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.