximum of Kähler function since otherwise metric defined by the second derivatives could have non-Euclidian signature. One cannot however exclude the possibility that in complex WCW coordinates the (1,1) restriction of the matrix defined by the second derivatives of Kähler function could be positive definite also for other than minima.
   

   It would seem that one cannot accept several roots for given zero modes since one cannot have maximum of Kähler function for all of them. This would allow only the the boundary of catastrophe region in which 2 or more roots co-incide. Positive definiteness of WCW metric would force quantum criticality.
   

   For given values of zero modes there would be single minimum and together with the cancellation of Gaussian and metric determinants this makes perturbation theory extremely simple since exponents of vacuum functional would cancel.
   
   

2. There is an infinite number of zero modes playing the role of control variables since the value of the induce Kähler form is symplectic invariant and there are also other symplectic invariants associated with the M⁴ degrees of freedom (carrying also the analog of Kähler form for the twistor lift of TGD and giving rise to CP breaking). One would have catastrophe theory with infinite number of control variables so that the number of catastrophes would be infinite so that standard catastrophe theory does not as such apply.
   
   
3. Therefore TGD would not be only a personal professional catastrophe but a catastrophe in much deeper sense. WCW would be a catastrophe surface for the functional gradient of the action defining Kähler function. WCW would consists of regions in which given zero modes would correspond to several minima. The region of zero mode space at which some roots identifiable as space-time surfaces co-incide would be analogous to the V-shaped cusp catastrophe and its higher-D generalizations. The question is whether one allows the entire catastrophe surface or whether one demands quantum criticality in the sense that only the union of singular sets at which roots co-incide is included.
   
   
4. For WCW as catastrophe surface the analog of V in the space of zero modes would correspond to a hierarchy of sub-WCWs consisting of preferred extremals satisfying the gauge conditions associated with a sub-algebra of supersymplectic algebra isomorphic to the full algebra. The sub-WCWs in the hierarchy of sub-WCWs within sub-WCWs would satisfy increasingly stronger gauge conditions and having decreasing dimension just like in the case of ordinary catastrophe. The lower the effective dimension, the higher the quantum criticality.
   
   
5. In ordinary catastrophe theory the effective number of behavior variables for given catastrophe can be reduced to some minimum number. In TGD framework this would correspond to the reduction of super-symplectic algebra to a finite-D Lie algebra or corresponding Kac-Moody (half-)algebra as modes of supersymplectic algebra with generators labelled by non-negative integer n modulo given integer m are eliminated as dynamical degrees of freedom by the gauge conditions: this would effectively leave only the modes smaller than m. The fractal hierarchy of these supersymplectic algebras would correspond to the decomposition of WCW as a catastrophe surface to pieces with varying dimension. The reduction of the effective dimension as two sheets of the catastrophe surface co-incide would mean transformation of some modes contributing to metric to zero modes.
   
   

RG invariance implies physical analogy with thermodynamics and gauge theories

One can consider coupling constant evolution and RG invariance from a new perspective based on the minimal surface property.

1. The critical values of Kähler coupling strength would correspond to quantum criticality of the S² part S(S²) of 6-D dimensionally reduced Kähler action for fixed values of zero modes. The relative S² rotation would serve as behavior variable. For its critical values the dimension of the critical manifold would be reduced, and keeping zero modes fixed a critical value of α_K would be selected from 1-D continuum.
   

   Quantum criticality condition might be fundamental since it implies the constancy of the value of the twistor lift of Kähler action for the space-time surfaces contributing to the scattering amplitudes. This would be crucial for number theoretical vision since the continuation of exponential to p-adic sectors is not possible in the generic case. One should however develop stronger arguments to exclude the continuous evolution of Kähler coupling strength in S² degrees of freedom for the real sector of the theory.
   
   

2. The extremals of twistor lift contain dependence on the rotation parameter for S² and this must be taken into account in coupling constant evolution along curve of S² connecting zeros of zeta since Kähler and volume term change with it. This can give an additional non-local term to the evolution equations coming from the dependence of the imbedding space coordinates of the preferred extremal on the evolution parameter. The derivative of the 6-D Kähler action is sum of two terms. The first term involves derivatives of α_K and of S(S²). Second term is sum of terms involving derivations of Kähler action and volume with respect to the evolution parameter. This is by chain rule proportional to the functional derivatives of total action with respect to imbedding space coordinates, and vanishes by field equations. It does not matter whether there is coupling between Kähler action and volume term.
   
   

Could one find interpretation for the miminal surface property which implies that field equations are separately satisfied for Kähler action and volume term?

1. Quantum TGD can be seen as a "complex" square root of thermodynamics. In thermodynamics one can define several thermodynamical functions. In particular, one can add to energy E as term pV to get enthalpy H= E+pV, which remains constant when entropy and pressures are kept constant. Could one do the same now?
   

   In TGD V replaced with volume action and p would be a coupling parameter analogous to pressure. The simplest replacement would give Kähler action as outcome. The replacement would allow RG invariance of the modified action only at critical points so that replacement would be possible only there. Furthermore, field equations must hold true separately for Kähler action and volume term everywhere.
   
   

2. The coupling between Kähler action and volume term could be non-trivial at singular sub-manifolds, where a transfer of conserved quantities between the two degrees of freedom would take place. The transfer would be proportional to the divergence of the canonical momentum current D_α(g^αβ∂_βh^k) assignable to the minimal surface and is conserved outside the singular sub-manifolds.
   

   Minimal surfaces provide a non-linear generalization of massless wave-equation for which the gradient of the field equals to conserved current. Therefore the interpretation could be that these singular manifolds are sources of the analogs of fields defined by M⁴ and CP₂ coordinates.
   

   In electrodynamics these singular manifolds would represented by charged particles. The simplest interpretation would be in terms of point like charges so that one would have line singularity. The natural identification of world line singularities would be as boundaries of string world sheets at the 3-D light-like partonic orbits between Minkowskian and Euclidian regions having induced 4-metric degenerating to 3-D metric would be a natural identification. These world lines indeed appear in twistor diagrams. Also string world sheets must be assumed and they are are natural candidates for the singular manifolds serving as sources of charges in 4-D context. Induced spinor fields would serve as a representation for these sources. These strings would generalize the notion of point like particle. Particles as 3-surfaces would be connected by flux tubes to a tensor network and string world sheets would connected fermion lines at the partonic 2-surfaces to an analogous network. This would be new from the standard model perspective.
   

   Singularities could also correspond to a discrete set of points having an interpretation as cognitive representation for the loci of initial and final states fermions at opposite boundaries of CD and at vertices represented topologically by partonic 2-surfaces at which partonic orbits meet. This interpretation makes sense if one interprets the imbedding space coordinates as analogs of propagators having delta singularities at these points. It is quite possible that also these contributions are present: one would have a hierarchy of delta function singularities associated with string worlds sheets, their boundaries and the ends of the boundaries at boundaries of CD, where string world sheet has edges.
   
   

3. There is also an interpretation of singularities suggested by the generalization of conformal invariance. String world sheets would be co-dimension 2 singularites analogous to poles of analytic function of two complex coordinates in 4-D complex space. String world sheets would be co-dimension 2 singularities analogous to poles at light-like 3-surfaces. The ends of the world lines could be analogous of poles of analytic function at partonic 2-surfaces.
   

   These singularities could provide to evolution equations what might be called matter contribution. This brings in mind evolution equations for n-point functions in QFT. The resolution of the overall singularity would decompose to 2-D, 1-D and 0-D parts just like cusp catastrophe. One can ask whether the singularities might allow interpretation as catastrophes.
   
   
   

4. The proposal for the analogs of thermodynamical functions suggests the following physical picture. More general thermodynamical functions are possible only at critical points and only if the extremals are miminal surfaces. The singularities should correspond to physical particles, fermions. Suppose that one considers entire scattering amplitude involving action exponential plus possible analog of pV term plus the terms associated with the fermions assigned with the singularities. Suppose that the coupling constant evolution from 6-D Kähler action is calculated without including the contribution of the singularities.
   

   The derivative of n-particle amplitude with respect to the evolution parameter contains a term coming from the action exponential receiving contributions only from the singularities and a term coming from the operators at singularities. RG invariance of the scattering amplitude would require that the two terms sum up to zero. In the thermodynamical picture the presence of particles in many particle scattering amplitude would force to add the analog of pressure term to the Kähler function: it would be determined by the zero energy state.
   

   One can of course ask how general terms can be added by requiring minimal surface property. Minimal surface property reduces to holomorphy, and can be true also for other kinds of actions such as the TGD analogs of electroweak and color actions that I considered originally as possible action candidates.
   

   This would have interpretation as an analog for YM equations in gauge theories. Space-time singularities as local failure of minimal surface property would correspond to sources for H coordinates as analogs of Maxwell's fields and sources currents would correspond to fermions currents.
   
   

See the article Does coupling constant evolution reduce to that of cosmological constant?.

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

Articles and other material related to TGD.

Cosmological Axis of Evil as a memory from primordial cosmology

Axis of evil is very interesting CMB anomaly (thanks for Sky Darmos for mentioning it in FB discussion). It has been even proposed that it forces Earth-centeredness. According to the Wikipedia article :

"The motion of the solar system, and the orientation of the plane of the ecliptic are aligned with features of the microwave sky, which on conventional thinking are caused by structure at the edge of the observable universe. Specifically, with respect to the ecliptic plane the "top half" of the CMB is slightly cooler than the "bottom half"; furthermore, the quadrupole and octupole axes are only a few degrees apart, and these axes are aligned with the top/bottom divide."

This is indeed really strange looking finding. To my view it does not however bring pre-Keplerian world view back but is related to the possibility of quantum coherence even in cosmological scales predicted by TGD. It would also reflect the situation during very early cosmology, which in TGD framework is cosmic string dominated.

1. The hierarchy of Planck constants h_eff=n×h₀ implies the existence of space-time sheets with arbitrary large size serving as quantum coherent regions. h_eff=h_gr assignable to flux tubes mediating gravitational interaction the value of h_eff can be gigantic. h_gr= GMm/v ₀, where M and m are masses such that M can be solar mass or even larger mass.
   
   
2. Cosmic strings dominated the very early TGD inspired cosmology. They have 2-D projections to M^4 and CP_2 so that GRT is not able to describe them. During the analog of inflationary period the dimension of M^4 projection became D=4 and cosmic strings became magnetic flux tubes. Ordinary GRT space-time emerged and GRT started to be a reasonable approximation as QFT limit of TGD.
   
   
3. Quantum coherence make possible long range correlations. One correlation of this kind could be occurrene of cosmic strings which are nearly parallel in even cosmic scales or more precisely nearly parallel at the time when the TGD counterpart of inflation occurred and the ordinary space-time emerged and cosmic strings thickened to magnetic flux tubes - a process directly corresponding to cosmic expansion. This time corresponds in standard cosmology the end of inflationary period.
   

   The volume that we observe via CMB now would correspond to a rather small volume at the end of the period when ordinary GRT space-time emerged and it is not too difficult to imagine that in this volume the cosmic strings would have formed a bundle nearly parallel cosmic strings. This property would have been preserved in good approximation during expansion. For instance, angular momentum conservation would have taken care of this if the galaxies along long cosmic strings had angular momenta in parallel: there is indeed evidence for this. Turning of cosmic string to a different direction would require a lot of angular momentum since also the galaxies should be turned at the same time.
   
   

4. Cosmic strings thicknened to flux tubes would contain galaxies - pearls in necklace is good metaphor. Galaxies would be local tangles of flux tubes with topology of dipole type magnetic field in reasonable approximation. Also stars and planets would have formed in the similar manner. This leads to a rather detailed model for galaxy formation. See for instance this.
   
   

See the chapter More about TGD and Cosmology or the article Breaking of CP, P, and T in cosmological scales in TGD Universe.

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

Articles and other material related to TGD.

Solution of renormalization group equation for flux tubes having minimum string tension and RG evolution in terms of Riemann zeta

The great surprise of the last year was that twistor induction allows large number of induced twistor structures. SO(3) acts as moduli space for the dimensional reductions of the 6-D Kähler action defining the twistor space of space-time surface as a 6-D surface in 12-D twistor space assignable to M⁴× CP₂. This 6-D surface has space-time surface as base and sphere S² as fiber. The area of the twistor sphere in induced twistor structure defines running cosmological constant and one can understand the mysterious smallness of cosmological constant.

This in turn led to the understanding of coupling constant evolution in terms of the flow changing the value of
cosmological constant defined by the area of the twistor sphere of space-time surface for induced twistor structure.

dlog(α_K)/ds = -[S(S²)/(S_K(X⁴)+S(S²)] dlog(S(S²))/ds .

Renormalization group equation for flux tubes having minimum string tension

It came as a further pleasant surprise that for a very important special case defined by the minima of the dimensionally reduce action consisting of Kähler magnetic part and volume term one can solve the renormalization group equations explicitly. For magnetic flux tubes for which one has S_K(X⁴)∝ 1/S and S_vol∝ S in good approximation, one has S_K(X⁴) =S_vol at minimum (say for the flux tubes with radius about 1 mm for the cosmological constant in cosmological scales). One can write

dlog(α_K)/ds = -1/2 dlog(S(S²))/ds ,

and solve the equation explicitly:

α_K,0/α_K = [S(S²)/S(S²)₀]^x , x=1/2 .

A more general situation would correspond to a model with x≠ 1/2: the deviation from x=1/2 could be interpreted as anomalous dimension. This allows to deduce numerically a formula for the value spectrum of α_K,0/α_K apart from the initial values.

The following considerations strongly suggest that this formula is not quite correct but applies only the real part of Kähler coupling strength. The following argument allows to deduce the imaginary part.

Could the critical values of α_K correspond to the zeros of Riemann Zeta?

Number theoretical intuitions strongly suggests that the critical values of 1/α_K could somehow correspond to zeros of Riemann Zeta. Riemann zeta is indeed known to be involved with critical systems.

The naivest ad hoc hypothesis is that the values of 1/α_K are actually proportional to the non-trivial zeros s=1/2+iy of zeta . A hypothesis more in line with QFT thinking is that they correspond to the imaginary parts of the roots of zeta. In TGD framework however complex values of α_K are possible and highly suggestive. In any case, one can test the hypothesis that the values of 1/α_K are proportional to the zeros of ζ at critical line. Problems indeed emerge.

1. The complexity of the zeros and the non-constancy of their phase implies that the RG equation can hold only for the imaginary part of s=1/2+it and therefore only for the imaginary part of the action. This suggests that 1/α_K is proportional to y. If 1/α_K is complex, RG equation implies that its phase RG invariant since the real and imaginary parts would obey the same RG equation.
   
   
2. The second - and much deeper - problem is that one has no reason for why dlog(α_K)/ds should vanish at zeros: one should have dy/ds=0 at zeros but since one can choose instead of parameter s any coordinate as evolution parameter, one can choose s=y so that one has dy/ds=1 and criticality condition cannot hold true. Hence it seems that this proposal is unrealistic although it worked qualitatively at numerical level.
   
   

It seems that it is better to proceed in a playful spirit by asking whether one could realize quantum criticality in terms of zeros of zeta.

1. The very fact that zero of zeta is in question should somehow guarantee quantum criticality. Zeros of ζ define the critical points of the complex analytic function defined by the integral
   

   X(s₀,s)= a∫_{Cs0→ s} ζ (s)ds ,
   

   where C_{s0→ s} is any curve connecting zeros of ζ, a is complex valued constant. Here s does not refer to s= sin(ε) introduced above but to complex coordinate s of Riemann sphere.
   

   By analyticity the integral does not depend on the curve C connecting the initial and final points and the derivative dS_c/ds= ζ(s) vanishes at the endpoints if they correspond to zeros of ζ so that would have criticality. The value of the integral for a closed contour containing the pole s=1 of ζ is non-vanishing so that the integral has two values depending on which side of the pole C goes.
   

   
   
   

2. The first guess is that one can define S_c as complex analytic function F(X) having interpretation as analytic continuation of the S² part of action identified as Re(S_c):
   

   S_c(S²)= F(X(s,s₀)) , & X(s,s₀)= ∫_{Cs0→ s} ζ (s)ds ,
   

   S(S²)= Re(S_c)= Re(F(X)) ,
   

   ζ(s)=0 , & Re(s₀)=1/2 .
   

   S_c(S²)=F(X) would be a complexified version of the Kähler action for S². s₀ must be at critical line but it is not quite clear whether one should require ζ(s₀)=0.
   

   The real valued function S(S²) would be thus extended to an analytic function S_c=F(X) such that the S(S²)=Re(S_c) would depend only on the end points of the integration path C_{s0→ s}. This is geometrically natural. Different integration paths at Riemann sphere would correspond to paths in the moduli space SO(3), whose action defines paths in S² and are indeed allowed as most general deformations. Therefore the twistor sphere could be identified Riemann sphere at which Riemann zeta is defined. The critical line and real axis would correspond to particular one parameter sub-groups of SO(3) or to more general one parameter subgroups.
   

   One would have
   

   α_K,0/α_K= (S_c/S₀)^1/2 .
   

   The imaginary part of 1/α_K (and in some sense also of the action S_c(S²)) would determined by analyticity somewhat like the real parts of the scattering amplitudes are determined by the discontinuities of their imaginary parts.
   
   

3. What constraints can one pose on F? F must be such that the value range for F(X) is in the value range of S(S²). The lower limit for S(S²) is S(S²)=0 corresponding to J_uΦ→ 0.
   

   The upper limit corresponds to the maximum of S(S²). If the one Kähler forms of M⁴ and S² have same sign, the maximum is 2× A, where A= 4π is the area of unit sphere. This is however not the physical case.
   

   If the Kähler forms of M⁴ and S² have opposite signs or if one has RP option, the maximum, call it S_max, is smaller. Symmetry considerations strongly suggest that the upper limit corresponds to a rotation of 2π in say (y,z) plane (s=sin(ε)= 1 using the previous notation).
   

   For s→ s₀ the value of S_c approaches zero: this limit must correspond to S(S²)=0 and J_uΦ→ 0. For Im(s)→ +/- ∞ along the critical line, the behavior of Re(ζ) (see this) strongly suggests that | X|→ ∞ . This requires that F is an analytic function, which approaches to a finite value at the limit |X| → ∞. Perhaps the simplest elementary function satisfying the saturation constraints is
   

   F(X) = S_maxtanh(-iX) .
   

   One has tanh(x+iy)→ +/- 1 for y→ +/- ∞ implying F(X)→ +/- S_max at these limits. More explicitly , one has tanh(-i/2-y)= [-1+exp(-4y)-2exp(-2y)(cos(1)-1)]/[1+exp(-4y)-2exp(-2y)(cos(1)-1)]. Since one has tanh(-i/2+0)= 1-1/cos(1)<0 and tanh(-i/2+∞)=1, one must have some finite value y=y₀>0 for which one has
   

   tanh(-i/2+y₀)=0 .
   

   The smallest possible lower bound s₀ for the integral defining X would naturally to s₀=1/2-iy₀ and would be below the real axis.
   
   

4. The interpretation of S(S²) as a positive definite action requires that the sign of S(S²)=Re(F) for a given choice of s₀= 1/2+iy₀ and for a propertly sign of y-y₀ at critical line should remain positive. One should show that the sign of S= a∫ Re(ζ)(s=1/2+it)dt is same for all zeros of ζ. The graph representing the real and imaginary parts of Riemann zeta along critical line s= 1/2+it (see this) shows that both the real and imaginary part oscillate and increase in amplitude. For the first zeros real part stays in good approximation positive but the the amplitude for the negative part increase be gradually. This suggests that S identified as integral of real part oscillates but preserves its sign and gradually increases as required.
   
   

A priori there is no reason to exclude the trivial zeros of ζ at s= -2n, n=1,2,....

1. The natural guess is that the function F(X) is same as for the critical line. The integral defining X would be along real axis and therefore real as also 1/α_K provided the sign of S_c is positive: for negative sign for S_c not allowed by the geometric interpretation the square root would give imaginary unit. The graph of the Riemann Zeta at real axis (real) is given in MathWorld Wolfram (see this).
   
   
2. The functional equation
   

   ζ(1-s)= ζ(s)Γ(s/2)/Γ((1-s)/2)
   

   allows to deduce information about the behavior of ζ at negative real axis. Γ((1-s)/2) is negative along negative real axis (for Re(s)≤ 1 actually) and poles at n+1/2. Its negative maxima approach to zero for large negative values of Re(s) (see this) whereas ζ(s) approaches value one for large positive values of s (see this). A cautious guess is that the sign of ζ(s) for s≤ 1 remains negative. If the integral defining the area is defined as integral contour directed from s<0 to a point s₀ near origin, S_c has positive sign and has a geometric interpretation.
   
   

3. The formula for 1/α_K would read as α_K,0/α_K(s=-2n) = (S_c/S₀)^1/2 so that α_K would remain real. This integration path could be interpreted as a rotation around z-axis leaving invariant the Kähler form J of S²(X⁴) and therefore also S=Re(S_c). Im(S_c)=0 indeed holds true. For the non-trivial zeros this is not the case and S=Re(S_c) is not invariant.
   
   
4. One can wonder whether one could distinguish between Minkowskian and Euclidian and regions in the sense that in Minkowskian regions 1/α_K correspond to the non-trivial zeros and in Euclidian regions to trivial zeros along negative real axis. The interpretation as different kind of phases might be appropriate.
   
   

What is nice that the hypothesis about equivalence of the geometry based and number theoretic approaches can be killed by just calculating the integral S as function of parameter s. The identification of the parameter s appearing in the RG equations is no unique. The identification of the Riemann sphere and twistor sphere could even allow identify the parameter t as imaginary coordinate in complex coordinates in SO(3) rotations around z-axis act as phase multiplication and in which metric has the standard form.

See the article TGD View about Coupling Constant Evolutionor the chapter with the same title.

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

Articles and other material related to TGD.