Thursday, February 14, 2019

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 JA of the volume action located at string world sheet?

  1. The divergences of JA are proportional to the trace of the second fundamental form H formed by the covariant derivatives of gradients ∂αhk of H-coordinates in the interior and vanish. The singular contribution at string world sheets is determined by the discontinuity of the isometry current JA and involves only the first derivatives ∂αhk.

  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 νR having no couplings to weak or color fields - be massless in 8-D or even 4-D sense? Could νR and νbarR 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?

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.

Wednesday, February 13, 2019

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 kth 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.

Tuesday, February 12, 2019

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.

Saturday, February 09, 2019

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.



Wednesday, January 30, 2019

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.

Sunday, January 27, 2019

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 Ai and Bi, 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 νLνbarR pairs are needed (lepton number for these pairs must vanish) for W-. For Z νbarLνR and νLνbarR 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.

Thursday, January 24, 2019

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.