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TGD: Physics as Infinite-Dimensional Geometry

Note: Newest contributions are at the top!

Year 2012

Could N=2 orN=4 SUSY have something to do with TGD?

N=4 SYM has been the theoretical laboratory of Nima and others. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form meant to be should be accessible to ordinary physicist.

N=4 SYM is definitely a completely exceptional theory and one cannot avoid the question whether it could in some sense be part of fundamental physics. In TGD framework right handed neutrinos have remained a mystery: whether one should assign space-time SUSY to them or not. Could they give rise to N=2 or N=4 SUSY with fermion number conservation?

My latest view is that fully covariantly constant right-handed neutrinos decouple from the dynamics completely. I will repeat first the earlier arguments which consider only fully covariantly constant right-handed neutrinos.

  1. N=1 SUSY is certainly excluded since it would require Majorana property not possible in TGD framework since it would require superposition of left and right handed neutrinos and lead to a breaking of lepton number conservation. Could one imagine SUSY in which both MEs between which particle wormhole contacts reside have N=2 SUSY which combine to form an N=4 SUSY?
  2. Right-handed neutrinos which are covariantly constant right-handed neutrinos in both M4 degrees of freedom cannot define a non-trivial theory as shown already earlier. They have no electroweak nor gravitational couplings and carry no momentum, only spin.

    The fully covariantly constant right-handed neutrinos with two possible helicities at given ME would define representation of SUSY at the limit of vanishing light-like momentum. At this limit the creation and annihilation operators creating the states would have vanishing anticommutator so that the oscillator operators would generate Grassmann algebra. Since creation and annihilation operators are hermitian conjugates, the states would have zero norm and the states generated by oscillator operators would be pure gauge and decouple from physics. This is the core of the earlier argument demonstrating that N=1 SUSY is not possible in TGD framework: LHC has given convincing experimental support for this belief.

Could massless right-handed neutrinos covariantly constant in CP2 degrees of freedom define N=2 or N=4 SUSY?

Consider next right-handed neutrinos, which are covariantly constant in CP2 degrees of freedom but have a light-like four-momentum. In this case fermion number is conserved but this is consistent with N=2 SUSY at both MEs with fermion number conservation. N=2 SUSYs could emerge from N=4 SUSY when one half of SUSY generators annihilate the states, which is a basic phenomenon in supersymmetric theories.

  1. At space-time level right-handed neutrinos couple to the space-time geometry - gravitation - although weak and color interactions are absent. One can say that this coupling forces them to move with light-like momentum parallel to that of ME. At the level of space-time surface right-handed neutrinos have a spectrum of excitations of four-dimensional analogs of conformal spinors at string world sheet (Hamilton-Jacobi structure).

    For MEs one indeed obtains massless solutions depending on longitudinal M2 coordinates only since the induced metric in M2 differs from the light-like metric only by a contribution which is light-like and contracts to zero with light-like momentum in the same direction. These solutions are analogs of (say) left movers of string theory. The dependence on E2 degrees of freedom is holomorphic. That left movers are only possible would suggest that one has only single helicity and conservation of fermion number at given space-time sheet rather than 2 helicities and non-conserved fermion number: two real Majorana spinors combine to single complex Weyl spinor.

  2. At imbedding space level one obtains a tensor product of ordinary representations of N=2 SUSY consisting of Weyl spinors with opposite helicities assigned with the ME. The state content is same as for a reduced N=4 SUSY with four N=1 Majorana spinors replaced by two complex N=2 spinors with fermion number conservation. This gives 4 states at both space-time sheets constructed from νR and its antiparticle. Altogether the two MEs give 8 states, which is one half of the 16 states of N=4 SUSY so that a degeneration of this symmetry forced by non-Majorana property is in question.

Is the dynamics of N=4 SYM possible in right-handed neutrino sector?

Could N=4 SYM be a part of quantum TGD? Could TGD be seen a fusion of a degenerate N=4 SYM describing the right-handed neutrino sector and string theory like theory describing the contribution of string world sheets carrying other leptonic and quark spinors? Or could one imagine even something simpler?

What is interesting that the net momenta assigned to the right handed neutrinos associated with a pair of MEs would correspond to the momenta assignable to the particles and obtained by p-adic mass calculations. It would seem that right-handed neutrinos provide a representation of the momenta of the elementary particles represented by wormhole contact structures. Does this mimircry generalize to a full duality so that all quantum numbers and even microscopic dynamics of defined by generalized Feynman diagrams (Euclidian space-time regions) would be represented by right-handed neutrinos and MEs? Could a generalization of N=4 SYM with non-trivial gauge group with proper choices of the ground states helicities allow to represent the entire microscopic dynamics?

  1. In the scattering of MEs induced by the dynamics of Kähler action the right-handed neutrinos play a passive role. Modified Dirac equation forces them to adopt the same direction of four-momentum as the MEs so that the scattering reduces to the geometric scattering for MEs as one indeed expects on basic of quantum classical correspondence. In νR sector the basic scattering vertex involves four MEs and could be a re-sharing of the right-handed neutrino content of the incoming two MEs between outgoing two MEs respecting fermion number conservation. Therefore N=4 SYM with fermion number conservation would represent the scattering of MEs at quantum level.
  2. N=4 SUSY would suggest that also in the degenerate case one obtains the full scattering amplitude as a sum of permutations of external particles followed by projections to the directions of light-like momenta and that BCFW bridge represents the analog of fundamental braiding operation. The decoration of permutations means that each external line is effectively doubled. Could the scattering of MEs can be interpreted in terms of these decorated permutations? Could the doubling of permutations by decoration relate to the occurrence of pairs of MEs.

    One can also consider the reverse of this. Could one construct massive states in N=4 SYM using pairs of momenta associated with particle with label k and its decorated copy with label k+n? Massive external particles obtained in this manner as bound states of massless ones could solve the IR divergence problem of N=4 SYM.

  3. The description of amplitudes in terms of leading singularities means picking up of the singular contribution by putting the fermionic propagators on mass shell. In the recent case it would give the inverse of massless Dirac propagator acting on the spinor at the end of the internal line annihilating it if it is a solution of Dirac equation.

    The only way out is a kind of cohomology theory in which solutions of Dirac equation represent exact forms. Dirac operator defines the exterior derivative d and virtual lines correspond to non-physical helicities with dΨ ≠ 0. Virtual fermions would be on mass-shell fermions with non-physical polarization satisfying d2Ψ=0. External particles would be those with physical polarization satisfying dΨ=0, and one can say that the Feynman diagrams containing physical helicities split into products of Feynman diagrams containing only non-physical helicities in internal lines.

  4. The fermionic states at wormhole contacts should define the ground states of SUSY representation with helicity +1/2 and -1/2 rather than spin 1 or -1 as in standard realization of N=4 SYM used in the article. This would modify the theory but the twistorial and Grassmannian description would remain more or less as such since it depends on light-likeneness and momentum conservation only.

3-vertices for sparticles are replaced with 4-vertices for MEs

In N=4 SYM the basic vertex is on mass-shell 3-vertex which requires that for real light-like momenta all 3 states are parallel. One must allow complex momenta in order to satisfy energy conservation and light-likeness conditions. This is strange from the point of view of physics although number theoretically oriented person might argue that the extensions of rationals involving also imaginary unit are rather natural.

The complex momenta can be expressed in terms of two light-like momenta in 3-vertex with one real momentum. For instance, the three light-like momenta can be taken to be p, k, p-ka, k= apR. Here p (incoming momentum) and pR are real light-like momenta satisfying p⋅ pR=0 with opposite sign of energy, and a is complex number. What is remarkable that also the negative sign of energy is necessary also now.

Should one allow complex light-like momenta in TGD framework? One can imagine two options.

  1. Option I: no complex momenta. In zero energy ontology the situation is different due to the presence of a pair of MEs meaning replaced of 3-vertices with 4-vertices or 6-vertices, the allowance of negative energies in internal lines, and the fact that scattering is of sparticles is induced by that of MEs. In the simplest vertex a massive external particle with non-parallel MEs carrying non-parallel light-like momenta can decay to a pair of MEs with light-like momenta. This can be interpreted as 4-ME-vertex rather than 3-vertex (say) BFF so that complex momenta are not needed. For an incoming boson identified as wormhole contact the vertex can be seen as BFF vertex.

    To obtain space-like momentum exchanges one must allow negative sign of energy and one has strong conditions coming from momentum conservation and light-likeness which allow non-trivial solutions (real momenta in the vertex are not parallel) since basically the vertices are 4-vertices. This reduces dramatically the number of graphs. Note that one can also consider vertices in which three pairs of MEs join along their ends so that 6 MEs (analog of 3-boson vertex) would be involved.

  2. Option II: complex momenta are allowed. Proceeding just formally, the (g4)1/2 factor in Kähler action density is imaginary in Minkowskian and real in Euclidian regions. It is now clear that the formal approach is correct: Euclidian regions give rise to Kähler function and Minkowskian regions to the analog of Morse function. TGD as almost topological QFT inspires the conjecture about the reduction of Kähler action to boundary terms proportional to Chern-Simons term. This is guaranteed if the condition jKμAμ=0 holds true: for the known extremals this is the case since Kähler current jK is light-like or vanishing for them. This would seem that Minkowskian and Euclidian regions provide dual descriptions of physics. If so, it would not be surprising if the real and complex parts of the four-momentum were parallel and in constant proportion to each other.

    This argument suggests that also the conserved quantities implied by the Noether theorem have the same structure so that charges would receive an imaginary contribution from Minkowskian regions and a real contribution from Euclidian regions (or vice versa). Four-momentum would be complex number of form P= PM+ iPE. Generalized light-likeness condition would give PM2=PE2 and PM⋅PE=0. Complexified momentum would have 6 free components. A stronger condition would be PM2=0=PE2 so that one would have two light-like momenta "orthogonal" to each other. For both relative signs energy PM and PE would be actually parallel: parametrization would be in terms of light-like momentum and scaling factor. This would suggest that complex momenta do not bring in anything new and Option II reduces effectively to Option I. If one wants a complete analogy with the usual twistor approach then PM2=PE2≠ 0 must be allowed.

Is SUSY breaking possible or needed?

It is difficult to imagine the breaking of the proposed kind of SUSY in TGD framework, and the first guess is that all these 4 super-partners of particle have identical masses. p-Adic thermodynamics does not distinguish between these states and the only possibility is that the p-adic primes differ for the spartners. But is the breaking of SUSY really necessary? Can one really distinguish between the 8 different states of a given elementary particle using the recent day experimental methods?

  1. In electroweak and color interactions the spartners behave in an identical manner classically. The coupling of right-handed neutrinos to space-time geometry however forces the right-handed neutrinos to adopt the same direction of four-momentum as MEs has. Could some gravitational effect allow to distinguish between spartners? This would be trivially the case if the p-adic mass scales of spartners would be different. Why this should be the case remains however an open question.
  2. In the case of unbroken SUSY only spin distinguishes between spartners. Spin determines statistics and the first naive guess would be that bosonic spartners obey totally different atomic physics allowing condensation of selectrons to the ground state. Very probably this is not true: the right-handed neutrinos are delocalized to 4-D MEs and other fermions correspond to wormhole contact structures and 2-D string world sheets.

    The coupling of the spin to the space-time geometry seems to provide the only possible manner to distinguish between spartners. Could one imagine a gravimagnetic effect with energy splitting proportional to the product of gravimagnetic moment and external gravimagnetic field B? If gravimagnetic moment is proportional to spin projection in the direction of B, a non-trivial effect would be possible. Needless to say this kind of effect is extremely small so that the unbroken SUSY might remain undetected.

  3. If the spin of sparticle be seen in the classical angular momentum of ME as quantum classical correspondence would suggest then the value of the angular momentum might allow to distinguish between spartners. Also now the effect is extremely small.

For background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation.

Scattering amplitudes and the positive Grassmannian

Perhaps I exaggerated a little bit in the previous posting, when I talked about declining theoretical physics. The work of Nima Arkani-Hamed and others represents something which makes me very optimistic and I would be happy if I could understand the horrible technicalities of their work. The article Scattering Amplitudes and the Positive Grassmannian by Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form, which should be accessible to ordinary physicist. Lubos has already discussed the article.

All scattering amplitudes have on shell amplitudes for massless particles as building bricks

The key idea is that all planar amplitudes can be constructed from on shell amplitudes: all virtual particles are actually real. In zero energy ontology I ended up with the representation of TGD analogs of Feynman diagrams using only mass shell massless states with both positive and negative energies. The enormous number of kinematic constraints eliminates UV and IR divergences and also the description of massive particles as bound states of massless ones becomes possible.

In TGD framework quantum classical correspondence requires a space-time correlate for the on mass shell property and it indeed exists. The mathematically ill-defined path integral over all 4-surfaces is replaced with a superposition of preferred extremals of Kähler action analogous to Bohr orbits, and one has only a functional integral over the 3-D ends at the light-like boundaries of causal diamond (Euclidian/Minkowskian space-time regions give real/imaginary Chern-Simons exponent to the vacuum functional). This would be obviously the deeper principle behind on mass shell representation of scattering amplitudes that Nima and others are certainly trying to identify. This principle in turn reduces to general coordinate invariance at the level of the world of classical worlds.

Quantum classical correspondence and quantum ergodicity would imply even stronger condition: the quantal correlation functions should be identical with classical correlation functions for any preferred extremal in the superposition: all preferred extremals in the superposition would be statistically equivalent (see the earlier posting). 4-D spin glass degeneracy of Kähler action however suggests that this is is probably too strong a condition applying only to building bricks of the superposition.

Minimal surface property is the geometric counterpart for masslessness and the preferred extremals are also minimal surfaces: this property reduces to the generalization of complex structure at space-time surfaces, which I call Hamilton-Jacobi structure for the Minkowskian signature of the induced metric. Einstein Maxwell equations with cosmological term are also satisfied.

Massless extremals and twistor approach

The decomposition M4=M2× E2 is fundamental in the formulation of quantum TGD, in the number theoretical vision about TGD, in the construction of preferred extremals, and for the vision about generalized Feynman diagrams. It is also fundamental in the decomposition of the degrees of string to longitudinal and transversal ones. An additional item to the list is that also the states appearing in thermodynamical ensemble in p-adic thermodynamics correspond to four-momenta in M2 fixed by the direction of the Lorentz boost. In twistor approach to TGD the possibility to decompose also internal lines to massless states at parallel space-time sheets is crucial.

Can one find a concrete identification for M2× E2 decomposition at the level of preferred extremals? Could these preferred extremals be interpreted as the internal lines of generalized Feynman diagrams carrying massless momenta? Could one identify the mass of particle predicted by p-adic thermodynamics with the sum of massless classical momenta assignable to two preferred extremals of this kind connected by wormhole contacts defining the elementary particle?

Candidates for this kind of preferred extremals indeed exist. Local M2× E2 decomposition and light-like longitudinal massless momentum assignable to M2 characterizes "massless extremals" (MEs, "topological light rays"). The simplest MEs correspond to single space-time sheet carrying a conserved light-like M2 momentum. For several MEs connected by wormhole contacts the longitudinal massless momenta are not conserved anymore but their sum defines a time-like conserved four-momentum: one has a bound states of massless MEs. The stable wormhole contacts binding MEs together possess Kähler magnetic charge and serve as building bricks of elementary particles. Particles are necessary closed magnetic flux tubes having two wormhole contacts at their ends and connecting the two MEs.

The sum of the classical massless momenta assignable to the pair of MEs is conserved even when they exchange momentum. Quantum classical correspondence requires that the conserved classical rest energy of the particle equals to the prediction of p-adic mass calculations. The massless momenta assignable to MEs would naturally correspond to the massless momenta propagating along the internal lines of generalized Feynman diagrams assumed in zero energy ontology. Masslessness of virtual particles makes also possible twistor approach. This supports the view that MEs are fundamental for the twistor approach in TGD framework.

Scattering amplitudes as representations for braids whose threads can fuse at 3-vertices

Just a little comment about the content of the article. The main message of the article is that non-equivalent contributions to a given scattering amplitude in N=4 SYM represent elements of the group of permutations of external lines - or to be more precise - decorated permutations which replace permutation group Sn with n! elements with its decorated version containing 2nn! elements. Besides 3-vertex the basic dynamical process is permutation having the exchange of neighboring lines as a generating permutation completely analogous to fundamental braiding. BFCW bridge has interpretation as a representations for the basic braiding operation.

This supports the TGD inspired proposal (TGD as almost topological QFT) that generalized Feynman diagrams are in some sense also knot or braid diagrams allowing besides braiding operation also two 3-vertices. The first 3-vertex generalizes the standard stringy 3-vertex but with totally different interpretation having nothing to do with particle decay: rather particle travels along two paths simultaneously after 1→2 decay. Second 3-vertex generalizes the 3-vertex of ordinary Feynman diagram (three 4-D lines of generalized Feynman diagram identified as Euclidian space-time regions meet at this vertex). I have discussed this vision in detail here. The main idea is that in TGD framework knotting and braiding emerges at two levels.

  1. At the level of space-time surface string world sheets at which the induced spinor fields (except right-handed neutrino, see this) are localized due to the conservation of electric charge can form 2-knots and can intersect at discrete points in the generic case. The boundaries of strings world sheets at light-like wormhole throat orbits and at space-like 3-surfaces defining the ends of the space-time at light-like boundaries of causal diamonds can form ordinary 1-knots, and get linked and braided. Elementary particles themselves correspond to closed loops at the ends of space-time surface and can also get knotted (for possible effects see this).

  2. One can assign to the lines of generalized Feynman diagrams lines in M2 characterizing given causal diamond. Therefore the 2-D representation of Feynman diagrams has concrete physical interpretation in TGD. These lines can intersect and what suggests itself is a description of non-planar diagrams (having this kind of intersections) in terms of an algebraic knot theory. A natural guess is that it is this knot theoretic operation which allows to describe also non-planar diagrams by reducing them to planar ones as one does when one constructs knot invariant by reducing the knot to a trivial one. Scattering amplitudes would be basically knot invariants.

"Almost topological" has also a meaning usually not assigned with it. Thurston's geometrization conjecture stating that geometric invariants of canonical representation of manifold as Riemann geometry, defined topological invariants, could generalize somehow. For instance, the geometric invariants of preferred extremals could be seen as topological or more refined invariants (symplectic, conformal in the sense of 4-D generalization of conformal structure). If quantum ergodicity holds true, the statistical geometric invariants defined by the classical correlation functions of various induced classical gauge fields for preferred extremals could be regarded as this kind of invariants for sub-manifolds. What would distinguish TGD from standard topological QFT would be that the invariants in question would involve length scale and thus have a physical content in the usual sense of the word!

For background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation.

Is there a connection between preferred extremals and AdS4/CFT correspondence?

The preferred extremals satisfy Einstein Maxwell equations with a cosmological constant and have negative curvature for negative value of Λ. 4-D space-times with hyperbolic metric provide canonical representation for a large class of four-manifolds and an interesting question is whether these spaces are obtained as preferred extremals and/or vacuum extremals.

4-D hyperbolic space with Minkowski signature is locally isometric with AdS4. This suggests a connection with AdS4/CFT correspondence of M-theory. The boundary of AdS would be now replaced with 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of complex coordinate at light-like boundary. AdS would presumably represent a special case of a more general family of space-time surfaces with constant Ricci scalar satisfying Einstein-Maxwell equations and generalizing the AdS4/CFT correspondence.

For the ordinary AdS5 correspondence empty M4 is identified as boundary. In the recent case the boundary of AdS4 is replaced with a 3-D light-like orbit of partonic 2-surface at which the signature of the induced metric changes. String world sheets have boundaries along light-like 3-surfaces and space-like 3-surfaces at the light-like boundaries of CD. The metric 2-dimensionality of the light-like surface makes possible generalization of 2-D conformal invariance with the light-like coordinate taking the role of hyper- complex coordinate at light-like 3-surface. AdS5× S5 of M-theory context is replaced by a 4-surface of constant Ricci scalar in 8-D imbedding space M4× CP2 satisfying Einstein-Maxwell equations. A generalization of AdS4/CFT correspondence would be in question. There is however a strong objection from cosmology: the accelerated expansion of the Universe requires positive value of Λ and favors De Sitter Space dS4 instead of AdS4.

These observations give motivations for finding whether AdS4 and/or AdS4 allows an imbedding as vacuum extremal to M4× S2⊂ M4× CP2, where S2 is a homologically trivial geodesic sphere of CP2. It is easy to guess the general form of the imbedding by writing the line elements of, M4, S2, and AdS4.

  1. The line element of M4 in spherical Minkowski coordinates (m,rM,θ,φ) reads as

    ds2= dm2-drM2-rM22 .

  2. Also the line element of S2 is familiar:

    ds2=- R2(dΘ2+sin2(θ)dΦ2) .

  3. By visiting in Wikipedia one learns that in spherical coordinate the line element of AdS4 is given by

    ds2= A(r)dt2-(1/A(r))dr2-r22 ,

    A(r)= 1+y2 , y = r/r0 .

  4. From these formulas it is easy to see that the ansatz is of the same general form as for the imbedding of Schwartschild-Nordstöm metric:

    m= Λ t+ h(y) , rM= r , Θ = s(y) , Φ= ω× (t+f(y)) .

    The non-trivial conditions on the components of the induced metric are given by

    gtt= Λ2-x2sin2(Θ) = A(r) ,

    gtr= 1/r0[Λ dh/dy -x2sin2(θ) df/dr]=0 ,

    grr= 1/r02[(dh/dy)2 -1- x2sin2(θ)(df/dy)2- R2(dΘ/dy)2]= -1/A(r) ,

    x=Rω .

By some simple algebraic manipulations one can derive expressions for sin(Θ), df/dr and dh/dr.
  1. For Θ(r) the equation for gtt gives the expression

    sin2(Θ)= P/x2 ,

    P= Λ2 -A =Λ2-1-y2 .

    The condition 0≤ sin2(Θ)≤ 1 gives the conditions

    2-x2-1)1/2 ≤ y≤ (Λ2-1)1/2 .

    Clearly only a spherical shell is possible.

  2. From the vanishing of gtr one obtains

    dh/dy = ( P/Λ)× df/dy ,

  3. The condition for grr gives

    (df/dy)2 =[r02/AP]× [A-1-R2(dΘ/dy)2] .

    Clearly, the right-hand side is positive if P≥ 0 holds true and RdΘ/dy is small. From this condition one can solved by expressing dΘ/dy using chain rule as

    (dΘ/dy)2=x2y2/[P (P-x2)] .

    One obtains

    (df/dy)2 = [Λ r02y2/AP]× [(1+y2)-1 -x2(R/r0)2 [P(P-x2)]-1)] .

    The right hand side of this equation is non-negative for certain range of parameters and variable y. Note that for r0>> R the second term on the right hand side can be neglected. In this case it is easy to integrate f(y).

The conclusion is that AdS4 allows a local imbedding as a vacuum extremal. Whether also an imbedding as a non-vacuum preferred extremal to homologically non-trivial geodesic sphere is possible, is an interesting question. The only modification in the case of De Sitter space dS4 is the replacement of the function A= 1+y2 appearing in the metric of AdS4 with A=1-y2. Also now the imbedded portion of the metric is a spherical shell. This brings in mind TGD inspired model for the final state of the star which is also a spherical shell. p-Adic length scale hypothesis motivates the conjecture that stars indeed have onion-like layered structure consisting of shells, whose radii are consistent with p-adic length scale hypothesis. This brings in mind also Titius-Bode law.

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?".

Could correlation functions, S-matrix, and coupling constant evolution be coded the statistical properties of preferred extremals?

Quantum classical correspondence states that all aspects of quantum states should have correlates in the geometry of preferred extremals. In particular, various elementary particle propagators should have a representation as properties of preferred extremals. This would allow to realize the old dream about being able to say something interesting about coupling constant evolution although it is not yet possible to calculate the M-matrices and U-matrix. Hitherto everything that has been said about coupling constant evolution has been rather speculative arguments except for the general vision that it reduces to a discrete evolution defined by p-adic length scales. General first principle definitions are much more valuable than ad hoc guesses even if the latter give rise to explicit formulas.

In quantum TGD and also at its QFT limit various correlation functions in given quantum state code for its properties. These correlation functions should have counterparts in the geometry of preferred extremals. Even more: these classical counterparts for a given preferred extremal ought to be identical with the quantum correlation functions for the superposition of preferred extremals.

  1. The marvelous implication of quantum ergodicity would be that one could calculate everything solely classically using the classical intuition - the only intuition that we have. Quantum ergodicity would also solve the paradox raised by the quantum classical correspondence for momentum eigenstates. Any preferred extremal in their superposition defining momentum eigenstate should code for the momentum characterizing the superposition itself. This is indeed possible if every extremal in the superposition codes the momentum to the properties of classical correlation functions which are identical for all of them.
  2. The only manner to possibly achieve quantum ergodicity is in terms of the statistical properties of the preferred extremals. It should be possible to generalize the ergodic theorem stating that the properties of statistical ensemble are represented by single space-time evolution in the ensemble of time evolutions. Quantum superposition of classical worlds would effectively reduce to single classical world as far as classical correlation functions are considered. The notion of finite measurement resolution suggests that one must state this more precisely by adding that classical correlation functions are calculated in a given UV and IR resolutions meaning UV cutoff defined by the smallest CD and IR cutoff defined by the largest CD present.
  3. The skeptic inside me immediately argues that TGD Universe is 4-D spin glass so that this quantum ergodic theorem must be broken. In the case of the ordinary spin classes one has not only statistical average for a fixed Hamiltonian but a statistical average over Hamiltonians. There is a probability distribution over the coupling parameters appearing in the Hamiltonian. Maybe the quantum counterpart of this is needed to predict the physically measurable correlation functions.

    Could this average be an ordinary classical statistical average over quantum states with different classical correlation functions? This kind of average is indeed taken in density matrix formalism. Or could it be that the square root of thermodynamics defined by ZEO actually gives automatically rise to this average? The eigenvalues of the "hermitian square root " of the density matrix would code for components of the state characterized by different classical correlation functions. One could assign these contributions to different "phases".

  4. Quantum classical correspondence in statistical sense would be very much like holography (now individual classical state represents the entire quantum state). Quantum ergodicity would pose a rather strong constraint on quantum states. This symmetry principle could actually fix the spectrum of zero energy states to a high degree and fix therefore the M-matrices given by the product of hermitian square root of density matrix and unitary S-matrix and unitary U-matrix having M-matrices as its orthonormal rows.
  5. In TGD inspired theory of consciousness the counterpart of quantum ergodicity is the postulate that the space-time geometry provides a symbolic representation for the quantum states and also for the contents of consciousness assignable to quantum jumps between quantum states. Quantum ergodicity would realize this strongly self-referential looking condition. The positive and negative energy parts of zero energy state would be analogous to the initial and final states of quantum jump and the classical correlation functions would code for the contents of consciousness like written formulas code for the thoughts of mathematician and provide a sensory feedback.
How classical correlation functions should be defined?
  1. General Coordinate Invariance and Lorentz invariance are the basic constraints on the definition. These are achieved for the space-time regions with Minkowskian signature and 4-D M4 projection if linear Minkowski coordinates are used. This is equivalent with the contraction of the indices of tensor fields with the space-time projections of M4 Killing vector fields representing translations. Accepting ths generalization, there is no need to restrict oneself to 4-D M4 projection and one can also consider also Euclidian regions identifiable as lines of generalized Feynman diagrams.

    Quantum ergodicity very probably however forces to restrict the consideration to Minkowskian and Euclidian space-time regions and various phases associated with them. Also CP2 Killing vector fields can be projected to space-time surface and give a representation for classical gluon fields. These in turn can be contracted with M4 Killing vectors giving rise to gluon fields as analogs of graviton fields but with second polarization index replaced with color index.

  2. The standard definition for the correlation functions associated with classical time evolution is the appropriate starting point. The correlation function GXY(τ) for two dynamical variables X(t) and Y(t) is defined as the average GXY(τ)=∫T X(t)Y(t+τ)dt/T over an interval of length T, and one can also consider the limit T→ ∞. In the recent case one would replace kenotau with the difference m1-m2=m of M4 coordinates of two points at the preferred extremal and integrate over the points of the extremal to get the average. The finite time interval T is replaced with the volume of causal diamond in a given length scale. Zero energy state with given quantum numbers for positive and negative energy parts of the state defines the initial and final states between which the fields appearing in the correlation functions are defined.
  3. What correlation functions should be considered? Certainly one could calculate correlation functions for the induced spinor connection given electro-weak propagators and correlation functions for CP2 Killing vector fields giving correlation functions for gluon fields using the description in terms of Killing vector fields. If one can uniquely separate from the Fourier transform uniquely a term of form Z/(p2-m2) by its momentum dependence, the coefficient Z can be identified as coupling constant squared for the corresponding gauge potential component and one can in principle deduce coupling constant evolution purely classically. One can imagine of calculating spinorial propagators for string world sheets in the same manner. Note that also the dependence on color quantum numbers would be present so that in principle all that is needed could be calculated for a single preferred extremal without the need to construct QFT limit and to introduce color quantum numbers of fermions as spin like quantum numbers (color quantum numbers corresponds to CP2 partial wave for the tip of the CD assigned with the particle).
  4. What about Higgs like field? TGD in principle allows scalar and pseudo-scalars which could be called Higgs like states. These states are however not necessary for particle massivation although they can represent particle massivation and must do so if one assumes that QFT limit exist. p-Adic thermodynamics however describes particle massivation microscopically.

    The problem is that Higgs like field does not seem to have any obvious space-time correlate. The trace of the second fundamental form is the obvious candidate but vanishes for preferred extremals which are both minimal surfaces and solutions of Einstein Maxwell equations with cosmological constant. If the string world sheets at which all spinor components except right handed neutrino are localized for the general solution ansatz of the modified Dirac equation, the corresponding second fundamental form at the level of imbedding space defines a candidate for classical Higgs field. A natural expectation is that string world sheets are minimal surfaces of space-time surface. In general they are however not minimal surfaces of the imbedding space so that one might achieve a microscopic definition of classical Higgs field and its vacuum expectation value as an average of one point correlation function over the string world sheet.

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Do geometric invariants of preferred extremals define topological invariants of space-time surface and code for quantum physics?".

Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric invariants are topological invariants

The recent progress in the understanding of the preferred extremals led to a reduction of the field equations to conditions stating for Euclidian signature the existence of Kähler metric. The resulting conditions are a direct generalization of corresponding conditions emerging for the string world sheet and stating that the 2-metric has only non-diagonal components in complex/hypercomplex coordinates. Also energy momentum of Kähler action and has this characteristic (1,1) tensor structure. In Minkowskian signature one obtains the analog of 4-D complex structure combining hyper-complex structure and 2-D complex structure.

The construction lead also to the understanding of how Einstein's equations with cosmological term follow as a consistency condition guaranteeing that the covariant divergence of the Maxwell's energy momentum tensor assignable to Kähler action vanishes. This gives T= kG+Λ g. By taking trace a further condition follows from the vanishing trace of T:

R = 4Λ/k .

That any preferred extremal should have a constant Ricci scalar proportional to cosmological constant is very strong prediction. Note however that both Λ and k∝ 1/G are both parameters characterizing one particular preferred extremal. One could of course argue that the dynamics allowing only constant curvature space-times is too simple. The point is however that particle can topologically condense on several space-time sheets meaning effective superposition of various classical fields defined by induced metric and spinor connection.

The following considerations demonstrate that preferred extremals can be seen as canonical representatives for the constant curvature manifolds playing central role inThurston's geometrization theorem known also as hyperbolization theorem implying that geometric invariants of space-time surfaces transform to topological invariants. The generalization of the notion of Ricci flow to Maxwell flow in the space of metrics and further to Kähler flow for preferred extremals in turn gives a rather detailed vision about how preferred extremals organize to one-parameter orbits. It is quite possible that Kähler flow is actually discrete. The natural interpretation is in terms of dissipation and self organization.

A. The geometrical invariants of space-time surfaces as topological invariants

An old conjecture inspired by the preferred extremal property is that the geometric invariants of the space-time surface serve as topological invariants. The reduction ofKähler action to 3-D Chern-Simons terms gives support for this conjecture as a classical counterpart for the view about TGD as almost topological QFT. The following arguments give a more precise content to this conjecture in terms of existing mathematics.

  1. It is not possible to represent the scaling of the induced metric as a deformation of the space-time surface preserving the preferred extremal property since the scale of CP2 breaks scale invariance. Therefore the curvature scalar cannot be chosen to be equal to one numerically. Therefore also the parameter R=4Λ/k and also Λ and k separately characterize the equivalence class of preferred extremals as is also physically clear.

    Also the volume of the space-time sheet closed inside causal diamond CD remains constant along the orbits of the flow and thus characterizes the space-time surface. Λ and even k∝ 1/G can indeed depend on space-time sheet and p-adic length scale hypothesis suggests a discrete spectrum for Λ/k expressible in terms of p-adic length scales: Λ/k ∝ 1/Lp2 with p≈ 2k favored by p-adic length scale hypothesis. During cosmic evolution the p-adic length scale would increase gradually. This would resolve the problem posed by cosmological constant in GRT based theories.

  2. One could also see the preferred extremals as 4-D counterparts of constant curvature 3-manifolds in the topology of 3-manifolds. An interesting possibility raised by the observed negative value of Λ is that most 4-surfaces are constant negative curvature 4-manifolds. By a general theorem coset spaces H4/Γ, where H4= SO(1,4)/SO(4) is hyperboloid of M5 and Γ a torsion free discrete subgroup of SO(1,4). Geometric invariants are therefore topological invariants. It is not clear to me, whether the constant value of Ricci scalar implies constant sectional curvatures and therefore hyperbolic space property. It could happen that the space of spaces with constant Ricci curvature contain a hyperbolic manifold as an especially symmetric representative. In any case, the geometric invariants of hyperbolic metric are topological invariants.

    By Mostow rigidity theorem finite-volume hyperbolic manifold is unique for D>2 and determined by the fundamental group of the manifold. Since the orbits under the Kähler flow preserve the curvature scalar the manifolds at the orbit must represent different imbeddings of one and hyperbolic 4-manifold. In 2-D case the moduli space for hyperbolic metric for a given genus g>0 is defined by Teichmueller parameters and has dimension 6(g-1). Obviously the exceptional character of D=2 case relates to conformal invariance. Note that the moduli space in question plays a key role in p-adic mass calculations \cite{allb}{elvafu}.

    In the recent case Mostow rigidity theorem could hold true for the Euclidian regions and maybe generalize also to Minkowskian regions. If so then both "topological" and "geometro" in "Topological GeometroDynamics" would be fully justified. The fact that geometric invariants become topological invariants also conforms with "TGD as almost topological QFT" and allows the notion of scale to find its place in topology. Also the dream about exact solvability of the theory would be realized in rather convincing manner.

These conjectures are the main result of this posting independent of whether the generalization of the Ricci flow discussed in the sequel exists as a continuous flow or possibly discrete sequence of iterates in the space of preferred extremals of Kähler action. My sincere hope is that the reader could grasp how far reaching these result really are.

B. Generalizing Ricci flow to Maxwell flow for 4-geometries and K\"ahler flow for space-time surfaces

The notion of Ricci flow has played a key part in the geometrization of topological invariants of Riemann manifolds. I certainly did not have this in mind when I choose to call my unification attempt "Topological Geometrodynamics" but this title strongly suggests that a suitable generalization of Ricci flow could play a key role in the understanding of also TGD.

B.1. Ricci flow and Maxwell flow for 4-geometries

The observation about constancy of 4-D curvature scalar for preferred extremals inspires a generalization of the well-known volume preserving Ricci flow introduced by Richard Hamilton and defined in the space of Riemann metrics as

dgαβ/dt= -2Rαβ+ (2/D)Ravggαβ .

Here Ravg denotes the average of the scalar curvature, and D is the dimension of the Riemann manifold. The flow is volume preserving in average sense as one easily checks (<gαβdgαβ/dt> =0). The volume preserving property of this flow allows to intuitively understand that the volume of a 3-manifold in the asymptotic metric defined by the Ricci flow is topological invariant. The fixed points of the flow serve as canonical representatives for the topological equivalence classes of 3-manifolds. These 3-manifolds (for instance hyperbolic 3-manifolds with constant sectional curvatures) are highly symmetric. This is easy to understand since the flow is dissipative and destroys all details from the metric.

What happens in the recent case? The first thing to do is to consider what might be called Maxwell flow in the space of all 4-D Riemann manifolds allowing Maxwell field.

  1. First of all, the vanishing of the trace of Maxwell's energy momentum tensor codes for the volume preserving character of the flow defined as

    dgαβ/dt= Tαβ .

    Taking covariant divergence on both sides and assuming that d/dt and Dα commute, one obtains that Tαβ is divergenceless.

    This is true if one assumes Einstein Maxwell equations with cosmological term. This gives

    dgαβ/dt= kGαβ+ Λ gαβ =k Rαβ + (-kR/2+Λ)gαβ .

    The trace of this equation gives that the curvature scalar is constant. Note that the value of the Kähler coupling strength plays a highly non-trivial role in these equations and it is quite possible that solutions exist only for some critical values of αK. Quantum criticality should fix the allow value triplets (G,Λ,αK) apart from overall scaling

    (G,Λ,αK)→ (xG,Λ/x, xαK) .

    Fixing the value of G fixes the values remaining parameters at critical points. The rescaling of the parameter t induces a scaling by x.

  2. By taking trace one obtains the already mentioned condition fixing the curvature to be constant, and one can write

    dgαβ/dt= kRαβ -Λ gαβ .

    Note that in the recent case Ravg=R holds true since curvature scalar is constant. The fixed points of the flow would be Einstein manifolds satisfying

    Rαβ= (Λ/k) gαβ .

  3. It is by no means obvious that continuous flow is possible. The condition that Einstein-Maxwell equations are satisfied might pick up from a completely general Maxwell flow a discrete subset as solutions of Einstein-Maxwell equations with a cosmological term. If so, one could assign to this subset a sequence of values tn of the flow parameter t.
  4. I do not know whether 3-dimensionality is somehow absolutely essential for getting the classification of closed 3-manifolds using Ricci flow. This ignorance allows me to pose some innocent questions. Could one have a canonical representation of 4-geometries as spaces with constant Ricci scalar? Could one select one particular Einstein space in the class four-metrics and could the ratio Λ/k represent topological invariant if one normalizes metric or curvature scalar suitably. In the 3-dimensional case curvature scalar is normalized to unity. In the recent case this normalization would give k= 4Λ in turn giving Rαβ= gαβ/4. Does this mean that there is only single fixed point in local sense, analogous to black hole toward which all geometries are driven by the Maxwell flow? Does this imply that only the 4-volume of the original space would serve as a topological invariant?

B.2. Maxwell flow for space-time surfaces

One can consider Maxwell flow for space-time surfaces too. In this case Kähler flow would be the appropriate term and provides families of preferred extremals. Since space-time surfaces inside CD are the basic physical objects are in TGD framework, a possible interpretation of these families would be as flows describing physical dissipation as a four-dimensional phenomenon polishing details from the space-time surface interpreted as an analog of Bohr orbit.

  1. The flow is now induced by a vector field jk(x,t) of the space-time surface having values in the tangent bundle of imbedding space M4× CP2. In the most general case one has Kähler flow without the Einstein equations. This flow would be defined in the space of all space-time surfaces or possibly in the space of all extremals. The flow equations reduce to

    hkl Dα jk(x,t) Dβhl= (1/2)Tαβ .

    The left hand side is the projection of the covariant gradient Dαjk(x,t) of the flow vector field jk(x,t) to the tangent space of the space-time surface. D α is covariant derivative taking into account that jk is imbedding space vector field. For a fixed point space-time surface this projection must vanish assuming that this space-time surface reachable. A good guess for the asymptotia is that the divergence of Maxwell energy momentum tensor vanishes and that Einstein's equations with cosmological constant are well-defined.

    Asymptotes corresponds to vacuum extremals. In Euclidian regions CP2 type vacuum extremals and in Minkowskian regions to any space-time surface in any 6-D sub-manifold M4× Y2, where Y2 is Lagrangian sub-manifold of CP2 having therefore vanishing induced Kähler form. Symplectic transformations of CP2 combined with diffeomorphisms of M4 give new Lagrangian manifolds. One would expect that vacuum extremals are approached but never reached at second extreme for the flow.

    If one assumes Einstein's equations with a cosmological term, allowed vacuum extremals must be Einstein manifolds. For CP2 type vacuum extremals this is the case. It is quite possible that these fixed points do not actually exist in Minkowskian sector, and could be replaced with more complex asymptotic behavior such as limit, chaos, or strange attractor.

  2. The flow could be also restricted to the space of preferred extremals. Assuming that Einstein Maxwell equations indeed hold true, the flow equations reduce to

    hklDα jk(x,t) ∂βhl= 1/2(kRαβ -Λ gαβ) .

    Preferred extremals would correspond to a fixed sub-manifold of the general flow in the space of all 4-surfaces.

  3. One can also consider a situation in which jk(x,t) is replaced with jk(h,t) defining a flow in the entire imbedding space. This assumption is probably too restrictive. In this case the equations reduce to

    (Dr jl(x,t)+Dljr)∂αhrβhl= kRαβ -Λ gαβ .

    Here Dr denotes covariant derivative. Asymptotia is achieved if the tensor Dkjl+Dkjl becomes orthogonal to the space-time surface. Note for that Killing vector fields of H the left hand side vanishes identically. Killing vector fields are indeed symmetries of also asymptotic states.

It must be made clear that the existence of a continuous flow in the space of preferred extremals might be too strong a condition. Already the restriction of the general Maxwell flow in the space of metrics to solutions of Einstein-Maxwell equations with cosmological term might lead to discretization, and the assumption about reprentability as 4-surface in M4 × CP2 would give a further condition reducing the number of solutions. On the other hand, one might consiser a possibility of a continuous flow in the space of constant Ricci scalar metrics with a fixed 4-volume and having hyperbolic spaces as the most symmetric representative.

B.3. Dissipation, self organization, transition to chaos, and coupling constant evolution

A beautiful connection with concepts like dissipation, self-organization, transition to chaos, and coupling constant evolution suggests itself.

  1. It is not at all clear whether the vacuum extremal limits of the preferred extremals can correspond to Einstein spaces except in special cases such as CP2 type vacuum extremals isometric with CP2. The imbeddability condition defines a constraint force which might well force asymptotically more complex situations such as limit cycles and strange attractors. In ordinary dissipative dynamics an external energy feed is essential prerequisite for this kind of non-trivial self-organization patterns. As a matter fact, the fact that the Kähler action equals to

    In the recent case the external energy feed could be replaced by the constraint forces due to the imbeddability condition. It is not too difficult to imagine that the flow (if it exists!) could define something analogous to a transition to chaos taking place in a stepwise manner for critical values of the parameter t. Alternatively, these discrete values could correspond to those values of t for which the preferred extremal property holds true for a general Maxwell flow in the space of 4-metrics. Therefore the preferred extremals of Kähler action could emerge as one-parameter (possibly discrete) families describing dissipation and self-organization at the level of space-time dynamics.

  2. For instance, one can consider the possibility that in some situations Einstein's equations split into two mutually consistent equations of which only the first one is independent

    xJανJνβ = Rαβ , LK= xJανJνβ= 4Λ ,

    x=1/16παK .

    Note that the first equation indeed gives the second one by tracing. This happens for CP2 type vacuum extremals.

    Kähler action density would reduce to cosmological constant which should have a continuous spectrum if this happens always. A more plausible alternative is that this holds true only asymptotically. In this case the flow equation could not lead arbitrary near to vacuum extremal, and one can think of situation in which LK= 4Λ defines an analog of limiting cycle or perhaps even strange attractor. In any case, the assumption would allow to deduce the asymptotic value of the action density which is of utmost importance from calculational point of view: action would be simply SK= 4Λ V4 and one could also say that one has minimal surface with Λ taking the role of string tension.

  3. One of the key ideas of TGD is quantum criticality implying that Kähler coupling strength is analogous to critical temperature. Second key idea is that p-adic coupling constant evolution represents discretized version of continuous coupling constant evolution so that each p-adic prime would correspond a fixed point of ordinary coupling constant evolution in the sense that the 4-volume characterized by the p-adic length scale remains constant. The invariance of the geometric and thus geometric parameters of hyperbolic 4-manifold under the Kähler flow would conform with the interpretation as a flow preserving scale assignable to a given p-adic prime. The continuous evolution in question (if possible at all!) might correspond to a fixed p-adic prime. Also the hierarchy of Planck constants relates to this picture naturally. Planck constant hbareff=nhbar corresponds to a multi-furcation generating n-sheeted structure and certainly affecting the fundamental group.
  4. One can of course question the assumption that a continuous flow exists. The property of being a solution of Einstein-Maxwell equations, imbeddability property, and preferred extremal property might allow allow only discrete sequences of space-time surfaces perhaps interpretable as orbit of an iterated map leading gradually to a fractal limit. This kind of discrete sequence might be also be selected as preferred extremals from the orbit of Maxwell flow without assuming Einstein-Maxwell equations. Perhaps the discrete p-adic coupling constant evolution could be seen in this manner and be regarded as an iteration so that the connection with fractality would become obvious too.

B.4 Does a 4-D counterpart of thermodynamics make sense?

The interpretation of the Kähler flow in terms of dissipation, the constancy of R, and almost constancy of LK suggest an interpretation in terms of 4-D variant of thermodynamics natural in zero energy ontology (ZEO), where physical states are analogs for pairs of initial and final states of quantum event are quantum superpositions of classical time evolutions. Quantum theory becomes a "square root" of thermodynamics so that 4-D analog of thermodynamics might even replace ordinary thermodynamics as a fundamental description. If so this 4-D thermodynamics should be qualitatively consistent with the ordinary 3-D thermodynamics.

  1. The first naive guess would be the interpretation of the action density LK as an analog of energy density e=E/V3 and that of R as the analog to entropy density s=S/V3. The asymptotic states would be analogs of thermodynamical equilibria having constant values of LK and R.
  2. Apart from an overall sign factor ε to be discussed, the analog of the first law de= Tds-pdV/V would be

    dLK = kdR +Λ dV4/V4 .

    One would have the correspondences S→ ε RV4, e→ ε LK and k→ T, p→ -Λ. k∝ 1/G indeed appears formally in the role of temperature in Einstein's action defining a formal partition function via its exponent. The analog of second law would state the increase of the magnitude of ε RV4 during the Kähler flow.

  3. One must be very careful with the signs and discuss Euclidian and Minkowskian regions separately. Concerning purely thermodynamic aspects at the level of vacuum functional Euclidian regions are those which matter.
    1. For CP2 type vacuum extremals LK ∝ E2+B2 , R=Λ/k, and Λ are positive. In thermodynamical analogy for ε=1 this would mean that pressure is negative.
    2. In Minkowskian regions the value of R=Λ/k is negative for Λ<0 suggested by the large abundance of 4-manifolds allowing hyperbolic metric and also by cosmological considerations. The asymptotic formula LK= 4Λ considered above suggests that also Kähler action is negative in Minkowskian regions for magnetic flux tubes dominating in TGD inspired cosmology: the reason is that the magnetic contribution to the action density LK∝ E2-B2 dominates.
Consider now in more detail the 4-D thermodynamics interpretation in Euclidian and Minkowskian regions assuming that the the evolution by quantum jumps has Kähler flow as a space-time correlate.
  1. In Euclidian regions the choice ε=1 seems to be more reasonable one. In Euclidian regions -Λ as the analog of pressure would be negative, and asymptotically (that is for CP2 type vacuum extremals) its value would be proportional to Λ ∝ 1/GR2, where R denotes CP2 radius defined by the length of its geodesic circle.

    A possible interpretation for negative pressure is in terms of string tension effectively inducing negative pressure (note that the solutions of the modified Dirac equation indeed assign a string to the wormhole contact). The analog of the second law would require the increase of RV4 in quantum jumps. The magnitudes of LK, R, V4 and Λ would be reduced and approach their asymptotic values. In particular, V4 would approach asymptotically the volume of CP2.

  2. In Minkowskian regions Kähler action contributes to the vacuum functional a phase factor analogous to an imaginary exponent of action serving in the role of Morse function so that thermodynamics interpretation can be questioned. Despite this one can check whether thermodynamic interpretation can be considered. The choice ε=-1 seems to be the correct choice now. -Λ would be analogous to a negative pressure whose gradually decreases. In 3-D thermodynamics it is natural to assign negative pressure to the magnetic flux tube like structures as their effective string tension defined by the density of magnetic energy per unit length. -R≥ 0 would entropy and -LK≥ 0 would be the analog of energy density.

    R=Λ/k and the reduction of Λ during cosmic evolution by quantum jumps suggests that the larger the volume of CD and thus of (at least) Minkowskian space-time sheet the smaller the negative value of Λ.

    Assume the recent view about state function reduction explaining how the arrow of geometric time is induced by the quantum jump sequence defining experienced time. According to this view zero energy states are quantum superpositions over CDs of various size scales but with common tip, which can correspond to either the upper or lower light-like boundary of CD. The sequence of quantum jumps the gradual increase of the average size of CD in the quantum superposition and therefore that of average value of V4. On the other hand, a gradual decrease of both -LK and -R looks physically very natural. If Kähler flow describes the effect of dissipation by quantum jumps in ZEO then the space-time surfaces would gradually approach nearly vacuum extremals with constant value of entropy density -R but gradually increasing 4-volume so that the analog of second law stating the increase of -RV4 would hold true.

  3. The interpretation of -R>0 as negentropy density assignable to entanglement is also possible and is consistent with the interpretation in terms of second law. This interpretation would only change the sign factor ε in the proposed formula. Otherwise the above arguments would remain as such.

For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric invariants are topological invariants".

Electron as a trefoil or something more general?

The progress in the understanding of the modified Dirac equation led to the conclusion that the mere conservation of em charge in spinorial manner rather than from theorem of Gauss leads the conclusion that the solutions of the modified Dirac equation must be localized at string world sheets and partonic 2-surfaces: right-handed neutrino is expection and delocalized into entire space-time surface. Looking more closely the implications of this led to the conclusion that every ordinary elementary fermion is accompanied by closed string. Ordinary elementary bosons can be accompanied by two such strings. If light-like wormhole throat orbots carry several fermions one can have several closed strings. These closed strings can get knotted and braided. I do not bother to type more but attach the abstract of a little article Electron as a trefoil or something more general?.

There have been suggestions that elementary particle could be braided structure and that standard model quantum numbers could be reduced to topology. In TGD framework this option does not look plausible. The braiding at the level of wormhole throat orbits is however in principle possible but need not be significant for the known elementary particles. In TGD framework elementary particle is identified as a closed Kähler magnetic flux tube carrying monopole magnetic field. This flux tube is accompanied by a closed string representing the end of string world sheet carrying induced spinor field. This string can be homologically non-trivial curve and can also get knotted. Bosons even braiding becomes possible and in the general case knotting, braiding, and non-trivial homology are possible. Therefore an extremely rich topological structure is predicted, which might corresponds to relatively low energy scale. Topological sum for knots and reconnection are the basic topological reactions for these strings and can change the knotting of the string. These reactions represent basic vertices for closed strings so that closed string model could give at least idea about the dynamics of knotting and un-knotting.

For details and background see the chapter Knots and TGD, or the already mentioned article .

Realization of large N SUSY in TGD

The generators large N SUSY algebras are obtained by taking fermionic currents for second quantized fermions and replacing either fermion field or its conjugate with its particular mode. The resulting super currents are conserved and define super charges. By replacing both fermion and its conjugate with modes one obtains c number valued currents. Therefore N=∞ SUSY - presumably equivalent with super-conformal invariance - or its finite N cutoff is realized in TGD framework and the challenge is to understand the realization in more detail.

Super-space viz. Grassmann algebra valued fields

Standard SUSY induces super-space extending space-time by adding anti-commuting coordinates as a formal tool. Many mathematicians are not enthusiastic about this approach because of the purely formal nature of anti-commuting coordinates. Also I regard them as a non-sense geometrically and there is actually no need to introduce them as the following little argument shows.

Grassmann parameters (anti-commuting theta parameters) are generators of Grassmann algebra and the natural object replacing super-space is this Grassmann algebra with coefficients of Grassmann algebra basis appearing as ordinary real or complex coordinates. This is just an ordinary space with additional algebraic structure: the mysterious anti-commuting coordinates are not needed. To me this notion is one of the conceptual monsters created by the over-pragmatic thinking of theoreticians.

This allows allows to replace field space with super field space, which is completely well-defined object mathematically, and leave space-time untouched. Linear field space is simply replaced with its Grassmann algebra. For non-linear field space this replacement does not work. This allows to formulate the notion of linear super-field just in the same manner as it is done usually.

The generators of super-symmetries in super-space formulation reduce to super translations , which anti-commute to translations. The super generators Qα and Qbardotβ of super Poincare algebra are Weyl spinors commuting with momenta and anti-commuting to momenta:

{Qα,Qbardotβ}=2σμαdotβPμ .

One particular representation of super generators acting on super fields is given by

Dα=i∂/∂θα ,

Ddotα=i∂/∂θbardotα+ θβσμβdotαμ

Here the index raising for 2-spinors is carried out using antisymmetric 2-tensor εαβ. Super-space interpretation is not necessary since one can interpret this action as an action on Grassmann algebra valued field mixing components with different fermion numbers.

Chiral superfields are defined as fields annihilated by Ddotα. Chiral fields are of form Ψ(xμ+iθbarσμθ, θ). The dependence on θbardotα comes only from its presence in the translated Minkowski coordinate annihilated by Ddotα. Super-space enthusiast would say that by a translation of M4 coordinates chiral fields reduce to fields, which depend on θ only.

The space of fermionic Fock states at partonic 2-surface as TGD counterpart of chiral super field

As already noticed, another manner to realize SUSY in terms of representations the super algebra of conserved super-charges. In TGD framework these super charges are naturally associated with the modified Dirac equation, and anti-commuting coordinates and super-fields do not appear anywhere. One can however ask whether one could identify a mathematical structure replacing the notion of chiral super field.

I have proposed that generalized chiral super-fields could effectively replace induced spinor fields and that second quantized fermionic oscillator operators define the analog of SUSY algebra. One would have N=∞ if all the conformal excitations of the induced spinor field restricted on 2-surface are present. For right-handed neutrino the modes are labeled by two integers and delocalized to the interior of Euclidian or Minkowskian regions of space-time sheet.

The obvious guess is that chiral super-field generalizes to the field having as its components many-fermions states at partonic 2-surfaces with theta parameters and their conjugates in one-one correspondence with fermionic creation operators and their hermitian conjugates.

  1. Fermionic creation operators - in classical theory corresponding anti-commuting Grassmann parameters - replace theta parameters. Theta parameters and their conjugates are not in one-one correspondence with spinor components but with the fermionic creation operators and their hermitian conjugates. One can say that the super-field in question is defined in the "world of classical worlds" (WCW) rather than in space-time. Fermionic Fock state at the partonic 2-surface is the value of the chiral super field at particular point of WCW.
  2. The matrix defined by the σμμ is replaced with a matrix defined by the modified Dirac operator D between spinor modes acting in the solution space of the modified Dirac equation. Since modified Dirac operator annihilates the modes of the induced spinor field, super covariant derivatives reduce to ordinary derivatives with respect the theta parameters labeling the modes. Hence the chiral super field is a field that depends on θm or conjugates θbarm only. In second quantization the modes of the chiral super-field are many-fermion states assigned to partonic 2-surfaces and string world sheets. Note that this is the only possibility since the notion of super-coordinate does not make sense now.
  3. It would seem that the notion of super-field does not bring anything new. This is not the case. First of all, the spinor fields are restricted to 2-surfaces. Second point is that one cannot assign to the fermions of the many-fermion states separate non-parallel or even parallel four-momenta. The many-fermion state behaves like elementary particle. This has non-trivial implications for propagators and a simple argument leads to the proposal that propagator for N-fermion partonic state is proportional to 1/pN. This would mean that only the states with fermion number equal to 1 or 2 behave like ordinary elementary particles.

How the fermionic anti-commutation relations are determined?

Understanding the fermionic anti-commutation relations is not trivial since all fermion fields except right-handed neutrino are assumed to be localized at 2-surfaces. Since fermionic conserved currents must give rise to well-defined charges as 3-D integrals the spinor modes must be proportional to a square root of delta function in normal directions. Furthermore, the modified Dirac operator must act only in the directions tangential to the 2-surface in order that the modified Dirac equation can be satisfied.

The square root of delta function can be formally defined by starting from the expansion of delta function in discrete basis for a particle in 1-D box. The product of two functions in x-space is convolution of Fourier transforms and the coefficients of Fourier transform of delta function are apart from a constant multiplier equal to 1: δ (x)= K∑n exp(inx/2π L). Therefore the Fourier transform of square root of delta function is obtained by normalizing the Fourier transform of delta function by N1/2, where N→ ∞ is the number of plane waves. In other words: (δ (x))1/2= (K/N)1/2 nexp(inx/2π L).

Canonical quantization defines the standard approach to the second quantization of the Dirac equation.

  1. One restricts the consideration to time=constant slices of space-time surface. Now the 3-surfaces at the ends of CD are natural slices. The intersection of string world sheet with these surfaces is 1-D whereas partonic 2-surfaces have 2-D Euclidian intersection with them.
  2. The canonical momentum density is defined by

    Πα= ∂ L/∂t Ψbarα= ΓtΨ ,

    Γt= ∂ LK/∂ (∂thkk .

    LK denotes Kähler action density: consistency requires DμΓμ=0, and this is guaranteed only by using the modified gamma matrices defined by Kähler action. Note that Γt contains also the (g4)1/2 factor. Induced gamma matrices would require action defined by four-volume. t is time coordinate varying in direction tangential to 2-surface.

  3. The standard equal time canonical anti-commutation relations state

    α,Ψbarβ}= δ3(x,y)δαβ .

Can these conditions be applied both at string world sheets and partonic 2-surfaces.

  1. String world sheets do not pose problems. The restriction of the modes to string world sheets means that the square root of delta function in the normal direction of string world sheet takes care of the normal dimensions and the dynamical part of anti-commutation relations is 1-dimensional just as in the case of strings.
  2. Partonic 2-surfaces are problematic. The (g4)1/2 factor in Γt implies that Γt approaches zero at partonic 2-surfaces since they belong to light-like wormhole throats at which the signature of the induced metric changes. Energy momentum tensor appearing in Γt involves two index raisings by induced metric so that it can grow without limit as one approaches partonic two-surface. Therefore it is quite possible that the limit is finite and the boundary conditions defined by the weak form of electric magnetic duality might imply that the limit is finite. The open question is whether one can apply canonical quantization at partonic 2-surfaces. One can also ask whether one can define induced spinor fields at wormhole throats only at the ends of string world sheets so that partonic 2-surface would be effectively discretized. This cautious conclusion emerged in the earlier study of the modified Dirac equation.
  3. Suppose that one can assume spinor modes at partonic 2-surfaces. 2-D conformal invariance suggests that the situation reduces to effectively one-dimensional also at the partonic two-surfaces. If so, one should pose the anti-commutation relations at some 1-D curves of the partonic 2-surface only. This is the only sensical option. The point is that the action of the modified Dirac operator is tangential so that also the canonical momentum current must be tangential and one can fix anti-commutations only at some set of curves of the partonic 2-surface.

One can of course worry what happens at the limit of vacuum extremals. The problem is that Γt vanishes for space-time surfaces reducing to vacuum extremals at the 2-surfaces carrying fermions so that the anti-commutations are inconsistent. Should one require - as done earlier- that the anti-commutation relations make sense at this limit and cannot therefore have the standard form but involve the scalar magnetic flux formed from the induced Kähler form by permuting it with the 2-D permutations symbl? The restriction to preferred extremals, which are always non-vacuum extremals, might allow to avoid this kind of problems automatically.

In the case of right-handed neutrino the situation is genuinely 3-dimensional and in this case non-vacuum extremal property must hold true in the regions where the modes of νR are non-vanishing. The same mechanism would save from problems also at the partonic 2-surfaces. The dynamics of induced spinor fields must avoid classical vacuum. Could this relate to color confinement? Could hadrons be surrounded by an insulating layer of Kähler vacuum?

For details and background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the same title.

M8-H duality, preferred extremals, criticality, and Mandelbrot fractals

M8-H duality (see this) represents an intriguing connection between number theory and TGD but the mathematics involved is extremely abstract and difficult so that I can only represent conjectures. In the following the basic duality is used to formulate a general conjecture for the construction of preferred extremals by iterative procedure. What is remarkable and extremely surprising is that the iteration gives rise to the analogs of Mandelbrot fractals and space-time surfaces can be seen as fractals defined as fixed sets of iteration. The analogy with Mandelbrot set can be also seen as a geometric correlate for quantum criticality.

M8-H duality

M8-H duality states the following. Consider a distribution of two planes M2(x) integrating to a 2-surface N2 with the property that a fixed 1-plane M1 defining time axis globally is contained in each M2(x) and therefore in N2. M1 defines real axis of octonionic plane M8 and M2(x) a local hyper-complex plane. Quaternionic subspaces with this property can be parameterized by points of CP2. Define quaternionic surfaces in M8 as 4-surfaces, whose tangent plane is quaternionic at each point x and contains the local hyper-complex plane M2(x) and is therefore labelled by a point s(x)∈ CP2. One can write these surfaces as union over 2-D surfaces associated with points of N2:

X4= ∪x∈ N2 X2(x)⊂ E6 .

These surfaces can be mapped to surfaces of M4× CP2 via the correspondence (m(x),e(x))→ (m,s(T(X4(x)). Also the image surface contains at given point x the preferred plane M2(x) ⊃ M1. One can also write these surfaces as union over 2-D surfaces associated with points of N2:

X4= ∪x∈ N2 X2(x)⊂ E2× CP2 .

One can also ask what are the conditions under which one can map surfaces X4= ∪x∈ N2 X2⊂ E2× CP2 to 4-surfaces in M8. The map would be given by (m,s)→ (m,T4(s) and the surface would be of the form as already described. The surface X4 must be such that the distribution of 4-D tangent planes defined in M8 is integrable and this gives complicated integrability conditions. One might hope that the conditions might hold true for preferred extremals satisfying some additional conditions.

One must make clear that these conditions do not allow most general possible surface. The point is that for preferred extremals with Euclidian signature of metric the M4 projection is 3-dimensional and involves light like projection. Here the fact that light-like line L⊂ M2 spans M2 in the sense that the complement of its orthogonal complement in M8 is M2. Therefore one could consider also more general solution ansatz for which one has

X4= ∪x∈ Lx⊂ N2 X3(x)⊂ E2× CP2 .

One can also consider co-quaternionic surfaces as surfaces for which tangent space is in the dual of a quaternionic subspace containing the preferred M2(x).

The integrability conditions

The integrability conditions are associated with the expression of tangent vectors of T(X4) as a linear combination of coordinate gradients ∇ mk, where mk denote the coordinates of M8. Consider the 4 tangent vectors ei) for the quaternionic tangent plane (containing M2(x)) regarded as vectors of M8. ei) have components ei)k, i=1,..,4, k=1,...,8. One must be able to express ei) as linear combinations of coordinate gradients ∇ mk:

ei)k= ei)ααmk .

Here xα and ek denote coordinates for X4 and M8. By forming inner products of of ei) one finds that matrix ei)α represents the components of vierbein at X4. One can invert this matrix to get ei)α satisfying




One can solve the coordinate gradients ∇ mk from above equation to get

αmk = ei)αei)k== Eαk .

The integrability conditions follow from the gradient property and state

DαEkβ= DβEkα .

One obtains 8× 6=48 conditions in the general case. The slicing to a union of two-surfaces labeled by M2(x) reduces the number of conditions since the number of coordinates mk reduces from 8 to 6 and one has 36 integrability conditions but still them is much larger than the number of free variables- essentially the six transversal coordinates mk.

For co-quaternionic surfaces one can formulate integrability conditions now as conditions for the existence of integrable distribution of orthogonal complements for tangent planes and it seems that the conditions are formally similar.

How to solve the integrability conditions and field equations for preferred extremals?

The basic idea has been that the integrability condition characterize preferred extremals so that they can be said to be quaternionic in a well-defined sense. Could one imagine solving the integrability conditions by some simple ansatz utilizing the core idea of M8-H duality? What comes in mind is that M8 represents tangent space of M4× CP2 so that one can assign to any point (m,s) of 4-surface X4⊂ M4× CP2 a tangent plane T4(x) in its tangent space M8 identifiable as subspace of complexified octonions in the proposed manner. Assume that s∈ CP2 corresponds to a fixed tangent plane containing M2x, and that all planes M2x are mapped to the same standard fixed hyper-octonionic plane M2⊂ M8, which does not depend on x. This guarantees that s corresponds to a unique quaternionic tangent plane for given M2(x).

Consider the map Tοs. The map takes the tangent plane T4 at point (m,e)∈ M4× E4 and maps it to (m,s1=s(T4))∈ M4× CP2. The obvious identification of quaternionic tangent plane at (m,s1) would be as T4. One would have Tοs=Id. One could do this for all points of the quaternion surface X4⊂ E4 and hope of getting smooth 4-surface X4⊂ H as a result. This is the case if the integrability conditions at various points (m,s(T4)(x))∈ H are satisfied. One could equally well start from a quaternionic surface of H and end up with integrability conditions in M8 discussed above. The geometric meaning would be that the quaternionic surface in H is image of quaternionic surface in M8 under this map.

Could one somehow generalize this construction so that one could iterate the map Tοs to get Tοs=Id at the limit? If so, quaternionic space-time surfaces would be obtained as limits of iteration for rather arbitrary space-time surface in either M8 or H. One can also consider limit cycles, even limiting manifolds with finite-dimension which would give quaternionic surfaces. This would give a connection with chaos theory.

  1. One could try to proceed by discretizing the situation in M8 and H. One does not fix quaternionic surface at either side but just considers for a fixed m2∈ M2(x) a discrete collection X {T4i⊃ M2(x) of quaternionic planes in M8. The points e2,i⊂ E2⊂ M2× E2=M4 are not fixed. One can also assume that the points si=s(T4i) of CP2 defined by the collection of planes form in a good approximation a cubic lattice in CP2 but this is not absolutely essential. Complex Eguchi-Hanson coordinates ξi are natural choice for the coordinates of CP2. Assume also that the distances between the nearest CP2 points are below some upper limit.
  2. Consider now the iteration. One can map the collection X to H by mapping it to the set s(X) of pairs ((m2,si). Next one must select some candidates for the points e2,i∈ E2⊂ M4 somehow. One can define a piece-wise linear surface in M4× CP2 consisting of 4-planes defined by the nearest neighbors of given point (m2,e2,i,si). The coordinates e2,i for E2⊂ M4 can be chosen rather freely. The collection (e2,i,i) defines a piece-wise linear surface in H consisting of four-cubes in the simplest case. One can hope that for certain choices of e2,i the four-cubes are quaternionic and that there is some further criterion allowing to choose the points e2,i uniquely. The tangent planes contain by construction M2(x) so that the product of remaining two spanning tangent space vectors (e3,e4) must give an element of M2 in order to achieve quaternionicity. Another natural condition would be that the resulting tangent planes are not only quaternionic but also as near as possible to the planes T4i. These conditions allow to find e2,i giving rise to geometrically determined quaternionic tangent planes as near as possible to those determined by si.
  3. What to do next? Should one replace the quaternionic planes T4i with geometrically determined quaternionic planes as near as possible to them and map them to points si slightly different from the original one and repeat the procedure? This would not add new points to the approximation, and this is an unsatisfactory feature.
  4. Second possibility is based on the addition of the quaternionic tangent planes obtained in this manner to the original collection of quaternionic planes. Therefore the number of points in discretization increases and the added points of CP2 are as near as possible to existing ones. One can again determine the points e2,i in such a manner that the resulting geometrically determined quaternionic tangent planes are as near as possible to the original ones. This guarantees that the algorithm converges.
  5. The iteration can be stopped when desired accuracy is achieved: in other words the geometrically determined quaternionic tangent planes are near enough to those determined by the points si. Also limit cycles are possible and would be assignable to the transversal coordinates e2i varying periodically during iteration. One can quite well allow this kind of cycles, and they would mean that e2 coordinate as a function of CP2 coordinates characterizing the tangent plane is many-valued. This is certainly very probable for solutions representable locally as graphs M4→ CP2. In this case the tangent planes associated with distant points in E2 would be strongly correlated which must have non-trivial physical implications. The iteration makes sense also p-adically and it might be that in some cases only p-adic iteration converges for some value of p.
It is not obvious whether the proposed procedure gives rise to a smooth or even continuous 4-surface. The conditions for this are geometric analogs of the above described algebraic integrability conditions for the map assigning to the surface in M4× CP2 a surface in M8. Therefore M8-H duality could express the integrability conditions and preferred extremals would be 4-surfaces having counterparts also in the tangent space M8 of H.

One might hope that the self-referentiality condition sοT=Id for the CP2 projection of (m,s) or its fractal generalization could solve the complicated integrability conditions for the map T. The image of the space-time surface in tangent space M8 in turn could be interpreted as a description of space-time surface using coordinates defined by the local tangent space M8. Also the analogy for the duality between position and momentum suggests itself.

Is there any hope that this kind of construction could make sense? Or could one demonstrate that it fails? If s would fix completely the tangent plane it would be probably easy to kill the conjecture but this is not the case. Same s corresponds for different planes M2x to different point tangent plane. Presumably they are related by a local G2 or SO(7) rotation. Note that the construction can be formulated without any reference to the representation of the imbedding space gamma matrices in terms of octonions. Complexified octonions are enough in the tangent space of M8.

Connection with Mandelbrot fractal and fractals as fixed sets for iteration

The occurrence of iteration in the construction of preferred extremals suggests a deep connection with the standard construction of 2-D fractals by iteration - about which Mandelbrot fractal is the canonical example. X2(x) (or X3(x)) could be identified as a union of orbits for the iteration of sοT. The appearance of the iteration map in the construction of solutions of field equation would answer positively to a long standing question whether the extremely beautiful mathematics of 2-D fractals could have some application at the level of fundamental physics according to TGD.

X2 (or X3) would be completely analogous to Mandelbrot set in the sense that it would be boundary separating points in two different basis of attraction. In the case of Mandelbrot set iteration would take points at the other side of boundary to origin on the other side and to infinity. The points of Mandelbrot set are permuted by the iteration. In the recent case sοT maps X2 (or X3) to itself. This map need not be diffeomorphism or even continuous map. The criticality of X2 (or X3) could be seen as a geometric correlate for quantum criticality.

In fact, iteration plays a very general role in the construction of fractals. Very general fractals can be defined as fixed sets of iteration and simple rules for iteration produce impressive representations for fractals appearing in Nature. Therefore it would be highly satisfactory if space-time surfaces would be in well-defined sense fixed sets of iteration. This would be also numerically beautiful aspect since fixed sets of iteration can be obtained as infinite limit of iteration for almost arbitrary initial set.

What is intriguing and challenging is that there are several very attractive approaches to the construction of preferred extremals and the challenge of unifying them still remains.

For details and background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the same title.

The analog of AdS5 duality in TGD framework

The generalization of AdS5 duality of N=4 SYMs to TGD framework is highly suggestive and states that string world sheets and partonic 2-surfaces play a dual role in the construction of M-matrices. In the following I give an argument providing a "proof" of this duality and also demonstrating that for singular string world sheets and partonic 2-surfaces perturbative description of generalized Feynman diagrams is especially simple since string effectively reduces to point like particles.

Some terminology first.

  1. Let us agree that string world sheets and partonic 2-surfaces refer to 2-surfaces in the slicing of space-time region defined by Hermitian structure or Hamilton-Jacobi structure.
  2. Let us also agree that singular string world sheets and partonic 2-surfaces are surfaces at which the effective metric defined by the anticommutators of the modified gamma matrices degenerates to effectively 2-D one.
  3. Braid strands at wormhole throats in turn would be loci at which the induced metric of the string world sheet transforms from Euclidian to Minkowskian as the signature of induced metric changes from Euclidian to Minkowskian.
AdS5 duality suggest that string world sheets are in the same role as string world sheets of 10-D space connecting branes in AdS5 duality for N=4 SYM. What is important is that there should exist a duality meaning two manners to calculate the amplitudes. What the duality could mean now?
  1. Also in TGD framework the first manner would be string model like description using string world sheets. The second one would be a generalization of conformal QFT at light-like 3-surfaces (allowing generalized conformal symmetry) defining the lines of generalized Feynman diagram. The correlation functions to be calculated would have points at the intersections of partonic 2-surfaces and string world sheets and would represent braid ends.
  2. General Coordinate Invariance (GCI) implies that physics should be codable by 3-surfaces. Light-like 3-surfaces define 3-surfaces of this kind and same applies to space-like 3-surfaces. There are also preferred 3-surfaces of this kind. The orbits of 2-D wormhole throats at which 4-metric degenerates to 3-dimensional one define preferred light-like 3-surfaces. Also the space-like 3-surfaces at the ends of space-time surface at light-like boundaries of causal diamonds (CDs) define preferred space-like 3-surfaces. Both light-like and space-like 3-surfaces should code for the same physics and therefore their intersections defining partonic 2-surfaces plus the 4-D tangent space data at them should be enough to code for physics. This is strong form of GCI implying effective 2-dimensionality. As a special case one obtains singular string world sheets at which the effective metric reduces to 2-dimensional and singular partonic 2-surfaces defining the wormhole throats. For these 2-surfaces situation could be especially simple mathematically.
  3. The guess inspired by strong GCI is that string world sheet -partonic 2-surface duality holds true. The functional integrals over the deformations of 2 kinds of 2-surfaces should give the same result so tthat functional integration over either kinds of 2-surfaces should be enough. Note that the members of a given pair in the slicing intersect at discrete set of points and these points define braid ends carrying fermion number. Discretization and braid picture follow automatically.
  4. Scattering amplitudes in the twistorial approach could be thus calculated by using any pair in the slicing - or only either member of the pair if the analog of AdS5 duality holds true as argued. The possibility to choose any pair in the slicing means general coordinate invariance as a symmetry of the Kähler metric of WCW and of the entire theory suggested already early: Kähler functions for difference choices in the slicing would differ by a real part of holomorphic function and give rise to same Kähler metric of "world of classical worlds" (WCW). For a general pair one obtains functional integral over deformations of space-time surface inducing deformations of 2-surfaces with only other kind 2-surface contributing to amplitude. This means the analog of stringy QFT: Minkowskian or Euclidian string theory depending on choice.
  5. For singular string world sheets and partonic 2-surfaces an enormous simplification results. The propagators for fermions and correlation functions for deformations reduce to 1-D instead of being 2-D: the propagation takes place only along the light-like lines at which the string world sheets with Euclidian signature (inside CP2 like regions) change to those with Minkowskian signature of induced metric. The local reduction of space-time dimension would be very real for particles moving along sub-manifolds at which higher dimensional space-time has reduced metric dimenson: they cannot get out from lower-D sub-manifold. This is like ending down to 1-D black hole interior and one would obtain the analog of ordinary Feynman diagrammatics. This kind of Feynman diagrammatics involving only braid strands is what I have indeed ended up earlier so that it seems that I can trust good intuition combined with a sloppy mathematics sometimes works;-).

    These singular lines represent orbits of point like particles carrying fermion number at the orbits of wormhole throats. Furthermore, in this representation the expansions coming from string world sheets and partonic 2-surfaces are identical automatically. This follows from the fact that only the light-like lines connecting points common to singular string world sheets and singular partonic 2-surfaces appear as propagator lines!

  6. The TGD analog of AdS5 duality of N=4 SUSYs would be trivially true as an identity in this special case, and the good guess is that it is true also generally. One could indeed use integral over either string world sheets or partonic 2-sheets to deduce the amplitudes.
What is important to notice that singularities of Feynman diagrams crucial for the Grassmannian approach of Nima and others would correspond at space-time level 2-D singularities of the effective metric defined by the modified gamma matrices defined as contractions of canonical momentum currents for Kähler action with ordinary gamma matrices of the imbedding space and therefore directly reflecting classical dynamics.

For background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

Twistor revolution and TGD

Lubos wrote a nice summary about the talk of Nima Arkani Hamed about twistor revolution in Strings 2012 and gave also a link to the talk. It seems that Nima and collaborators are ending to a picture about scattering amplitudes which strongly resembles that provided bt generalized Feynman diagrammatics in TGD framework

TGD framework is much more general than N=4 SYM and is to it same as general relativity for special relativity whereas the latter is completely explicit. Of course, I cannot hope that TGD view could be taken seriously - at least publicly. One might hope that these approaches could be combined some day: both have a lot to give for each other. Below I compare these approaches.

The origin of twistor diagrammatics

In TGD framework zero energy ontology forces to replace the idea about continuous unitary evolution in Minkowski space with something more general assignable to causal diamonds (CDs), and S-matrix is replaced with a square root of density matrix equal to a hermitian l square root of density matrix multiplied by unitary S-matrix. Also in twistor approach unitarity has ceased to be a star actor. In p-Adic context continuous unitary time evolution fails to make sense also mathematically.

Twistor diagrammatics involves only massless on mass shell particles on both external and internal lines. Zero energy ontology (ZEO) requires same in TGD: wormhole lines carry parallely moving massless fermions and antifermions. The mass shell conditions at vertices are enormously powerful and imply UV finiteness. Also IR finiteness follows if external particles are massive.

What one means with mass is however a delicate matter. What does one mean with mass? I have pondered 35 years this question and the recent view is inspired by p-adic mass calculations and ZEO, and states that observed mass is in a well-defined sense expectation value of longitudinal mass squared for all possible choices of M2 ⊂ M4 characterizing the choices of quantization axis for energy and spin at the level of "world of classical worlds" (WCW) assignable with given causal diamond CD.

The choice of quantization axis thus becomes part of the geometry of WCW. All wormhole throats are massless but develop non-vanishing longitudinal mass squared. Gauge bosons correspond to wormhole contacts and thus consist of pairs of massless wormhole throats. Gauge bosons could develop 4-D mass squared but also remain massless in 4-D sense if the throats have parallel massless momenta. Longitudinal mass squared is however non-vanishing andp-adic thermodynamics predicts it.

The emergence of 2-D sub-dynamics at space-time level

Nima et al introduce ordering of the vertices in 4-D case. Ordering and related braiding are however essentially 2-D notions. Somehow 2-D theory must be a part of the 4-D theory also at space-time level, and I understood that understanding this is the challenge of the twistor approach at this moment.

The twistor amplitude can be represented as sum over the permutations of n external gluons and all diagrams corresponding to the same permutation are equivalent. Permutations are more like braidings since they carry information about how the permutation proceeded as a homotopy. Yang-Baxter equation emerge. The allowed braidings are minimal braidings in the sense that the repetitions of permutations of two adjacent vertices are not considered to be separate. Minimal braidings reduce to ordinary permutations. Nima also talks about affine braidings which I interpret as analogs of Kac-Moody algebras meaning that one uses projective representations which for Kac-Moody algebra mean non-trivial central extension. Perhaps the condition is that the square of a permutation permuting only two vertices which each other gives only a non-trivial phase factor. Lubos suggests an alternative interpretation for "affine" which would select only special permutations and cannot be therefore correct.

There are rules of identifying the permutation associated with a given diagram involving only basic 3-gluon vertex with white circle and its conjugate. Lubos explains this "Mickey Mouse in maze" rule in his posting in detail: to determine the image p(n) of vertex n in the permutation put a mouse in the maze defined by the diagram and let it run around obeying single rule: if the vertex is black turn right and if the vertex is white turn left. Eventually the mouse ends up to external vertex. The mouse cannot end up with loop: if it would do it, the rule would force it to run back to n after the full loop and one would have fixed point: p(n)=n. The reduction in the number of diagrams is enormous: the infinity of different diagrams reduces to n! diagrams!

  1. In TGD framework string world sheets and partonic 2-surfaces (or either or these if they are dual notions as conjectured) at space-time surface would define the sought for 2-D theory, and one obtains indeed perturbative expansion with fermionic propagator defined by the inverse of the modified Dirac operator and bosonic propagator defined by the correlation function for small deformations of the string world sheet. The vertices of twistor diagrams emerge as braid ends defining the intersections of string world sheets and partonic 2-surfaces.

    String model like description becomes part of TGD and the role of string world sheets in X4 is highly analogous to that of string world sheets connecting branes in AdS5× S5 of N=4 SYM. In TGD framework 10-D AdS5× S5 is replaced with 4-D space-time surface in M4× CP2. The meaning of the analog of AdS5 duality in TGD framework should be understood. In particular, could it be that the descriptions involving string world sheets on one hand and partonic 2-surfaces - or 3-D orbits of wormhole throats defining the generalized Feynman diagram- on the other hand are dual to each other. I have conjectured something like this earlier but it takes some time for this kind of issues to find their natural answer.

  2. As described in the article, string world sheets and partonic 2-surfaces emerge directly from the construction of the solutions of the modified Dirac equation by requiring conservation of em charge. This result has been conjectured already earlier but using other less direct arguments. 2-D "string world sheets" as sub-manifolds of the space-time surface make the ordering possible, and guarantee the finiteness of the perturbation theory involving n-point functions of a conformal QFT for fermions at wormhole throats and n-point functions for the deformations of the space-time surface. Conformal invariance should dictate these n-point functions to a high degree. In TGD framework the fundamental 3-vertex corresponds to joining of light-like orbits of three wormhole contacts along their 2-D ends (partonic 2-surfaces).

The emergence of Yangian symmetry

Yangian symmetry associated with the conformal transformations of M4 is a key symmetry of Grassmannian approach. Is it possible to derive it in TGD framework?

  1. TGD indeed leads to a concrete representation of Yangian algebra as generalization of color and electroweak gauge Kac-Moody algebra using general formula discussed in Witten's article about Yangian algebras (see the article).
  2. Article discusses also a conjecture about 2-D Hodge duality of quantized YM gauge potentials assignable to string world sheets with Kac-Moody currents. Quantum gauge potentials are defined only where they are needed - at string world sheets rather than entire 4-D space-time.
  3. Conformal scalings of the effective metric defined by the anticommutators of the modified gamma matrices emerges as realization of quantum criticality. They are induced by critical deformations (second variations not changing Kähler action) of the space-time surface. This algebra can be generalized to Yangian using the formulas in Witten's article (see the article).
  4. Critical deformations induce also electroweak gauge transformations and even more general symmetries for which infinitesimal generators are products of U(n) generators permuting n modes of the modified Dirac operator and infinitesimal generators of local electro-weak gauge transformations. These symmetries would relate in a natural manner to finite measurement resolution realized in terms of inclusions of hyperfinite factors with included algebra taking the role of gauge group transforming to each other states not distinguishable from each other.
  5. How to end up with Grassmannian picture in TGD framework? This has inspired some speculations in the past. From Nima's lecture one however learns that Grassmannian picture emerges as a convenient parametrization. One starts from the basic 3-gluon vertex or its conjugate expressed in terms of twistors. Momentum conservation implies that with the three twistors λi or their conjugates are proportional to each other (depending on which is the case one assigns white or black dot with the vertex). This constraint can be expressed as a delta function constraint by introducing additional integration variables and these integration variables lead to the emergence of the Grassmannian Gn,k where n is the number of gluons, and k the number of positive helicity gluons.

    Since only momentum conservation is involved, and since twistorial description works because only massless on mass shell virtual particles are involved, one is bound to end up with the Grassmannian description also in TGD.

Problems of the twistor approach from TGD point of view

Twistor approach has also its problems and here TGD suggests how to proceed. Signature problem is the first problem.

  1. Twistor diagrammatics works in a strict mathematical sense only for M2,2 with metric signature (1,1,-1,-1) rather than M4 with metric signature (1,-1,-1,-1). Metric signature is wrong in the physical case. This is a real problem which must be solved eventually.
  2. Effective metric defined by anticommutators of the modified gamma matrices (to be distinguished from the induced gamma matrices) could solve that problem since it would have the correct signature in TGD framework (see the article). String world sheets and partonic 2-surfaces would correspond to the 2-D singularities of this effective metric at which the even-even signature (1,1,1,1) changes to even-even signature (1,1,-1,-1). Space-time at string world sheet would become locally 2-D with respect to effective metric just as space-time becomes locally 3-D with respect to the induced metric at the light-like orbits of wormhole throats. String world sheets become also locally 1-D at light-like curves at which Euclidian signature of world sheet in induced metric transforms to Minkowskian.
  3. Twistor amplitudes are indeed singularities and string world sheets implied in TGD framework by conservation of em charge would represent these singularities at space-time level. At the end of the talk Nima conjectured about lower-dimensional manifolds of space-time as representation of space-time singularities. Note that string world sheets and partonic 2-surfaces have been part of TGD for years. TGD is of course to N=4 SYM what general relativity is for the special relativity. Space-time surface is dynamical and possesses induced and effective metrics rather than being flat.
Second limitation is that twistor diagrammatics works only for planar diagrams. This is a problem which must be also fixed sooner or later.
  1. This perhaps dangerous and blasphemous statement that I will regret it some day but I will make it;-). Nima and others have not yet discovered that M2 ⊂ M4 must be there but will discover it when they begin to generalize the results to non-planar diagrams and realize that Feynman diagrams are analogous to knot diagrams in 2-D plane (with crossings allowed) and that this 2-D plane must correspond to M2⊂ M4. The different choices of causal diamond CD correspond to different choices of M2 representing choice of quantization axes 4-momentum and spin. The integral over these choices guarantees Lorentz invariance. Gauge conditions are modified: longitudinal M2 projection of massless four-momentum is orthogonal to polarization so that three polarizations are possible: states are massive in longitudinal sense.
  2. In TGD framework one replaces the lines of Feynman diagrams with the light-like 3-surfaces defining orbits of wormhole throats. These lines carry many fermion states defining braid strands at light-like 3-surfaces. There is internal braiding associated with these braid strands. String world sheets connect fermions at different wormhole throats with space-like braid strands. The M2 projections of generalized Feynman diagrams with 4-D "lines" replaced with genuine lines define the ordinary Feynman diagram as the analog of braid diagram. The conjecture is that one can reduce non-planar diagrams to planar diagrams using a procedure analogous to the construction of knot invariants by un-knotting the knot in Alexandrian manner by allowing it to be cut temporarily.
  3. The permutations of string vertices emerge naturally as one constructs diagrams by adding to the interior of polygon sub-polygons connected to the external vertices. This corresponds to the addition of internal partonic two-surfaces. There are very many equivalent diagrams of this kind. Only permutations matter and the permutation associated with a given diagram of this kind can be deduced by the Mickey-Mouse rule described explicitly by Lubos. A connection with planar operads is highly suggestive and also conjecture already earlier in TGD framework.

For background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

About the definition of Hamilton-Jacobi structure

I have talked in previous postings a lot about Hamilton-Jacobi structure without bothering to write detailed definitions. In the following I discuss the notion in more detail. Thanks for Hamed who asked for more detailed explanation.

Hermitian and hyper-Hermitian structures

The starting point is the observation that besides the complex numbers forming a number field there are hyper-complex numbers. Imaginary unit i is replaced with e satisfying e2=1. One obtains an algebra but not a number field since the norm is Minkowskian norm x2-y2, which vanishes at light-cone x=y so that light-like hypercomplex numbers x+/- e) do not have inverse. One has "almost" number field.

Hyper-complex numbers appear naturally in 2-D Minkowski space since the solutions of a massless field equation can be written as f=g(u=t-ex)+h(v=t+ex) whith e2=1 realized by putting e=1. Therefore Wick rotation relates sums of holomorphic and antiholomorphic functions to sums of hyper-holomorphic and anti-hyper-holomorphic functions. Note that u and v are hyper-complex conjugates of each other.

Complex n-dimensional spaces allow Hermitian structure. This means that the metric has in complex coordinates (z1,....,zn) the form in which the matrix elements of metric are nonvanishing only between zi and complex conjugate of zj. In 2-D case one obtains just ds2=gzz*dzdz*. Note that in this case metric is conformally flat since line element is proportional to the line element ds2=dzdz* of plane. This form is always possible locally. For complex n-D case one obtains ds2=gij*dzidzj*. gij*=(gji*)* guaranteing the reality of ds2. In 2-D case this condition gives gzz*= (gz*z)*.

How could one generalize this line element to hyper-complex n-dimensional case? In 2-D case Minkowski space M2 one has ds2= guvdudv, guv=1. The obvious generalization would be the replacement ds2=guivjduidvj. Also now the analogs of reality conditions must hold with respect to ui↔ vi.

Hamilton-Jacobi structure

Consider next the path leading to Hamilton-Jacobi structure.

4-D Minkowski space M4=M2× E2 is Cartesian product of hyper-complex M2 with complex plane E2, and one has ds2= dudv+ dzdz* in standard Minkowski coordinates. One can also consider more general integrable decompositions of M4 for which the tangent space TM4=M4 at each point is decomposed to M2(x)× E2(x). The physical analogy would be a position dependent decomposition of the degrees of freedom of massless particle to longitudinal ones (M2(x): light-like momentum is in this plane) and transversal ones (E2(x): polarization vector is in this plane). Cylindrical and spherical variants of Minkowski coordinates define two examples of this kind of coordinates (it is perhaps a good exercize to think what kind of decomposition of tangent space is in question in these examples). An interesting mathematical problem highly relevant for TGD is to identify all possible decompositions of this kind for empty Minkowski space.

The integrability of the decomposition means that the planes M2(x) are tangent planes for 2-D surfaces of M4 analogous to Euclidian string world sheet. This gives slicing of M4 to Minkowskian string world sheets parametrized by euclidian string world sheets. The question is whether the sheets are stringy in a strong sense: that is minimal surfaces. This is not the case: for spherical coordinates the Euclidian string world sheets would be spheres which are not minimal surfaces. For cylindrical and spherical coordinates hower M2(x) integrate to plane M2 which is minimal surface.

Integrability means in the case of M2(x) the existence of light-like vector field J whose flow lines define a global coordinate. Its existence implies also the existence of its conjugate and together these vector fields give rise to M2(x) at each point. This means that one has J= Ψ∇ Φ: Φ indeed defines the global coordinate along flow lines. In the case of M2 either the coordinate u or v would be the coordinate in question. This kind of flows are called Beltrami flows. Obviously the same holds for the transversal planes E2.

One can generalize this metric to the case of general 4-D space with Minkowski signature of metric. At least the elements guv and gzz* are non-vanishing and can depend on both u,v and z,z*. They must satisfy the reality conditions gzz*= (gzz*)* and guv= (gvu)* where complex conjugation in the argument involves also u↔ v besides z↔ z*.

The question is whether the components guz, gvz, and their complex conjugates are non-vanishing if they satisfy some conditions. They can. The direct generalization from complex 2-D space would be that one treats u and v as complex conjugates and therefore requires a direct generalization of the hermiticity condition

guz= (gvz*)*, gvz= (guz*)* .

This would give complete symmetry with the complex 2-D (4-D in real sense) spaces. This would allow the algebraic continuation of hermitian structures to Hamilton-Jacobi structures by just replacing i with e for some complex coordinates.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title.

The importance of being light-like

The singular geometric objects associated with the space-time surface have become increasingly important in TGD framework. In particular, the recent progress has made clear that these objects might be crucial for the understanding of quantum TGD. The singular objects are associated not only with the induced metric but also with the effective metric defined by the anti-commutators of the modified gamma matrices appearing in the modified Dirac equation and determined by the Kähler action.

The singular objects associated with the induced metric

Consider first the singular objects associated with the induced metric.

  1. At light-like 3-surfaces defined by wormhole throats the signature of the induced metric changes from Euclidian to Minkowskian so that 4-metric is degenerate. These surfaces are carriers of elementary particle quantum numbers and the 4-D induced metric degenerates locally to 3-D one at these surfaces.
  2. Braid strands at light-like 3-surfaces are most naturally light-like curves: this correspond to the boundary condition for open strings. One can assign fermion number to the braid strands. Braid strands allow an identification as curves along which the Euclidian signature of the string world sheet in Euclidian region transforms to Minkowskian one. Number theoretic interpretation would be as a transformation of complex regions to hyper-complex regions meaning that imaginary unit i satisfying i2=-1 becomes hyper-complex unit e satisfying e2=1. The complex coordinates (z,z*) become hyper-complex coordinates (u=t+ex, v=t-ex) giving the standard light-like coordinates when one puts e=1.

The singular objects associated with the effective metric

There are also singular objects assignable to the effective metric. According to the simple arguments already developed, string world sheets and possibly also partonic 2-surfaces are singular objects with respect to the effective metric defined by the anti-commutators of the modified gamma matrices rather than induced gamma matrices. Therefore the effective metric seems to be much more than a mere formal structure.

  1. For instance, quaternionicity of the space-time surface could allow an elegant formulation in terms of the effective metric avoiding the problems due to the Minkowski signature. This is achieved if the effective metric has Euclidian signature ε × (1,1,1,1), ε=+/- 1 or a complex counterpart of the Minkowskian signature ε (1,1,-1,-1).
  2. String world sheets and perhaps also partonic 2-surfaces could be understood as singularities of the effective metric. What happens that the effective metric with Euclidian signature ε ×(1,1,1,1) transforms to the signature ε (1,1,-1,-1) (say) at string world sheet so that one would have the degenerate signature ε×(1,1,0,0) at the string world sheet.

    What is amazing is that this works also number theoretically. It came as a total surprise to me that the notion of hyper-quaternions as a closed algebraic structure indeed exists. The hyper-quaternionic units would be given by (1,I, iJ,iK), where i is a commuting imaginary unit satisfying i2=-1. Hyper-quaternionic numbers defined as combinations of these units with real coefficients do form a closed algebraic structure which however fails to be a number field just like hyper-complex numbers do. Note that the hyper-quaternions obtained with real coefficients from the basis (1,iI,iJ,iK) fail to form an algebra since the product is not hyper-quaternion in this sense but belongs to the algebra of complexified quaternions. The same problem is encountered in the case of hyper-octonions defined in this manner. This has been a stone in my shoe since I feel strong disrelish towards Wick rotation as a trick for moving between different signatures.

  3. Could also partonic 2-surfaces correspond to this kind of singular 2-surfaces? In principle, 2-D surfaces of 4-D space intersect at discrete points just as string world sheets and partonic 2-surfaces do so that this might make sense. By complex structure the situation is algebraically equivalent to the analog of plane with non-flat metric allowing all possible signatures (ε12) in various regions. At light-like curve either ε1 or ε2 changes sign and light-like curves for these two kinds of changes can intersect as one can easily verify by drawing what happens. At the intersection point the metric is completely degenerate and simply vanishes.
  4. Replacing real 2-dimensionality with complex 2-dimensionality, one obtains by the universality of algebraic dimension the same result for partonic 2-surfaces and string world sheets. The braid ends at partonic 2-surfaces representing the intersection points of 2-surfaces of this kind would have completely degenerate effective metric so that the modified gamma matrices would vanish implying that energy momentum tensor vanishes as also the induced Kähler field.
  5. The effective metric suffers a local conformal scaling in the critical deformations identified in the proposed manner. Since ordinary conformal group acts on Minkowski space and leaves the boundary of light-cone invariant, one has two conformal groups. It is not however clear whether the $M^4$ conformal transformations can act as symmetries in TGD, where the presence of the induced metric in K\"ahler action breaks $M^4$ conformal symmetry. As found, also in TGD framework the Kac-Moody currents assigned to the braid strands generate Yangian: this is expected to be true also for the Kac-Moody counterparts of the conformal algebra associated with quantum criticality. On the other hand, in twistor program one encounters also two conformal groups and the space in which the second conformal group acts remains somewhat mysterious object. The Lie algebras for the two conformal groups generate the conformal Yangian and the integrands of the scattering amplitudes are Yangian invariants. Twistor approach should apply in TGD if zero energy ontology is right. Does this mean a deep connection?

    What is also intriguing that twistor approach in principle works in strict mathematical sense only at signatures ε × (1,1,-1-1) and the scattering amplitudes in Minkowski signature are obtained by analytic continuation. Could the effective metric give rise to the desired signature? Note that the notion of massless particle does not make sense in the signature ε × (1,1,1,1).

These arguments provide genuine a support for the notion of quaternionicity and suggest a connection with the twistor approach.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title.

Quantum criticality and electro-weak gauge symmetries

Quantum criticality is one of the basic guiding principles of Quantum TGD. What it means mathematically is however far from clear.

  1. What is obvious is that quantum criticality implies quantization of Kähler coupling strength as a mathematical analog of critical temperature so that the theory becomes mathematically unique if only single critical temperature is possible. Physically this means the presence of long range fluctuations characteristic for criticality and perhaps assignable to the effective hierarchy of Planck constants having explanation in terms of effective covering spaces of the imbedding space. This hierarchy follows from the vacuum degeneracy of Kähler action, which in turn implies 4-D spin-glass degeneracy. It is easy to interpret the degeneracy in terms of criticality.
  2. At more technical level one would expect criticality to corresponds deformations of a given preferred extremal defining a vanishing second variation of Kähler action. This is analogous to the vanishing of also second derivatives of potential function at extremum in certain directions so that the matrix defined by second derivatives does not have maximum rank. Entire hierarchy of criticalities is expected and a good finite-dimensional model is provided by the catastrophe theory of Thom. Cusp catastrophe is the simplest catastrophe one can think of, and here the folds of cusp where discontinuous jump occurs correspond to criticality with respect to one control variable and the tip to criticality with respect to both control variables.
  3. I have discussed what criticality could mean for modified Dirac action (see this) and claimed that it leads to the existence of additional conserved currents defined by the variations which do not affect the value of Kähler action. These arguments are far from being mathematically rigorous and the recent view about the solutions of the modified Dirac equation predicting that the spinor modes are restricted to 2-D string world sheets requires a modification of these arguments.
In the following these arguments are updated. The unexpected result is that critical deformations induce conformal scalings of the modified metric and electro-weak gauge transformations of the induced spinor connection at X2. Therefore holomorphy brings in the Kac-Moody symmetries associated with isometries of H (gravitation and color gauge group) and quantum criticality those associated with the holonomies of H (electro-weak-gauge group) as additional symmetries.

The variation of modes of the induced spinor field in a variation of space-time surface respecting the preferred extremal property

Consider first the variation of the induced spinor field in a variation of space-time surface respecting the preferred extremal property. The deformation must be such that the deformed modified Dirac operator D annihilates the modified mode. By writing explicitly the variation of the modified Dirac action (the action vanishes by modified Dirac equation) one obtains deformations and requiring its vanishing one obtains

δ Ψ=D-1(δ D)Ψ .

D-1 is the inverse of the modified Dirac operator defining the analog of Dirac propagator and δ D defines vertex completely analogous to γkδ Ak in gauge theory context. The functional integral over preferred extremals can be carried out perturbatively by expressing Δ D in terms of δ hk and one obtains stringy perturbation theory around X2 associated with the preferred extremal defining maximum of Kähler function in Euclidian region and extremum of Kähler action in Minkowskian region (stationary phase approximation).

What one obtains is stringy perturbation theory for calculating n-points functions for fermions at the ends of braid strands located at partonic 2-surfaces and representing intersections of string world sheets and partonic 2-surfaces at the light-like boundaries of CDs. δ D- or more precisely, its partial derivatives with respect to functional integration variables - appear atthe vertices located anywhere in the interior of X2 with outcoming fermions at braid ends. Bosonic propagators are replaced with correlation functions for δ hk. Fermionic propagator is defined by D-1.

After 35 years or hard work this provides for the first time a reasonably explicit formula for the N-point functions of fermions. This is enough since by bosonic emergence(se this) these N-point functions define the basic building blocks of the scattering amplitudes. Note that bosonic emergence states that bosons corresponds to wormhole contacts with fermion and antifermion at the opposite wormhole throats.

What critical modes could mean for the induced spinor fields?

What critical modes could mean for the induced spinor fields at string world sheets and partonic 2-surfaces. The problematic part seems to be the variation of the modified Dirac operator since it involves gradient. One cannot require that covariant derivative remains invariant since this would require that the components of the induced spinor connection remain invariant and this is quite too restrictive condition. Right handed neutrino solutions delocalized into entire X2 are however an exception since they have no electro-weak gauge couplings and in this case the condition is obvious: modified gamma matrices suffer a local scaling for critical deformations:

δ Γμ = Λ(x)Γμ .

This guarantees that the modified Dirac operator D is mapped to Λ D and still annihilates the modes of νR labelled by conformal weight, which thus remain unchanged.

What is the situation for the 2-D modes located at string world sheets? The condition is obvious. Ψ suffers an electro-weak gauge transformation as does also the induced spinor connection so that Dμ is not affected at all. Criticality condition states that the deformation of the space-time surfaces induces a conformal scaling of Γμ at X2, It might be possible to continue this conformal scaling of the entire space-time sheet but this might be not necessary and this would mean that all critical deformations induced conformal transformations of the effective metric of the space-time surface defined by {Γμ, Γν}=2 Gμν. Thus it seems that effective metric is indeed central concept (recall that if the conjectured quaternionic structure is associated with the effective metric, it might be possible to avoid problem related to the Minkowskian signature in an elegant manner).

Note that one can consider even more general action of critical deformation: the modes of the induced spinor field would be mixed together in the infinitesimal deformation besides infinitesimal electroweak gauge transformation, which is same for all modes. This would extend electroweak gauge symmetry. Modified Dirac equation holds true also for these deformations. One might wonder whether the conjecture dynamically generated gauge symmetries assignable to finite measurement resolution could be generated in this manner.

Thus the critical deformations would induce conformal scalings of the effective metric and dynamical electro-weak gauge transformations. Electro-weak gauge symmetry would be a dynamical symmetry restricted to string world sheets and partonic 2-surfaces rather than acting at the entire space-time surface. For 4-D delocalized right-handed neutrino modes the conformal scalings of the effective metric are analogous to the conformal transformations of M4 for N=4 SYMs. Also ordinary conformal symmetries of M4 could be present for string world sheets and could act as symmetries of generalized Feynman graphs since even virtual wormhole throats are massless. An interesting question is whether the conformal invariance associated with the effective metric is the analog of dual conformal invariance in N=4 theories.

Critical deformations of space-time surface are accompanied by conserved fermionic currents. By using standard Noetherian formulas one can write

Jμi= Ψbar Γμδi Ψ + δi ΨbarΓμΨ .

Here δ Ψi denotes derivative of the variation with respect to a group parameter labeled by i. Since δ Ψi reduces to an infinitesimal gauge transformation of Ψ induced by deformation, these currents are the analogs of gauge currents. The integrals of these currents along the braid strands at the ends of string world sheets define the analogs of gauge charges. The interpretation as Kac-Moody charges is also very attractive and I have proposed that the 2-D Hodge duals of gauge potentials could be identified as Kac-Moody currents. If so, the 2-D Hodge duals of J would define the quantum analogs of dynamical electro-weak gauge fields and Kac-Moody charge could be also seen as non-integral phase factor associated with the braid strand in Abelian approximation (the interpretation in terms of finite measurement resolution is discussed earlier).

One can also define super currents by replacing Ψbar or Ψ by a particular mode of the induced spinor field as well as c-number valued currents by performing the replacement for both Ψbar and Ψ. As expected, one obtains a super-conformal algebra with all modes of induced spinor fields acting as generators of super-symmetries restricted to 2-D surfaces. The number of the charges which do not annihilate physical states as also the effective number of fermionic modes could be finite and this would suggest that the integer N for the supersymmetry in question is finite. This would conform with the earlier proposal inspired by the notion of finite measurement resolution implying the replacement of the partonic 2-surfaces with collections of braid ends.

Note that Kac-Moody charges might be associated with "long" braid strands connecting different wormhole throats as well as short braid strands connecting opposite throats of wormhole contacts. Both kinds of charges would appear in the theory.

What is the interpretation of the critical deformations?

Critical deformations bring in an additional gauge symmetry. Certainly not all possible gauge transformations are induced by the deformations of preferred extremals and a good guess is that they correspond to holomorphic gauge group elements as in theories with Kac-Moody symmetry. What is the physical character of this dynamical gauge symmetry?

  1. Do the gauge charges vanish? Do they annihilate the physical states? Do only their positive energy parts annihilate the states so that one has a situation characteristic for the representation of Kac-Moody algebras. Or could some of these charges be analogous to the gauge charges associated with the constant gauge transformations in gauge theories and be therefore non-vanishing in the absence of confinement. Now one has electro-weak gauge charges and these should be non-vanishing. Can one assign them to deformations with a vanishing conformal weight and the remaining deformations to those with non-vanishing conformal weight and acting like Kac-Moody generators on the physical states?
  2. The simplest option is that the critical Kac-Moody charges/gauge charges with non-vanishing positive conformal weight annihilate the physical states. Critical degrees of freedom would not disappear but make their presence known via the states labelled by different gauge charges assignable to critical deformations with vanishing conformal weight. Note that constant gauge transformations can be said to break the gauge symmetry also in the ordinary gauge theories unless one has confinement.
  3. The hierarchy of quantum criticalities suggests however entire hierarchy of electro-weak Kac-Moody algebras. Does this mean a hierarchy of electro-weak symmetries breakings in which the number of Kac-Moody generators not annihilating the physical states gradually increases as also modes with a higher value of positive conformal weight fail to annihilate the physical state?

    The only manner to have a hierarchy of algebras is by assuming that only the generators satisfying n mod N=0 define the sub-Kac-Moody algebra annihilating the physical states so that the generators with n mod N≠ 0 would define the analogs of gauge charges. I have suggested for long time ago the relevance of kind of fractal hierarchy of Kac-Moody and Super-Virasoro algebras for TGD but failed to imagine any concrete realization.

    A stronger condition would be that the algebra reduces to a finite dimensional algebra in the sense that the actions of generators Qn and Qn+kN are identical. This would correspond to periodic boundary conditions in the space of conformal weights. The notion of finite measurement resolution suggests that the number of independent fermionic oscillator operators is proportional to the number of braid ends so that an effective reduction to a finite algebra is expected.

    Whatever the correct interpretation is, this would obviously refine the usual view about electro-weak symmetry breaking.

These arguments suggests the following overall view. The holomorphy of spinor modes gives rise to Kac-Moody algebra defined by isometries and includes besides Minkowskian generators associated with gravitation also SU(3) generators associated with color symmetries. Vanishing second variations in turn define electro-weak Kac-Moody type algebra.

Note that criticality suggests that one must perform functional integral over WCW by decomposing it to an integral over zero modes for which deformations of X4 induce only an electro-weak gauge transformation of the induced spinor field and to an integral over moduli corresponding to the remaining degrees of freedom.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title.

Constraints on super-conformal invariance from p-adic mass calculations and ZEO

The generalization of super-conformal symmetry to 4-D context is basic element of quantum TGD. Several variants for the realization of supersymmetry has been proposed. Especially problematic has been the question about whether the counterpart of standard SUSY is realized in TGD framework or not. Thanks to the progress made in the understanding of preferred extremals of Kähler action and of solutions of modified Dirac equation, it has become to develop the vision in considerable detail. As a consequence some existing alternative visions have been eliminated from consideration. One of them the proposal that Equivalence Principle might have realization in terms of coset representations. Second one is the idea that right-handed neutrino generating space-time supersymmetry might be in color partial wave so that sparticles would be colored: this would explain why sparticles are not observed at LHC. The new view providing a possible alternative explanation for the absence of sparticles has been discussed in previous posting.

Concerning the general understanding of super-conformal invariance in TGD framework, an important physical constraint comes from the success p-adic thermodynamics: superconformal invariance indeed forms a core element of p-adic mass calculations (see this, especially this).

  1. The first thing that one can get worried about relates to the extension of conformal symmetries. If the conformal symmetries generalize to D=4, how can one take seriously the results of p-adic mass calculations based on 2-D conformal invariance? There is no reason to worry. The reduction of the conformal invariance to 2-D one for the preferred extremals takes care of this problem. This however requires that the fermionic contributions assignable to string world sheets and/or partonic 2-surfaces - Super- Kac-Moody contributions - should dictate the elementary particle masses. For hadrons also symplectic contributions should be present (see this). This is a valuable hint in attempts to identify the mathematical structure in more detail.
  2. Zero Energy Ontology (ZEO) suggests that all particles, even virtual ones correspond to massless wormhole throats carrying fermions. As a consequence, twistor approach would work and the kinematical constraints to vertices would allow th cancellation of both UV and IR divergences. This would suggest that the p-adic thermal expectation value is for the longitudinal M2 momentum squared (the definition of CD selects M1⊂ M2⊂ M4 as also does number theoretic vision). Also propagator would be determined by M2 momentum. Lorentz invariance would be obtained by integration of the moduli for CD including also Lorentz boosts of CD. This is definitely something new from standard physics point of view, but suggested already by the first p-adic mass calculations. The fact that parton distributions in hadrons are functions of longitudinal momentum fraction, and the division of tangent space of M4 to longitudinal and transversal parts for gauge bosons suggests the same. Also number theoretical vision and the need to assign to realize the choice of spin quantization axis and time like direction defining the rest system in quantum measurement theory at the level of WCW geometry also this.
  3. In the original approach one allows states with arbitrary large values of L0 as physical states. Usually one would require that L0 annihilates the states. In the calculations however mass squared was assumed to be proportional L0 apart from vacuum contribution. This is a questionable assumption. ZEO suggests that total mass squared vanishes and that one can decompose mass squared to a sum of longitudinal and transversal parts. If one can do the same decomposition to longitudinal and transverse parts also for the Super Virasoro algebra then one can calculate longitudinal mass squared as a p-adic thermal expectation in the transversal super-Virasoro algebra and only states with L0=0 would contribute and one would have conformal invariance in the standard sense.
  4. The assumption motivated by Lorentz invariance has been that mass squared is replaced with conformal weight in thermodynamics, and that one first calculates the thermal average of the conformal weight and then equates it with mass squared. This assumption is somewhat ad hoc. ZEO however suggests an alternative interpretation in which one has zero energy states for which longitudinal mass squared of positive energy state derive from p-adic thermodynamics. Thermodynamics - or rather, its square root - would become part of quantum theory in ZEO. M-matrix is indeed product of hermitian square root of density matrix multiplied by unitary S-matrix and defines the entanglement coefficients between positive and negative energy parts of zero energy state.
  5. The crucial constraint is that the number of super-conformal tensor factors is N=5: this suggests that thermodynamics applied in Super-Kac-Moody degrees of freedom assignable to string world sheets is enough, when one is interested in the masses of fermions and gauge bosons. Super-symplectic degrees of freedom can also contribute and determine the dominant contribution to baryon masses. Should also this contribution obey p-adic thermodynamics in the case when it is present? Or does the very fact that this contribution need not be present mean that it is not thermal? The symplectic contribution should correspond to hadronic p-adic length prime rather the one assignable to (say) u quark. Hadronic p-adic mass squared and partonic p-adic mass squared cannot be summed since primes are different. If one accepts the basic rules (see this), longitudinal energy and momentum are additive as indeed assumed in perturbative QCD.
  6. Calculations work if the vacuum expectation value of the mass squared must be assumed to be tachyonic. There are two options depending on whether one whether p-adic thermodynamics gives total mass squared or longitudinal mass squared.
    1. One could argue that the total mass squared has naturally tachyonic ground state expectation since for massless extremals longitudinal momentum is light-like and transversal momentum squared is necessary present and non-vanishing by the localization to topological light ray of finite thickness of order p-adic length scale. Transversal degrees of freedom would be modeled with a particle in a box.
    2. If longitudinal mass squared is what is calculated, the condition would require that transversal momentum squared is negative so that instead of plane wave like behavior exponential damping would be required. This would conform with the localization in transversal degrees of freedom.

  7. For preferred extremals Einstein's equations with cosmological term are satisfied as a consistency condition guaranteing algebraization of the equations. Hence Equivalence Principle holds true. But what about possible quantum realization of Equivalence Principle in this framework? A possible quantum counterpart of Equivalence Principle could be that the longitudinal parts of the imbedding space mass squared operator for a given massless state equals to that for d'Alembert operator assignable to the modified Dirac action. The attempts to formulate this in more precise manner however seem to produce only troubles.
To sum up, the basic new element inspired by ZEO is that p-adic mass calculations are for the longitudinal momentum squared and that elementary particles and even hadrons are basically massless. This assumption looks certainly strange at first. Thank so the integration over boosts of CDs it is however consistent with Lorentz invariance and also allows to understand p-adic thermodynamics as thermodynamics for the transversal part of scaling generator L0. ZEO allows indeed superposition of pairs of positive and negative energies with different momenta for positive energy state without a loss of Lorentz invariance. Also p-adic thermodynamics finds a natural interpretation in terms of M-matrix defining square root of hermitian density matrix.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equationarticle with the same title.

The role of the right-handed neutrino in TGD based view about SUSY

The general ansatz for the preferred extremals of Kähler action and application of the conservation of em charge to the modified Dirac equation have led to a rather detailed view about classical and TGD and allowed to build a bridge between general vision about super-conformal symmetries in TGD Universe and field equations.

  1. Equivalence Principle realized as Einstein's equations in all scales follows directly from the general assatz for preferred extremals implying that space-time surface has either hermitian or Hamilton-Jacobi structure (which of them depends on the signature of the induced metric).
  2. The general structure of Super Virasoro representations can be understood: super-symplectic algebra is responsible for the non-perturbative aspects of QCD and determines also the ground states of elementary particles determining their quantum numbers.
  3. Super-Kac-Moody algebras associated with isometries and holonomies dictate standard model quantum numbers and lead to a massivation by p-adic thermodynamics: the crucial condition that the number of tensor factors in Super-Virasoro represention is 5 is satisfied.
  4. One can understand how the Super-Kac-Moody currents assignable to stringy world sheets emerging naturally from the conservation of em charge defined as their string world sheet Hodge duals gauge potentials for standard model gauge group and also their analogs for gravitons. Also the conjecture Yangian algebra generated by Super-Kac-Moody charges emerges naturally.
  5. One also finds that right handed neutrino is in a very special role because of its lacking couplings in electroweak sector and its role as a generator of the least broken SUSY. All other modes of induced spinor field are restricted to 2-D string world sheets and partonic 2-surfaces. Right-handed neutrino allows also the mode delocalized to entire space-time surface or perhaps only to the Euclidian regions defined by the 4-D line of the generalized Feynman diagrams.

    In fact, in the following the possibility that the resulting sparticles cannot be distinguished from particles since the presence of right handed neutrino is not seen in the interactions and does not manifest itself in different spin structures for the couplings of particle and sparticle. This could explain the failure to detect spartners at LHC. Intermediate gauge boson decay widths however require that sparticles are dark in the sense of having non-standard value of Planck constant. Another variant of the argument assumes that 4-D right handed neutrinos are associated with space-time regions of Minkowskian signature and SUSY is defined for many-particle states rather than single particle states. It should be emphasized that TGD predicts that all fermions act as generators of badly broken supersymmetries at partonic 2-surfaces but these super-symmetries could correspond to much higher mass scale as that associated with the delocalized right-handed neutrino. The following piece of text summarizes the argument.

A highly interesting aspect of Super-Kac-Moody symmetry is the special role of right handed neutrino.

  1. Only right handed neutrino allows besides the modes restricted to 2-D surfaces also the 4D modes delocalized to the entire space-time surface. The first ones are holomorphic functions of single coordinate and the latter ones holomorphic functions of two complex/Hamilton-Jacobi coordinates. OnlyνR has the full D=4 counterpart of the conformal symmetry and the localization to 2-surfaces has interpretation as super-conformal symmetry breaking halving the number of super-conformal generators.
  2. This forces to ask for the meaning of super-partners. Are super-partners obtained by adding νR neutrino localized at partonic 2-surface or delocalized to entire space-time surface or its Euclidian or Minkowskian region accompanying particle identified as wormhole throat? Only the Euclidian option allows to assign right handed neutrino to a unique partonic 2-surface. For the Minkowskian regions the assignment is to many particle state defined by the partonic 2-surfaces associated with the 3-surface. Hence for spartners the 4-D right-handed neutrino must be associated with the 4-D Euclidian line of the generalized Feynman diagram.
  3. The orthogonality of the localized and de-localized right handed neutrino modes requires that 2-D modes correspond to higher color partial waves at the level of imbedding space. If color octet is in question, the 2-D right handed neutrino as the candidate for the generator of standard SUSY would combine with the left handed neutrino to form a massive neutrino. If 2-D massive neutrino acts as a generator of super-symmetries, it is is in the same role as badly broken supers-ymmeries generated by other 2-D modes of the induced spinor field (SUSY with rather large value of N) and one can argue that the counterpart of standard SUSY cannot correspond to this kind of super-symmetries. The right-handed neutrinos delocalized inside the lines of generalized Feynman diagrams, could generate N=2 variant of the standard SUSY.

1. How particle and right handed neutrino are bound together?

Ordinary SUSY means that apart from kinematical spin factors sparticles and particles behave identically with respect to standard model interactions. These spin factors would allow to distinguish between particles and sparticles. But is this the case now?

  1. One can argue that 2-D particle and 4-D right-handed neutrino behave like independent entities, and because νR has no standard model couplings this entire structure behaves like a particle rather than sparticle with respect to standard model interactions: the kinematical spin dependent factors would be absent.
  2. The question is also about the internal structure of the sparticle. How the four-momentum is divided between the νR and and 2-D fermion. If νR carries a negligible portion of four-momentum, the four-momentum carried by the particle part of sparticle is same as that carried by particle for given four-momentum so that the distinctions are only kinematical for the ordinary view about sparticle and trivial for the view suggested by the 4-D character of νR.
Could sparticle character become manifest in the ordinary scattering of sparticle?
  1. If νR behaves as an independent unit not bound to the particle it would continue un-scattered as particle scatters: sparticle would decay to particle and right-handed neutrino. If νR carries a non-negligible energy the scattering could be detected via a missing energy. If not, then the decay could be detected by the interactions revealing the presence of νR. νR can have only gravitational interactions. What these gravitational interactions are is not however quite clear since the proposed identification of gravitational gauge potentials is as duals of Kac-Moody currents analogous to gauge potentials located at the boundaries of string world sheets. Does this mean that 4-D right-handed neutrino has no quantal gravitational interactions? Does internal consistency require νR to have a vanishing gravitational and inertial masses and does this mean that this particle carries only spin?
  2. The cautious conclusion would be following: if delocalized νR and parton are un-correlated particle and sparticle cannot be distinguished experimentally and one might perhaps understand the failure to detect standard SUSY at LHC. Note however that the 2-D fermionic oscillator algebra defines badly broken large N SUSY containing also massive (longitudinal momentum square is non-vanishing) neutrino modes as generators.

2. Taking a closer look on sparticles

It is good to take a closer look at the delocalized right handed neutrino modes.

  1. At imbedding space level that is in cm mass degrees of freedom they correspond to covariantly constant CP2 spinors carrying light-like momentum which for causal diamond could be discretized. For non-vanishing momentum one can speak about helicity having opposite sign for νR and νRbar. For vanishing four-momentum the situation is delicate since only spin remains and Majorana like behavior is suggestive. Unless one has momentum continuum, this mode might be important and generate additional SUSY resembling standard N=1 SUSY.
  2. At space-time level the solutions of modified Dirac equation are holomorphic or anti-holomorphic.
    1. For non-constant holomorphic modes these characteristics correlate naturally with fermion number and helicity of νR . One can assign creation/annihilation operator to these two kinds of modes and the sign of fermion number correlates with the sign of helicity.
    2. The covariantly constant mode is naturally assignable to the covariantly constant neutrino spinor of imbedding space. To the two helicities one can assign also oscillator operators {a+/-,a+/-}. The effective Majorana property is expressed in terms of non-orthogonality of νR and and νRbar translated to the the non-vanishing of the anti-commutator {a+,a-}= {a-,a+}=1. The reduction of the rank of the 4× 4 matrix defined by anti-commutators to two expresses the fact that the number of degrees of freedom has halved. a+=a- realizes the conditions and implies that one has only N=1 SUSY multiplet since the state containing both νR and νRbar is same as that containing no right handed neutrinos.
    3. One can wonder whether this SUSY is masked totally by the fact that sparticles with all possible conformal weights n for induced spinor field are possible and the branching ratio to n=0 channel is small. If momentum continuum is present, the zero momentum mode might be equivalent to nothing.

What can happen in spin degrees of freedom in super-symmetric interaction vertices if one accepts this interpretation? As already noticed, this depends solely on what one assumes about the correlation of the four-momenta of particle and νR.

  1. For SUSY generated by covariantly constant νR and νRbar there is no neutrino four-momentum involved so that only spin matters. One cannot speak about the change of direction for νR. In the scattering of sparticle the direction of particle changes and introduces different spin quantization axes. νR retains its spin and in new system it is superposition of two spin projections. The presence of both helicities requires that the transformation νR→ νRbar happens with an amplitude determined purely kinematically by spin rotation matrices. This is consistent with fermion number conservation modulo 2. N=1 SUSY based on Majorana spinors is highly suggestive.
  2. For SUSY generated by non-constant holomorphic and anti-holomorphic modes carrying fermion number the behavior in the scattering is different. Suppose that the sparticle does not split to particle moving in the new direction and νR moving in the original direction so that also νR or νRbar carrying some massless fraction of four-momentum changes its direction of motion. One can form the spin projections with respect to the new spin axis but must drop the projection which does not conserve fermion number. Therefore the kinematics at the vertices is different. Hence N=2 SUSY with fermion number conservation is suggestive when the momentum directions of particle and νR are completely correlated.
  3. Since right-handed neutrino has no standard model couplings, p-adic thermodynamics for 4-D right-handed neutrino must correspond to a very low p-adic temperature T=1/n. This implies that the excitations with higher conformal weight are effectively absent and one would have N =1 SUSY effectively.

    The simplest assumption is that particle and sparticle correspond to the same p-adic mass scale and have degenerate masses: it is difficult to imagine any good reason for why the p-adic mass scales should differ. This should have been observed -say in decay widths of weak bosons - unless the spartners correspond to large hbar phase and therefore to dark matter . Note that for the badly broken 2-D N=2 SUSY in fermionic sector this kind of almost degeneracy cannot be excluded and I have considered an explanation for the mysterious X and Y mesons in terms of this degeneracy (see this).

  4. LHC suggests that one does not have N=1 SUSY in standard sense. Could spartners correspond to dark matter with a large value of Planck constant and same mass? Or could 4-D right-handed neutrino exists only in the Minkowskian regions where they define superpartners of many particle states rather than single particle states? Could the reason be that for CP2 type vacuum extremals modified gamma matrices vanish identically? Could this be used to argue that 4-D right-handed neutrinos cannot appear in the lines of generalize Feynman graphs which involve deformations of CP2 vacuum extremals?

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

The emergence of Yangian symmetry and gauge potentials as duals of Kac-Moody currents

Yangian symmetry plays a key role in N=4 super-symmetric gauge theories. What is special in Yangian symmetry is that the algebra contains also multi-local generators. In TGD framework multi-locality would naturally correspond to that with respect to partonic 2-surfaces and string world sheets and the proposal has been that the Super-Kac-Moody algebras assignable to string worlds sheets could generalize to Yangian.

Witten has written a beautiful exposition of Yangian algebras (see this). Yangian is generated by two kinds of generators JA and QA by a repeated formation of commutators. The number of commutations tells the integer characterizing the multi-locality and provides the Yangian algebra with grading by natural numbers. Witten describes a 2-dimensional QFT like situation in which one has 2-D situation and Kac-Moody currents assignable to real axis define the Kac-Moody charges as integrals in the usual manner. It is also assumed that the gauge potentials defined by the 1-form associated with the Kac-Moody current define a flat connection:

μjAν- ∂νjAν +[jAμ,jAν]=0 .

This condition guarantees that the generators of Yangian are conserved charges. One can however consider alternative manners to obtain the conservation.

  1. The generators of first kind - call them JA - are just the conserved Kac-Moody charges. The formula is given by

    JA= ∫-∞ dxjA0(x,t) .

  2. The generators of second kind contain bi-local part. They are convolutions of generators of first kind associated with different points of string described as real axis. In the basic formula one has integration over the point of real axis.

    QA= fABC-∞ dx ∫xdy jB0(x,t)jC0(y,t)- 2∫-∞ jAxdx .

    These charges are indeed conserved if the curvature form is vanishing as a little calculation shows.

How to generalize this to the recent context?

  1. The Kac-Moody charges would be associated with the braid strands connecting two partonic 2-surfaces - Strands would be located either at the space-like 3-surfaces at the ends of the space-time surface or at light-like 3-surfaces connecting the ends. Modified Dirac equation would define Super-Kac-Moody charges as standard Noether charges. Super charges would be obtained by replacing the second quantized spinor field or its conjugate in the fermionic bilinear by particular mode of the spinor field. By replacing both spinor field and its conjugate by its mode one would obtain a conserved c-number charge corresponding to an anti-commutator of two fermionic super-charges. The convolution involving double integral is however not number theoretically attactive whereas single 1-D integrals might make sense.

  2. An encouraging observation is that the Hodge dual of the Kac-Moody current defines the analog of gauge potential and exponents of the conserved Kac-Moody charges could be identified as analogs for the non-integrable phase factors for the components of this gauge potential. This identification is precise only in the approximation that generators commute since only in this case the ordered integral P(exp(i∫ Adx)) reduces to P(exp(i∫ Adx)).Partonic 2-surfaces connected by braid strand would be analogous to nearby points of space-time in its discretization implying that Abelian approximation works. This conforms with the vision about finite measurement resolution as discretization in terms partonic 2-surfaces and braids.

    This would make possible a direct identification of Kac-Moody symmetries in terms of gauge symmetries. For isometries one would obtain color gauge potentials and the analogs of gauge potentials for graviton field (in TGD framework the contraction with M4 vierbein would transform tensor field to 4 vector fields). For Kac-Moody generators corresponding to holonomies one would obtain electroweak gauge potentials. Note that super-charges would give rise to a collection of spartners of gauge potentials automatically. One would obtain a badly broken SUSY with very large value of N defined by the number of spinor modes as indeed speculated earlier (see this).

  3. The condition that the gauge field defined by 1-forms associated with the Kac-Moody currents are trivial looks unphysical since it would give rise to the analog of topological QFT with gauge potentials defined by the Kac-Moody charges. For the duals of Kac-Moody currents defining gauge potentials only covariant divergence vanishes implying that curvature form is

    Fαβ= εαβ [jμ, jμ] ,

    so that the situation does not reduce to topological QFT unless the induced metric is diagonal. This is not the case in general for string world sheets.

  4. It seems however that there is no need to assume that jμ defines a flat connection. Witten mentions that although the discretization in the definition of JA does not seem to be possible, it makes sense for QA in the case of G=SU(N) for any representation of G. For general G and its general representation there exists no satisfactory definition of Q. For certain representations, such as the fundamental representation of SU(N), the definition of QA is especially simple. One just takes the bi-local part of the previous formula:

    QA= fABCi<jJBiJCj .

    What is remarkable that in this formula the summation need not refer to a discretized point of braid but to braid strands ordered by the label i by requiring that they form a connected polygon. Therefore the definition of JA could be just as above.

  5. This brings strongly in mind the interpretation in terms of twistor diagrams. Yangian would be identified as the algebra generated by the logarithms of non-integrable phase factors in Abelian approximation assigned with pairs of partonic 2-surfaces defined in terms of Kac-Moody currents assigned with the modified Dirac action. Partonic 2-surfaces connected by braid strand would be analogous to nearby points of space-time in its discretization. This would fit nicely with the vision about finite measurement resolution as discretization in terms partonic 2-surfaces and braids.

The resulting algebra satisfies the basic commutation relations



plus the rather complex Serre relations described in Witten's article).

The connection between Kac-Moody symmetries and gauge symmetries is suggestive and in this case it would be realized in terms of 2-D Hodge duality. Also finite measurement resolution realized in the sense that the points at the ends of given braid strand are regarded to be effectively infinitesimally close so that the gauge algebra is effectively Abelian is essential. Yangian symmetry is crucial for the success of the twistor approach. Zero energy ontology implies that generalized Feynman diagrams contain only massless partonic 2-surfaces with propagators defined by longitudinal momentum components defined in terms of M2⊂ M4 characterizing given causal diamond. There there are excellent hopes that twistor approach applies also in TGD framework. Note that also the conformal transformations of M4 might allow Yangian variants.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title.

The recent vision about preferred extremals and solutions of the modified Dirac equation

The understanding of preferred exrremals of preferred extremals of Kähler action and solutions of the modified Dirac equation has increased dramatically during last months and I have been busily deducing the consequences for the quantum TGD. This process led also to a new chapter of the book about physics as WCW geometry. I attach below the extended abstract of the chapter.

During years several approaches to what preferred extremals of Kähler action and solutions of the modified Dirac equation could be have been proposed and the challenge is to see whether at least some of these approaches are consistent with each other. It is good to list various approaches first.

  1. For preferred extremals generalization of conformal invariance to 4-D situation is very attractive approach and leads to concrete conditions formally similar to those encountered in string model. In particular, Einstein's equations with cosmological constant follow as consistency conditions and field equations reduce to a purely algebraic statements analogous to those appearing in equations for minimal surfaces if one assumes that space-time surface has Hermitian structure or its Minkowskian variant Hamilton-Jacobi structure. The older approach based on basic heuristics for massless equations, on effective 3-dimensionality, and weak form of electric magnetic duality, and Beltrami flows is also promising. An alternative approach is inspired by number theoretical considerations and identifies space-time surfaces as associative or co-associative sub-manifolds of octonionic imbedding space.

    The basic step of progress was the realization that the known extremals of Kähler action - certainly limiting cases of more general extremals - can be deformed to more general extremals having interpretation as preferred extremals.

    1. The generalization boils down to the condition that field equations reduce to the condition that the traces Tr(THk) for the product of energy momentum tensor and second fundamental form vanish. In string models energy momentum tensor corresponds to metric and one obtains minimal surface equations. The equations reduce to purely algebraic conditions stating that T and Hk have no common components. Complex structure of string world sheet makes this possible.

      Stringy conditions for metric stating gzz=gz*z*=0 generalize. The condition that field equations reduce to Tr(THk)=0 requires that the terms involving Kähler gauge current in field equations vanish. This is achieved if Einstein's equations hold true. The conditions guaranteeing the vanishing of the trace in turn boil down to the existence of Hermitian structure in the case of Euclidian signature and to the existence of its analog - Hamilton-Jacobi structure - for Minkowskian signature. These conditions state that certain components of the induced metric vanish in complex coordinates or Hamilton-Jacobi coordinates.

      In string model the replacement of the imbedding space coordinate variables with quantized ones allows to interpret the conditions on metric as Virasoro conditions. In the recent case generalization of classical Virasoro conditions to four-dimensional ones would be in question. An interesting question is whether quantization of these conditions could make sense also in TGD framework at least as a useful trick to deduce information about quantum states in WCW degrees of freedom.

      The interpretation of the extended algebra as Yangian suggested previously to act as a generalization of conformal algebra in TGD Universe is attractive. There is also the conjecture that preferred extremals could be interpreted as quaternionic of co-quaternionic 4-surface of the octonionic imbedding space with octonionic representation of the gamma matrices defining the notion of tangent space quanternionicity.

  2. There are also several approaches for solving the modified Dirac equation. The most promising approach is assumes that the solutions are restricted on 2-D stringy world sheets and/or partonic 2-surfaces. This strange looking view is a rather natural consequence of both strong form of holography and of number theoretic vision, and also follows from the notion of finite measurement resolution having discretization at partonic 2-surfaces as a geometric correlate. The conditions stating that electric charge is conserved for preferred extremals is an alternative very promising approach. One expects that stringy approach based on 4-D generalization of conformal invariance or its 2-D variant at 2-D preferred surfaces should also allow to understand the modified Dirac equation. In accordance with the earlier conjecture, all modes of the modified Dirac operator generate badly broken super-symmetries. Right-handed neutrino allows also holomorphic modes delocalized at entire space-time surface and the delocalization inside Euclidian region defining the line of generalized Feynman diagram is a good candidate for the right-handed neutrino generating the least broken super-symmetry. This super-symmetry seems however to differ from the ordinary one in that νR is expected to behave like a passive spectator in the scattering.
The question whether these various approaches are mutually consistent is discussed. It indeed turns out that the approach based on the conservation of electric charge leads under rather general assumptions to the proposal that solutions of the modified Dirac equation are localized on 2-dimensional string world sheets and/or partonic 2-surfaces. Einstein's equations are satisfied for the preferred extremals and this implies that the earlier proposal for the realization of Equivalence Principle is not needed. This leads to a considerable progress in the understanding of super Virasoro representations for super-symplectic and super-Kac-Moody algebra. In particular, the proposal is that super-Kac-Moody currents assignable to string world sheets define duals of gauge potentials and their generalization for gravitons: in the approximation that gauge group is Abelian - motivated by the notion of finite measurement resolution - the exponents for the sum of KM charges would define non-integrable phase factors. One can also identify Yangian as the algebra generated by these charges. The approach allows also to understand the special role of the right handed neutrino in SUSY according to TGD.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title.

How quaternicity of space-time could be consistent with Hermitian/Hamilton-Jacobi structure?

The recent progress in the understanding of preferred extremals of Kähler action suggests also an interesting connection to the number theoretic vision about field equations. In particular, it might be possible to understand how one can have Hermitian/Harmilton-Jacobi structure simultaneously with quaternionic structure and how quaternionic structure is possible for the Minkowskian signature of the induced metric.

One can imagine two manners of introducing octonionic and quaternionic structures. The first one is based on the introduction of octonionic representation of gamma matrices and second on the notion of octonion real-analycity.

  1. If quaternionic structure is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions represented in terms of gamma matrices. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure in Minkowskian regions or to introduce what I have called hyper-quaternionic structure by imbedding the space-time surface to a sub-space $M^8$ of complexified octonions. This is admittedly artificial.
  2. The octonionic representation effectively replaces SO(7,1) as tangent space group with G2 and means selection of preferred M2⊂ M4 having interpretation complex plane of octonionic space. A more general condition is that the tangent space of space-time surface at each point contains preferred sub-space M2(x)⊂ M4 forming an integrable distribution. The same condition is involved with the definition of Hamilton-Jacobi structure. What puts bells ringing is that the modified Dirac equation for the octonionic representation of gamma matrices allows the conservation of electromagnetic charge in the proposed sense a observed for years ago. One can ask whether the conditions on the charged part of energy momentum tensor could relate to the reduction of SO(7,1) to G2.
  3. Octonionic gamma matrices appear also in the proposal stating that space-time surfaces are quaternionic in the sense that tangent space of the space-time surface is quaternionic in the sense that induced octonionic gamma matrices generate a quaternionic sub-space at a given point of space-time time. Besides this the already mentioned additional condition stating that the tangent space contains preferred sub-space M2⊂ M4 or integrable distribution of this kind of sub-spaces is required. It must be emphasized that induced rather than modified gamma matrices are natural in these conditions.
The definition of quaternionicity in terms of gamma matrices looks more promising. This however raises two questions.
  1. Could the quaternionicity of the space-time surface together with a preferred distribution of tangent planes M2(x)⊂ M4 or E2(x)⊂ CP2 be equivalent with the reduction of the field equations to the analogs of minimal surface equations stating that certain components of the induced metric in complex/Hamilton-Jacobi coordinates vanish in turn guaranteeing that field equations reduce to algebraic identifies following from the fact that energy momentum tensor and second fundamental form have no common components? This should be the case if one requires that the two solution ansätze are equivalent.
  2. Can the conditions for the modified Dirac equation select complex of co-complex 2-sub-manifold of space-time surface identified as quaternionic or co-quaternionic 4-surface? Could the conditions stating the vanishing of charged energy momentum currents state that the spinor fields are localized to complex or co-complex (hyper-complex or co-hypercomplex) 2-surfaces?
One should assign to the space-time sheets both quaternionic and Hermitian or Hamilton-Jacobi structure. There are two structures involved. Euclidian metric is an essential aspect of what it is to be quaternionic or octonions. It however seems that one can assign to the induced metric only Hermitian or Hamilton-Jacobi structure. This leads to a serious of innocent questions.
  1. Could these two structures be associated with energy momentum tensor and metric respectively? Or perhaps vice versa? Anti-commutators of the modified gamma matrices define an effective metric as

    Gαβ=TαμTμβ .

    This effective metric should have a deep physical and mathematical meaning but this meaning has remained a mystery. Note that iT takes the role of Kähler form for Kähler metric in this expression with imaginary unit plus symmetry replacing antisymmetry. The neutral and charged parts of T would be analogous to quaternionic imaginary units and real part but one would have sum over the squares of all of them rather than only single square as for Kähler metric.

  2. Could G be assigned with the quaternionic structure and induced metric to the Hermitian/Hamilton-Jacobi structures? Could the neutral and charged components of the energy momentum tensor somehow correspond to quaternionic units?
The basic potential problem with the assignment of quaternionic structure to the induced gamma matrices is the signature of the metric in Minkowskian regions.
  1. If quaternionic structrures is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions.
  2. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure or to introduce hyper-quaternionic structure by imbedding the space-time surface to a sub-space M8 of complexified octonions. This is admittedly artificial.

Could one pose the additional requirement that the signature of the effective metric G defined by the modified gamma marices (and to be distinguished from Einstein tensor) is Euclidian in the sense that all four eigenvalues of this tensor would have same sign.

  1. For the induced metric the projections of gamma matrices are given by

    Γαae , e= eakα hk .

    For the modified gamma matrices their analogs would be given by

    Γαa E , E= eTμα .

    Projection would be followed by multiplication with energy momentum tensor. One cannot induce G from any metric defined in the imbedding space but the notion of tangent space quaternionicity is well-defined.

  2. What quaternionic structure for G could mean? One can imagine several options.
    1. For the ordinary complex structure metric has vanishing diagonal components and the infinitesimal line element ds2=gzz*dzdz*. Could this formula generalize to


      The generalization would be a direct generalization of conformal invariance to 4-D context stating that 4-metric is quaternion-conformally equivalent to flat metric. This would give additional strong condition on energy momentum tensor:

      G= TαμTμβ= T2δαβ .

      The proportionality to Euclidian metric means in Minkowskian realm that the G is of form G= T2(2uαuβ -gαβ. Here u is time-like vector field satisfying uαuα=1 and having interpretation as local four-velocity (in Robertson-Walker cosmology similar situation is encountered). The eigen value problem in the form Gαβxβ= λ xα makes sense and eigenvectors would be u with eigenvalue λ=T2 and three vectors orthogonal to u with eigenvalue -T2. This requires integrable flow defined by u and defining a preferred time coordinate. In number theoretic vision this kind of time coordinate is introduced and corresponds to the direction assignable to the octonionic real unit. Note that the vanishing of charged projections of the energy momentum tensor does not imply a reduction of the rank of T so that this option might work.

    2. Quaternionicity could mean also the structure of hyper-Kähler manifold. Metric and Kähler form for Kähler manifold are generalized to metric representing quaternion real unit and three covariantly constant Kähler forms Ii obeying the multiplication rules for quaternions. The necessary condition is that the holonomy group equals to SU(2) identifiable as automorphism group of quaternions. One can also define quaternionic structure: there would exist three antisymmetric tensors, whose squares give the negative of the metric. CP2 allows quaternionic structure in this sense and only one of these forms is covariantly constant.

      Could space-time surface allow Hyper-Kähler or quaternionic structure somehow induced from that of CP2? This does not work for G. G is quadratic in energy momentum tensor and therefore involves four power of J rather than being square of projection of J or two other quaternionic imaginary units of CP2. One can of course ask whether the induced quaternionic units could obey the multiplication of quaternionic units and have same square given by the projection of CP2 metric. In this case CP2 metric would define the effective metric and would be indeed Euclidian. For the ansatz for preferred extremals with Minkowskian signature CP2 projection is at most 3-dimensional but also in this case the imaginary units might allow a realization as projections.

For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle? or the article. For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle? or the article.

The recent vision about preferred extremals and solutions of the modified Dirac equation

During last months a considerable progress in understanding of both the preferred extremals of Kähler action and solutions of modified Dirac equation has taken place and there are good reasons to believe that various approaches are converging. Instead of trying to describe the results in detail here I just give the abstract of the article The recent vision about preferred extremals and solutions of the modified Dirac equation. The text appears also in the chapter Does the modified Dirac action define the fundamental variational principle? of the online book "TGD: Physics as Infinite-dimensional Geometry". Here is the short abstract.

During years several approaches to what preferred extremals of Kähler action and solutions of the modified Dirac equation could be have been proposed and the challenge is to see whether at least some of these approaches are consistent with each other. It is good to list various approaches first.

  1. For preferred extremals generalization of conformal invariance to 4-D situation is very attractive approach and leads to concrete conditions formally similar to those encountered in string model (see this). In particular, Einstein's equations with cosmological constant follow as consistency conditions and field equations reduce to a purely algebraic statements analogous to those appearing in equations for minimal surfaces if one assumes that space-time surface has Hermitian structure or its Minkowskian variant Hamilton-Jacobi structure. The older approach based on basic heuristics for massless equations, on effective 3-dimensionality, and weak form of electric magnetic duality, and Beltrami flows is also promising. An alternative approach is inspired by number theoretical considerations and identifies space-time surfaces as associative or co-associative sub-manifolds of octonionic imbedding space (see this).
  2. There are also several approaches for solving the modified Dirac equation. The most promising approach is assumes that the solutions are restricted on 2-D stringy world sheets and/or partonic 2-surfaces. This strange looking view is a rather natural consequence of both strong form of holography and of number theoretic vision, and also follows from the notion of finite measurement resolution having discretization at partonic 2-surfaces as a geometric correlate. The conditions stating that electric charge is conserved for preferred extremals is an alternative very promising approach. One expects that stringy approach based on 4-D generalization of conformal invariance or its 2-D variant at 2-D preferred surfaces should also allow to understand the modified Dirac equation.
In the following the question whether these various approaches are mutually consistent is discussed. It indeed turns out that the approach based on the conservation of electric charge leads under rather general assumptions to the proposal that solutions of the modified Dirac equation are localized on 2-dimensional string world sheets and/or partonic 2-surfaces. One can also apply the approach at imbedding space level to the solutions of ordinary Dirac equation and this might be actually relevant for the representations of symplectic group of δ M4+/-× CP2. The implication seems to be that only color singlet representation is allowed for leptons and color triplet/antitriplet for quarks/anti-quarks. The result is physically very nice but would almost totally eliminate spinorial CP2 partial waves.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article already mentioned.

Under what conditions electric charge is conserved for the modified Dirac equation?

One might think that talking about the conservation of electric charge at 21st century is a waste of time. In TGD framework this is certainly not the case.

  1. In quantum field theories there are two manners to define em charge: as electric flux over 2-D surface sufficiently far from the source region or in the case of spinor field quantum mechanically as combination of fermion number and vectorial isospin. The latter definition is quantum mechanically more appropriate.
  2. There is however a problem. In standard approach to gauge theory Dirac equation in presence of charged classical gauge fields does not conserve electric charge: electron is transformed to neutrino and vice versa. Quantization solves the problem since the non-conservation can be interpreted in terms of emission of gauge bosons. In TGD framework this does not work since one does not have path integral quantization anymore. Preferred extremals carry classical gauge fields and the question whether em charge is conserved arises. Heuristic picture suggests that em charge must be conserved. This condition might be actually one of the conditions defining what it is to be a preferred extremal. It is not however trivial whether this kind of additional condition can be posed.
What does the conservation of em charge imply in the case of the modified Dirac equation? The obvious guess that the em charged part of the modified Dirac operator must annihilate the solutions, turns out to be correct as the following argument demonstrates.
  1. Em charge as coupling matrix can be defined as a linear combination Q= aI+bI3, I3=JklΣkl, where I is unit matrix and I3 vectorial isospin matrix, Jkl is the Kähler form of CP2, Σkl denotes sigma matrices, and a and b are numerical constants different for quarks and leptons. Q is covariantly constant in M4× CP2 and its covariant derivatives at space-time surface are also well-defined and vanish.
  2. The modes of the modified Dirac equation should be eigen modes of Q. This is the case if the modified Dirac operator D commutes with Q. The covariant constancy of Q can be used to derive the condition

    [D,Q] Ψ= D1Ψ=0 , D= ΓμDμ ,

    D1=[D,Q]=Γμ1Dμ , Γμ1= [ Γμ,Q] .

    Note that here Γμ denotes modified gamma matrices rather than ordinary induced gamma matrices defined by contractions of energy momentum tensor with induced gamma matrices. Covariant constancy of J is absolutely essential: without it the resulting conditions would not be so simple.

    It is easy to find that also [D1,Q]Ψ=0 and its higher iterates [Dn,Q]Ψ=0, Dn= [Dn-1,Q] must be true. The solutions of the modified Dirac equation would have an additional symmetry.

  3. The commutator D1=[D,Q] reduces to a sum of terms involving the commutators of the vectorial isospin I3=JklΣkl with the CP2 part of the gamma matrices:

    D1=[Q , D]= [I3r] ∂μsr Tαμ .

    In standard complex coordinates in which U(2) acts linearly the complexified gamma matrices can be chosen to be eigenstates of vectorial isospin. Only the charged flat space complexified gamma matrices ΓA denoted by Γ+ and Γ- possessing charges +1 and -1 contribute to the right hand side. Therefore the additional Dirac equation D1Ψ=0 states

    D1Ψ=[Q , D]Ψ= I3(A)eAr ΓAμsr TαμDαΨ

    = (e+r Γ+-e-rΓ-)∂μsr TαμDαΨ =0 .

    The next condition is

    D2Ψ=[Q , D1]Ψ=(e+r Γ++e-rΓ-) ∂μsr TαμDαΨ =0 .

    The remaining conditions give nothing new.

  4. These equations imply two separate equations for the two charged gamma matrices

    D+Ψ = Γ+T+αDαΨ=0

    D-Ψ = Γ-T-αDαΨ=0

    T+/-α= e+/-rμsr Tαμ .

    These conditions state what one might have expected: the charged part of the modified Dirac operator annihilates separately the solutions. The reason is that the classical W fields are proportional to er+/-.

    The above equations can be generalized to define a decomposition of the energy momentum tensor to charged and neutral components in terms of vierbein projections. The equations state that the analogs of the modified Dirac equation defined by charged components of the energy momentum tensor are satisfied separately.

  5. In complex coordinates one expects that the two equations are complex conjugates of each other for Euclidian signature. For the Minkowskian signature an analogous condition should hold true. The dynamics enters the game in an essential manner: whether the equations can be satisfied depends on the coefficients a and b in the expression T= aG+bg implied by Einstein's equations in turn guaranteeing that the solution ansatz generalizing minimal surface solutions holds true (see this).

  6. As a result one obtains three separate Dirac equations corresponding to the the neutral part D0Ψ=0 and charged parts D+/-Ψ=0 of the modified Dirac equation. By acting on the equations with these Dirac operators one obtains that also the commutators [D+,D-], [D0,D+/-] and also higher commutators obtained from these annihilate the induced spinor field model. Therefore entire - possibly- infinite-dimensional - algebra would annihilate the induced spinor fields. In string model the counterpart of Dirac equation when quantized gives rise to Super-Virasoro conditions. This analogy would suggest that modified Dirac equation gives rise to the analog of Super-Virasoro conditions in 4-D case. But what the higher conditions mean? Obviously these conditions resemble Virasoro conditions Ln|phys>=0 and their supersymmetric generalizations and might indeed correspond to a generalization of these conditions just as the field equations for preferred extremals could correspond to the Virasoro conditions if one takes seriously the analogy with the quantized string.
What could this additional symmetry mean from the point of view of the solutions of the modified Dirac equation? The field equations for the preferred extremals of Kähler action reduce to purely algebraic conditions in the same manner as the field equations for the minimal surfaces in string model. Could this happen also for the modified Dirac equation and could the condition on charged part of the Dirac operator help to achieve this?
  1. CP2 type vacuum extremals serve as a convenient test case. In this case the modified Dirac equation reduces to the ordinary Dirac equation in CP2. One can construct the solutions of the ordinary Dirac equation from covariantly constant right-handed neutrino spinor playing the role of fermionic vacuum annihilated by the second half of complexified gamma matrices. Dirac equation reduces to Laplace equation for a scalar function and solution can be constructed from this "vacuum" by multiplying with the spherical harmonics of CP2 and applying Dirac operator (see this). Similar construction works quite generally thanks to the existence of covariantly constant right handed neutrino spinor. Spherical harmonics of CP2 are only replaced with those of space-time surface possessing either hermitian structure of Hamilton-Jacobi structure (corresponding to Euclidian and Minkowskian signatures of the induced metric, see this).
  2. Could the properties of the preferred extremals make it possible to satisfy the additional conditions for the modified Dirac operator? The identical vanishing of the charged components of the energy momentum tensor would be an obvious manner to reduce the conditions as algebraic identifies. Note however that it could be an un-necessarily strong condition. In any case, the vanishing of charged components of the energy momentum tensor looks like a natural weakening of the conditions stating the vanishing of charged components of the induced gauge field.
There is also an interesting potential connection to the number theoretic vision about field equations.
  1. The octonionic representation effectively replaces SO(7,1) as tangent space group with G2 and means selection of preferred M2⊂ M4 having interpretation complex plane of octonionic space (see this). A more general condition is that the tangent space of space-time surface at each point contains preferred sub-space M2(x)⊂ M4 forming an integrable distribution. The same condition is involved with the definition of Hamilton-Jacobi structure. What puts bells ringing is that the modified Dirac equation for the octonionic representation of gamma matrices allows the conservation of electromagnetic charge in the proposed sense a observed for years ago. One can ask whether the conditions on the charged part of energy momentum tensor could relate to the reduction of SO(7,1) to G2.
  2. Octonionic gamma matrices appear also in the proposal stating that space-time surfaces are quaternionic in the sense that tangent space of the space-time surface is quaternionic in the sense that induced octonionic gamma matrices generate a quaternionic sub-space at a given point of space-time time. Besides this the already mentioned additional condition stating that the tangent space contains preferred sub-space M2⊂ M4 or integrable distribution of this kind of sub-spaces is required. It must be emphasized that induced rather than modified gamma matrices are natural in these conditions.
This raises two questions.
  1. Could the quaternionicity of the space-time surface be equivalent with the reduction of the field equations to "stringy" field equations (minimal surface equations) stating that certain components of the induced metric in complex/Hamilton-Jacobi coordinates vanish in turn guaranteeing that field equations reduce to algebraic identifies following from the fact that energy momentum tensor and second fundamental form have no common components? This should be the case if one requires that the two solution ansätze are equivalent.
  2. Could the above conditions for the modified Dirac equation imply that the space-time surface is quaternionic? Or could these conditions be seen as consistency conditions making the modified Dirac equation associated with the ordinary and octonionic gamma matrices equivalent at space-time level in the sense that the induced gamma matrices or modified gamma matrices are equivalent?

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title.

What could be the counterpart of T-duality in TGD framework?

Stephen Crowell sent me a book of Michel Lapidus about zeros of Riemann zeta and also about his own ideas in this respect. The book has been written in avery lucid manner and looks very interesting. The big idea is that the T-duality of string models could correspond to the functional equation for Riemann zeta relating the values of zeta at different sides of the critical line. T-duality is formulated for strings in space Md× S1 or its generalization replacing S1 with higher-dimensional torus and generalized to fractal strings. Duality states that the transformation R→ 1/R with suitable unit for R defined by string tension is a duality: the physics for these different values of R is same. Intuitively this is due to the fact that the contributions of the string modes representing n-fold winding and those representing vibrations labelled by integer n are transformed to each other in the transformation R→ 1/R.

Lapidus is a mathematician and mathematicians often do not care too much about the physical meaning of the numbers. For a physicist like me it is extremely painful to type the equation R→ 1/R without explicitly explaining that it should actually read as R→ R02/R, where R0 is length unit, which must represent fundamental length scale remaining invariant under the duality transformation. Only after this physicist could reluctantly put R0=1 but still would feel himself guilty of unforgivable sloppiness. R0=1 simplifies the formulas but one must not forget that there are three scales involved rather than only two. The question inspired by this nitpicking is how the physics in the length scales R1 and R relates to the physics in length scale R. Are dualities - or perhaps holography like relations in question- so that T-duality would follow from these dualities?

Could one replace winding number with magnetic charge and T-duality with canonical identification?

How could one generalize T-duality to TGD framework? One should identify the counterpart of the winding number, the three fundamental scales, and say something about the duality transformation itself.

  1. In TGD Universe partonic 2-surfaces are the basic object. Partonic 2-surface is not strings and the only reasonable generalization for winding number is as Kähler magnetic charge representing the analog of winding of the partonic 2-surface around magnetically charged 2-sphere of CP2. Magnetic charge tells how many times partonic 2-surface wraps around the homologically non-trivial geodesic sphere with unit magnetic charge. If the generalization of T-duality holds true, one would expect that the contributions of the oscillations and windings of the partonic two-surface to ground state energy must be transformable to each other by the counterpart of the transformation R→ R02/R - or something akin to that. Also less concrete and more general interpretations are possible, and below the most plausible interpretation will be considered.

  2. The duality R→ R_0^2/R= R1 gives R_0 as a geometric mean R_0=(RR1)1/2 of the scales R and R1. What are these three length scales in TGD Universe? The obvious candidate for R is CP2 size scale. p-Adic mass calculations imply that the primary p-adic length scale Lp,1= p1/2R is of order of Compton length of the elementary particle characterized by the p-adic prime p. The secondary p-adic length scale Lp,2= pR in turn defines the size scale of causal diamond (CD) assignable to the magnetic body of the elementary particle characterized by prime p. For instance, for electron this scale corresponds to .1 seconds, a fundamental biological time scale.

    One indeed has Lp,1= (Lp,2R)1/2, and CP2 scale and CD length scale are dual to each other if T-duality holds true. Therefore the duality would relate physics at CP2 scale - counterpart of Planck length in TGD framework - and in biological scales and would have direct relevance to quantum biology. One has an infinite hierarchy of p-adic length scales and each of them would give rise to one particular instance of the T-duality. Adeles would provide appropriate formulation of T-duality in TGD framework. The corresponding mass scales would be hbar/R, hbar/p1/2R and hbar/pR. The third scale corresponds to a scale, which for electron corresponds to the 10 Hz frequency in the case of photons. The duality would suggest that the physics associated with the frequencies in EEG scale related to the communications from the biological body to magnetic body is dual to the physics in CP2 scale.

    Note that one cannot exclude alternative variants of T-duality. In particular, Planck scale and CP2 length scale as candidates R1 and R could be considered.

  3. What is the interpretation of these three length scales? CP2 length scale corresponds naturally to the size scale of wormhole contacts. They are Euclidian regions of space-time surface and represent lines of generalized Feynman graphs. Both general arguments and the construction of elementary bosons forces to assign to these regions braid strands playing a role of Euclidian strings. Parallel translation along the strands is essential in the construction of fermionic bilinears as invariant under general coordinate transformations and gauge transformations. The ends of these strands carry fermion and anti-fermion numbers. The counterpart of string tension involved appearing in stringy mass formula implied by super-conformal invariance is indeed determined by R and p-adic thermodynamics leads to a detailed and successful calculations for elementary particle masses using only p-adic thermodynamics, super-conformal invariance, and p-adic length scale hypothesis as basic assumptions.

  4. The wormhole throats carrying fermion number are Kähler magnetic monopoles and the wormhole must be accompanied by a second wormhole throat carrying opposite magnetic charge and also a neutrino pair neutralizing the weak isospin so that weak massivation takes place. The end of the flux tube containing the neutrino pair is virtually non-existent at low energies. The length scale for this string must correspond to Compton length for elementary particle given essentially by primary p-adic length scale Lp,1. The more restrictive assumption that this length scale corresponds to the Compton length of weak bosons looks un-necessarily restrictive and looks also un-natural.

  5. The excitations with mass scale hbar/pR would correspond to excitations assignable to entire CD, maybe assignable to the flux tubes of the magnetic bodies of elementary particles defining also string like objects but in macroscopic scales. For electron the scale is of order of the circumference of Earth. This dynamics would naturally correspond to the dynamics in Minkowskian space-time regions. The dynamics at intermediate length scale would be intermediate between the Euclidian and Minkowskian dynamics and reduce to that for light-like orbits of partonic 2-surfaces with metric intermediate between Minkowskian and Euclidian.

  6. A natural interpretation for T-duality in this sense is in terms of strong form of holography. The interior dynamics at length scale R resp. pR assigned to Euclidian resp. Minkowskian regions of space-time surface corresponds by holography to the dynamics of light-like orbits of partonic 2-surfaces identified as wormhole throats. Therefore the dynamics in Euclidian and Minkowskian regions are dual to each other. Therefore T-duality in TGD sense would follow from the possibility of having both Euclidian and Minkowskian holography. Strong form of holography in turn reduces to strong form of General Coordinate Invariance, which has turned out to be extremely powerful principle in TGD framework.

Is the physics of life dual to the physics in CP2 scale?

The duality of life with elementary particle physics at CP2 length scale - the TGD counterpart of Planck scale - looks rather far-fetched idea. There is however already earlier support for this idea.

  1. p-Adic physics is physics of cognition, and one can say that living systems are in the algebraic intersection of real and p-adic worlds: the intersection of cognition and matter. Canonical identification maps p-adic physics to real physics. This map takes p-adic integers which are small in p-adic sense to larger integers in real sense and thus maps long real scales to short real scales. Clearly this map is jhighly analogous to the T-duality. p-Adic length scales are indeed explicitly related with the above identification of the T-duality so that canonical identification might be involved with T-duality.

    If this interpretation is correct, cognitive p-adic representations in long real length scales would give representations for the physics in short length scales. EEG range of frequencies allowing communication to the magnetic bodies is absolutely essential for brain function. CDs would correspond to the real physics scale associated with the cognitive representations. These cognitive representations are indeed exactly what our science is building so that T-duality would make also scientist as a part of the big vision!

  2. The model for dark nucleons as three quark states led to one of the greatest surprises of my professional life. Under rather general conditions the three quark states for nucleon are in one-one correspondence with the DNA, RNA, tRNA codons, and aminoacids for vertebrate genetic code and there is natural physical correspondence between DNA triplets and aminoacids. This suggests that genetic code is realized at the level of hadrons and that living matter is a kind of emulation for it, or that living matter is representation for matter at hadron level. This leads to rather far reaching speculations about biological evolution - not as random process - but a process analogous R&D applied in industry. New genes would be continually tested at the level of dark matter and the modifications of genome could be carried out if there is a transcription process transforming dark DNA to ordinary DNA.

  3. The secondary p-adic mass scale of electron corresponds to the 10 Hz frequency, which defines a fundamental biorhythm. Also to current quark masses, which are actually not so well-known but are in MeV range, one can assign biologically interesting time scales in millisecond range. This suggests that all elementary particles induce physics in macroscopic time scales via their CD:s containing their magnetic bodies.

The unavoidable and completely crazy looking question raised by T-duality is whether there is intelligent life in the Euclidian realm below the CP2 length scale - inside the lines of generalized Feynman graphs. This kind of possibility cannot be avoided if one takes holography absolutely seriously. In purely mathematical sense TGD suggests even stronger form of holography based on the notion of infinite primes. In this holography the number theoretic anatomy of given space-time point is infinitely complex and evolves. The notion of quantum mathematics replacing numbers by Hilbert spaces representing ordinary arithmetics in terms of direct sum and tensor product suggest the same. Space-time point would be in this picture its own infinitely complex Universe - the Platonia.

Could one get expression for Kähler coupling strength from restricted form of modular invariance?

The contributions to the exponent of the vacuum functional, which is proportional to Kähler action for preferred extremal, are real resp. imaginary in Euclidian resp. Minkowskian regions. Under rather general assumptions (weak form of electric-magnetic duality defining boundary conditions at wormhole throats plus additional intuitively plausible assumption) these contributions are proportional to the same Chern-Simons term but with possibly different constant of proportionality.

These terms sum up to a Chern-Simons term with a coefficient analogous to the complex inverse gauge coupling

τ=θ/2π + i4π/g2K .

The real part would correspond to Kähler function coming from Euclidian regions defining the lines of generalized Feynman diagrams and imaginary part to Minkowskian regions. There are could arguments suggesting that With the conventions that I have used θ/2π is counterpart for 1/αK and there are good arguments that it corresponds to finte structure constant in electron length scale. Furthermore, T-duality would suggest that τ is proportional to 1+i so that one would have

θ/2π = 4π/g2K .

This condition would fit nicely with the fact that Chern-Simons contributions from Minkowskian and Euclidian regions are identical. If this equation holds true the modular transformations must reduce to those leaving this relationship invariant and can only permute the complex and real parts and thus leave τ invariant. One could also interpret this value of τ as physically especially interesting representation and assign to all values of τ related by modular transformation an isotropy group leaving it fixed. All other physically equivalent values would be obtained as SL(2,Z) orbit of this value.

The counterpart of T-duality should somehow relate dynamics in Minkowskian and Euclidian regions and this raises the question whether it corresponds to τ→ iτ and is represented by some duality transformation

τ→ (aτ+b)/(cτ+d) ,

where (a,b;c,d) defines a unimodular matrix (ad-bc=1) with integer elements, that is in SL(2,Z). The electric-magnetic duality τ→ -1/τ and the shift τ→ τ+1 are the generators of this group. It is not however quite clear whether they can be regarded as gauge symmetries in TGD framework. If they are gauge symmetries, then the critical values of Kähler coupling strength defined as fixed points of coupling constant evolution must form an orbit of SL(2,Z). It could be also that modular symmetry is broken to a subgroup of SL(2,Z) and this subgroup leaves τ invariant in the case of minimal symmetry.

  1. τ→ iτ would permute Euclidian and Minkowskian regions with each other and is therefore a candidate for the T-duality. This condition cannot be satisfied in generic case but one can ask whether for some special choices of τ these transformations could generate a non-trivial sub-group of modular transformations. This subgroup

    To see whether this is the case let us write explicitly the condition τ→ iτ:

    (aτ +b)/(cτ+d)= θ/2π+ i/αK , αK= g2K/4π .

    The condition allows to solve τ as

    τ= ((a-id)/2c)× [1+ε1(1+ 4ibc/(d-ia)2)1/2)] , ε1=+/- 1 .

  2. For

    d=ε a , ε=+/- 1

    implying a2-bc=1, the solution simplifies since the argument of square root is real. One has

    τ= (a/2c)(1-ε i)[1+ε1 (1-ε (a-1/a)1/2)1/2] .

    The imaginary and real parts of τ are identical: this might allow an interpretation in terms of the fact that Chern-Simons terms from two regions are identical (normal derivatives are however discontinuous at wormhole throat). Certainly this is a rather strong prediction.

  3. Does this mean that SL(2,C) is broken down to the 4-element isotropy group generated by this transformation? If so, a the condition just deduced could allow to deduce additional constraints on the value of Kähler coupling strength, which is in principle fixed by the criticality condition to have only finite number of values? By the earlier arguments - related to p-adic mass calculations and the heuristic formula for the gravitational constant - the value of Kähler coupling strength is in a good approximation equal to fine structure constant at electron length scale:

    αK =gK2/4π≈ α , 1/α≈ 137.035999084 .

  4. One obtains the following estimate for a/2c from the estimate for αK by considering the imaginary part of τ:

    (a/2c)[1+ε1 (1-ε(a-1/a)1/2)1/2]= 1/αK .

    At the limit a→ ∞ one has

    (a/2c)[[1+ε1 (1-ε)1/2 ]= 1/αK .

    The simplest option at this limit corresponds to ε=1 giving a/2c≈ 137.035999084 .

    Note that a/2c=137 is not allowed by determinant condition so that the deviation of αK from 1/137 is predicted. One must have a>137× 2c≥ 2× 137. This implies

    1+ε1(1-(a-1/a)1/2)1/2 =1+.0026 ε1 .

    By expanding the square root in first order to Taylor series one obtains the condition

    (a/2c)× (1+ε1/23/2× 137 c)≈ 1/αK .

    For large enough values of a and c it is possible to have arbitrary good approximation to fine structure constant. Note that the integers a and c cannot have common factors since this together with determinant condition a2-bc=1 would lead to contradiction.

For a pdf version see the chapter Miscellaneous Topics or the article What could be the counterpart of T-duality in TGD framework?.

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