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Towards M-Matrix

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Year 2015

Is non-associative physics and language possible only in many-sheeted space-time?

In Thinking Allowed Original there was very interesting link added by Ulla about the possibility of non- associative quantum mechanics.

Also I have been forced to consider this possibility.

  1. The 8-D imbedding space of TGD has octonionic tangent space structure and octonions are non-associative. Octonionic quantum theory however has serious mathematical difficulties since the operators of Hilbert space are by definition associative. The representation of say octonionic multiplication table by matrices is possible but is not faithful since it misses the associativity. More concretely, so called associators associated with triplets of representation matrices vanish. One should somehow transcend the standard quantum theory if one wants non-associative physics.
  2. Associativity therefore seems to be fundamental in quantum theory as we understand it recently. Associativity is indeed a fundamental and highly non-trivial constraint on the correlation functions of conformal field theories. In TGD framework classical physics is an exact part of quantum theory so that quantum classical correspondence suggests that associativity could play a highly non-trivial role in classical TGD.

    The conjecture is that associativity requirement fixes the dynamics of space-time sheets - preferred extremals of Kähler action - more or less uniquely. One can endow the tangent space of 8-D imbedding H=M4× CP2 space at given point with octonionic structure: the 8 tangent vectors of the tangent space basis obey octonionic multiplication table.

    Space-time realized as n-D surface in 8-D H must be either associative or co-associative: this depending on whether the tangent space basis or normal space basis is associative. The maximal dimension of space-time surface is predicted to be the observed dimension D=4 and tangent space or normal space allows a quaternionic basis.

  3. There are also other conjectures (see this) about what the preferred extremals of Kähler action defining space-time surfaces are.
    1. A very general conjecture states that strong form of holography allows to determine space-time surfaces from the knowledge of partonic 2-surfaces and 2-D string world sheets.
    2. Second conjecture involves quaternion analyticity and generalization of complex structure to quaternionic structure involving generalization of Cauchy-Riemann conditions.
    3. M8-M4× CP2 duality stating that space-time surfaces can be regarded as surface in either M8 or M4× CP2 is a further conjecture.
    4. Twistorial considerations select M4× CP2 as a completely unique choice since M4 and CP2 are the only spaces allowing twistor space with Kähler structure. The conjecture is that preferred extremals can be identified base spaces of 6-D sub-manifolds of the product CP3× SU(3)/U(1)× U(1) of twistor spaces of M4 and CP2 having the property that it makes sense to speak about induced twistor structure.
    The "super(optimistic)" conjecture is that all these conjectures are equivalent.
One must be of course very cautious in order to not draw too strong conclusions. Above one considers quantum physics at the level of single space-time sheet. What about many-sheeted space-time? Could non-associative physics emerge in TGD via many-sheeted space-time? To answer this question one must first understand what non-associativity means.
  1. In non-associative situation brackets matter. A(BC) is different from (AB)C. From schooldays or at least from the first year calculus course one recalls the algorithm: when calculating the expression involving brackets one first finds the innermost brackets and calculates what is inside them, then proceed to the next innermost brackets, etc... In computer programs the realization of the command sequences involving brackets is called parsing and compilers perform it. Parsing involves decomposition of program to modules calling modules calling.... Quite generally, the analysis of linguistic expressions involves parsing. Bells start to ring as one realizes that parsings form a hierarchy as also do the space-time sheets!
  2. More concretely, there is hierarchy of brackets and there is also a hierarchy of space-time sheets, perhaps labelled by p-adic primes. B and C inside brackets form (BC), something analogous to a bound state or chemical compound. In TGD this something could correspond to a "glueing" space-time sheets B and C at the same larger space-time sheet. More concretely, (BC) could correspond to braided pair of flux tubes B and C inside larger flux tube, whose presence is expressed as brackets (..). As one forms A(BC) one puts flux tube A and flux tube (BC) containing braided flux tubes B and C inside larger flux tube. For (AB)C flux one puts tube (AB) containing braided flux tubes A and B and tube C inside larger flux tube. The outcomes are obviously different. A
  3. Non-associativity in this sense would be a key signature of many-sheeted space-time. It should show itself in say molecular chemistry, where putting on same sheet could mean formation of chemical compound AB from A and B. Another highly interesting possibility is hierarchy of braids formed from flux tubes: braids can form braids, which in turn can form braids,... Flux tubes inside flux tubes inside... Maybe this more refined breaking of associativity could underly the possible non-associativity of biochemistry: biomolecules looking exactly the same would differ in subtle manner.
  4. What about quantum theory level? Non-associativity at the level of quantum theory could correspond to the breaking of associativity for the correlation functions of n fields if the fields are not associated with the same space-time sheet but to space-time sheets labelled by different p-adic primes. At QFT limit of TGD giving standard model and GRT the sheets are lumped together to single piece of Minkowski space and all physical effects making possible non-associativity in the proposed sense are lost. Language would be thus possible only in TGD Universe! My nasty alter ego wants to say now something - my sincere apologies: in superstring Universe communication of at least TGD has indeed turned out to be impossible! If superstringy universe allows communications at all, they must be uni-directional!
Non-associativity is an essentially linguistic phenomenon and relates therefore to cognition. p-Adic physics labelled by p-adic primes fusing with real physics to form adelic physics are identified as the physics of cognition in TGD framework.
  1. Could many-sheeted space-time of TGD provides the geometric realization of language like structures? Could sentences and more complex structures have many-sheeted space-time structures as geometrical correlates? p-Adic physics as physics of cognition would suggests that p-adic primes label the sheets in the parsing hierarchy. Could bio-chemistry with hierarchy of magnetic flux tubes added, realize the parsing hierarchies?
  2. DNA is a language and might provide a key example about parsing hierarchy. The mystery is that human DNA and DNAs of most simplest creatures do not differ much. Our cousins have almost identical DNA with us. Why do we differ so much? Could the number of parsing levels be the reason- p-adic primes labelling space-time sheets? Could our DNA language be much more structured than that of our cousins. At the level of concrete language the linguistic expressions of our cousin are indeed simple signals rather than extremely complex sentences of old-fashioned German professor forming a single lecture each. Could these parsing hierarchies realize themselves as braiding hierarchies of magnetic flux tubes physically and more abstractly as the parsing hierarchies of social structures. Indeed, I have proposed that the presence of collective levels of consciousness having hierarchy of magnetic bodies as a space-time correlates distinguishes us from our cousins so that this explanation is consistent with more quantitative one relying on language.
  3. I have also proposed that intronic portion of DNA is crucial for understanding why we differ so much from our cousins (see this and this). How does this view relate to the above proposal? In the simplest model for DNA as topological quantum computer introns would be connected by flux tubes to the lipids of nuclear and cell membranes. This would make possible topological quantum computations with the braiding of flux tubes defining the topological quantum computer program.

    Ordinary computer programs rely on computer language. Same should be true about quantum computer programs realized as braidings. Now the hierarchical structure of parsings would correspond to that of braidings: one would have braids, braids of braids, etc... This kind of structure is also directly visible as the multiply coiled structure of DNA. The braids beginning from the intronic portion of DNA would form braided flux tubes inside larger braided flux tubes inside.... defining the parsing of the topological quantum computer program.

    The higher the number of parsing levels, the higher the position in the evolutionary hierarchy. Each braiding would define one particular fundamental program module and taking this kind of braided flux tubes and braiding them would give a program calling these programs as sub-programs.

  4. The phonemes of language would have no meaning to us (at our level of self hierarchy) but the words formed by phonemes and involving at basic level the braiding of "phoneme flux tubes" would have. Sentences and their substructures would in turn involve braiding of "word flux tubes". Spoken language would correspond to a temporal sequence of braidings of flux tubes at various hierarchy levels.
  5. The difference between us and our cousins (or other organisms) would not be at the level of visible DNA but at the level of magnetic body. Magnetic bodies would serve as correlates also for social structures and associated collective levels of consciousness. The degree of braiding would define the level in the evolutionary hierarchy. This is of course the basic vision of TGD inspired quantum biology and quantum bio-chemistry in which the double formed by organism and environment is completed to a triple by adding the magnetic body.
p-Adic hierarchy is not the only hierarchy in TGD Universe: there is also the hierarchy of Planck constants heff=n× h giving rise to a hierarchy of intelligences. What is the relationship between these hierarchies?
  1. I have proposed that speech and music are fundamental aspects of conscious intelligence and that DNA realizes what I call bio-harmonies in quite concrete sense (see this and this): DNA codons would correspond to 3-chords. DNA would both talk and sing. Both language and music are highly structured. Could the relation of heff hierarchy to language be same as the relation of music to speech?
  2. Are both musical and linguistic parsing hierarchies present? Are they somehow dual? What does parsing mean for music? How musical sounds could combine to form the analog of two braided strand? Depending on situation we hear music both as separate notes and as chords as separate notes fuse in our mind to a larger unit like phonemes fuse to a word.

    Could chords played by single instrument correspond to braidings of flux tubes at the same level? Could the duality between linguistic and musical intelligence (analogous to that between function and its Fourier transform) be very concrete and detailed and reflect itself also as the possibility to interpret DNA codons both as three letter words and as 3-chords (see this)?

See the new chapter Is Non-Associative Physics and Language Possible Only in Many-Sheeted Space-Time?.

Does Riemann Zeta Code for Generic Coupling Constant Evolution?

The understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During years I have made several attempts to understand coupling evolution.

  1. The first idea dates back to the discovery of WCW Kähler geometry defined by Kähler function defined by Kähler action (this happened around 1990) (see this). The only free parameter of the theory is Kähler coupling strength αK analogous to temperature parameter αK postulated to be is analogous to critical temperature. Whether only single value or entire spectrum of of values αK is possible, remained an open question.

    About decade ago I realized that Kähler action is complex receiving a real contribution from space-time regions of Euclidian signature of metric and imaginary contribution from the Minkoswkian regions. Euclidian region would give Kähler function and Minkowskian regions analog of QFT action of path integral approach defining also Morse function. Zero energy ontology (ZEO) (see this) led to the interpretation of quantum TGD as complex square root of thermodynamics so that the vacuum functional as exponent of Kähler action could be identified as a complex square root of the ordinary partition function. Kähler function would correspond to the real contribution Kähler action from Euclidian space-time regions. This led to ask whether also Kähler coupling strength might be complex: in analogy with the complexification of gauge coupling strength in theories allowing magnetic monopoles. Complex αK could allow to explain CP breaking. I proposed that instanton term also reducing to Chern-Simons term could be behind CP breaking

  2. p-Adic mass calculations for 2 decades ago (see this) inspired the idea that length scale evolution is discretized so that the real version of p-adic coupling constant would have discrete set of values labelled by p-adic primes. The simple working hypothesis was that Kähler coupling strength is renormalization group (RG) invariant and only the weak and color coupling strengths depend on the p-adic length scale. The alternative ad hoc hypothesis considered was that gravitational constant is RG invariant. I made several number theoretically motivated ad hoc guesses about coupling constant evolution, in particular a guess for the formula for gravitational coupling in terms of Kähler coupling strength, action for CP2 type vacuum extremal, p-adic length scale as dimensional quantity (see this). Needless to say these attempts were premature and a hoc.
  3. The vision about hierarchy of Planck constants heff=n× h and the connection heff= hgr= GMm/v0, where v0<c=1 has dimensions of velocity (see this>) forced to consider very seriously the hypothesis that Kähler coupling strength has a spectrum of values in one-one correspondence with p-adic length scales. A separate coupling constant evolution associated with heff induced by αK∝ 1/hbareff ∝ 1/n looks natural and was motivated by the idea that Nature is theoretician friendly: when the situation becomes non-perturbative, Mother Nature comes in rescue and an heff increasing phase transition makes the situation perturbative again.

    Quite recently the number theoretic interpretation of coupling constant evolution (see this> or this in terms of a hierarchy of algebraic extensions of rational numbers inducing those of p-adic number fields encouraged to think that 1/αK has spectrum labelled by primes and values of heff. Two coupling constant evolutions suggest themselves: they could be assigned to length scales and angles which are in p-adic sectors necessarily discretized and describable using only algebraic extensions involve roots of unity replacing angles with discrete phases.

  4. Few years ago the relationship of TGD and GRT was finally understood (see this>) . GRT space-time is obtained as an approximation as the sheets of the many-sheeted space-time of TGD are replaced with single region of space-time. The gravitational and gauge potential of sheets add together so that linear superposition corresponds to set theoretic union geometrically. This forced to consider the possibility that gauge coupling evolution takes place only at the level of the QFT approximation and αK has only single value. This is nice but if true, one does not have much to say about the evolution of gauge coupling strengths.
  5. The analogy of Riemann zeta function with the partition function of complex square root of thermodynamics suggests that the zeros of zeta have interpretation as inverses of complex temperatures s=1/β. Also 1/αK is analogous to temperature. This led to a radical idea to be discussed in detail in the sequel.

    Could the spectrum of 1/αK reduce to that for the zeros of Riemann zeta or - more plausibly - to the spectrum of poles of fermionic zeta ζF(ks)= ζ(ks)/ζ(2ks) giving for k=1/2 poles as zeros of zeta and as point s=2? ζF is motivated by the fact that fermions are the only fundamental particles in TGD and by the fact that poles of the partition function are naturally associated with quantum criticality whereas the vanishing of ζ and varying sign allow no natural physical interpretation.

    The poles of ζF(s/2) define the spectrum of 1/αK and correspond to zeros of ζ(s) and to the pole of ζ(s/2) at s=2. The trivial poles for s=2n, n=1,2,.. correspond naturally to the values of 1/αK for different values of heff=n× h with n even integer. Complex poles would correspond to ordinary QFT coupling constant evolution. The zeros of zeta in increasing order would correspond to p-adic primes in increasing order and UV limit to smallest value of poles at critical line. One can distinguish the pole s=2 as extreme UV limit at which QFT approximation fails totally. CP2 length scale indeed corresponds to GUT scale.

  6. One can test this hypothesis. 1/αK corresponds to the electroweak U(1) coupling strength so that the identification 1/αK= 1/αU(1) makes sense. One also knows a lot about the evolutions of 1/αU(1) and of electromagnetic coupling strength 1/αem= 1/[cos2WU(1). What does this predict?

    It turns out that at p-adic length scale k=131 (p≈ 2k by p-adic length scale hypothesis, which now can be understood number theoretically (see this ) fine structure constant is predicted with .7 per cent accuracy if Weinberg angle is assumed to have its value at atomic scale! It is difficult to believe that this could be a mere accident because also the prediction evolution of αU(1) is correct qualitatively. Note however that for k=127 labelling electron one can reproduce fine structure constant with Weinberg angle deviating about 10 per cent from the measured value of Weinberg angle. Both models will be considered.

  7. What about the evolution of weak, color and gravitational coupling strengths? Quantum criticality suggests that the evolution of these couplings strengths is universal and independent of the details of the dynamics. Since one must be able to compare various evolutions and combine them together, the only possibility seems to be that the spectra of gauge coupling strengths are given by the poles of ζF(w) but with argument w=w(s) obtained by a global conformal transformation of upper half plane - that is Möbius transformation (see this) with real coefficients (element of GL(2,R)) so that one as ζF((as+b)/(cs+d)). Rather general arguments force it to be and element of GL(2,Q), GL(2,Z) or maybe even SL(2,Z) (ad-bc=1) satisfying additional constraints. Since TGD predicts several scaled variants of weak and color interactions, these copies could be perhaps parameterized by some elements of SL(2,Z) and by a scaling factor K.

    Could one understand the general qualitative features of color and weak coupling contant evolutions from the properties of corresponding Möbius transformation? At the critical line there can be no poles or zeros but could asymptotic freedom be assigned with a pole of cs+d and color confinement with the zero of as+b at real axes? Pole makes sense only if Kähler action for the preferred extremal vanishes. Vanishing can occur and does so for massless extremals characterizing conformally invariant phase. For zero of as+b vacuum function would be equal to one unless Kähler action is allowed to be infinite: does this make sense?. One can however hope that the values of parameters allow to distinguish between weak and color interactions. It is certainly possible to get an idea about the values of the parameters of the transformation and one ends up with a general model predicting the entire electroweak coupling constant evolution successfully.

To sum up, the big idea is the identification of the spectra of coupling constant strengths as poles of ζF((as+b/)(cs+d)) identified as a complex square root of partition function with motivation coming from ZEO, quantum criticality, and super-conformal symmetry; the discretization of the RG flow made possible by the p-adic length scale hypothesis p≈ kk, k prime; and the assignment of complex zeros of ζ with p-adic primes in increasing order. These assumptions reduce the coupling constant evolution to four real rational or integer valued parameters (a,b,c,d). One can say that one of the greatest challenges of TGD has been overcome.

See the new chapter Does Riemann Zeta Code for Generic Coupling Constant Evolution? or the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?.

Algebraic universality and the value of Kähler coupling strength

With the development of the vision about number theoretically universal view about functional integration in WCW , a concrete vision about the exponent of Kähler action in Euclidian and Minkowskian space-time regions. The basic requirement is that exponent of Kähler action belongs to an algebraic extension of rationals and therefore to that of p-adic numbers and does not depend on the ordinary p-adic numbers at all - this at least for sufficiently large primes p. Functional integral would reduce in Euclidian regions to a sum over maxima since the troublesome Gaussian determinants that could spoil number theoretic universality are cancelled by the metric determinant for WCW.

The adelically exceptional properties of Neper number e, Kähler metric of WCW, and strong form of holography posing extremely strong constraints on preferred extremals, could make this possible. In Minkowskian regions the exponent of imaginary Kähler action would be root of unity. In Euclidian space-time regions expressible as power of some root of e which is is unique in sense that ep is ordinary p-adic number so that e is p-adically an algebraic number - p:th root of ep.

These conditions give conditions on Kähler coupling strength αK= gK2/4π (hbar=1)) identifiable as an analog of critical temperature. Quantum criticality of TGD would thus make possible number theoretical universality (or vice versa).

  1. In Euclidian regions the natural starting point is CP2 vacuum extremal for which the maximum value of Kähler action is

    SK= π2/2gK2= π/8αK .

    The condition reads SK=q =m/n if one allows roots of e in the extension. If one requires minimal extension of involving only e and its powers one would have SK=n. One obtains

    1/αK= 8q/π ,

    where the rational q=m/n can also reduce to integer. One cannot exclude the possibiity that q depends on the algebraic extension of rationals defining the adele in question.

    For CP2 type extremals the value of p-adic prime should be larger than pmin=53. One can consider a situation in which large number of CP2 type vacuum extremals contribute and in this case the condition would be more stringent. The condition that the action for CP2 extremal is smaller than 2 gives

    1/αK≤ 16/π ≈ 5.09 .

    It seems there is lower bound for the p-adic prime assignable to a given space-time surface inside CD suggesting that p-adic prime is larger than 53× N, where N is particle number.

    This bound has not practical significance. In condensed matter particle number is proportional to (L/a)3 - the volume divided by atomic volume. On basis p-adic mass calculations p-adic prime can be estimated to be of order (L/R)2. Here a is atomic size of about 10 Angstroms and R CP2 "radius. Using R≈ 104 LPlanck this gives as upper bound for the size L of condensed matter blob a completely super-astronomical distance L≤ a3/R2 ∼ 1025 ly to be compared with the distance of about 1010 ly travelled by light during the lifetime of the Universe. For a blackhole of radius rS= 2GM with p∼ (2GM/R)2 and consisting of particles with mass above M≈ hbar/R one would obtain the rough estimate M>(27/2)× 10-12mPlanck ∼ 13.5× 103 TeV trivially satisfied.

  2. The physically motivated expectation from earlier arguments - not necessarily consistent with the recent ones - is that the value αK is quite near to fine structure constant at electron length scale: αK≈ αem≈ 137.035999074(44).

    The latter condition gives n=54=2× 33 and 1/αK≈ 137.51. The deviation from the fine structure constant is Δ α/α= 3× 10-3 -- .3 per cent. For n=53 one obtains 1/αK= 134.96 with error of 1.5 per cent. For n=55 one obtains 1/αK= 150.06 with error of 2.2 per cent. Is the relatively good prediction could be a mere accident or there is something deeper involved?

What about Minkowskian regions? It is difficult to say anything definite. For cosmic string like objects the action is non-vanishing but proportional to the area A of the string like object and the conditions would give quantization of the area. The area of geodesic sphere of CP2 is proportional to π. If the value of gK is same for Minkowskian and Euclidian regions, gK2∝ π2 implies SK ∝ A/R2π so that A/R2∝ π2 is required.

This approach leads to different algebraic structure of αK than the earlier arguments.

  1. αK is rational multiple of π so that gK2 is proportional to π2. At the level of quantum TGD the theory is completely integrable by the definition of WCW integration(!) and there are no radiative corrections in WCW integration. Hence αK does not appear in vertices and therefore does not produce any problems in p-adic sectors.
  2. This approach is consistent with the proposed formula relating gravitational constant and p-adic length scale. G/Lp2 for p=M127 would be rational power of e now and number theoretically universally. A good guess is that G does not depend on p. As found this could be achieved also if the volume of CP2 type extremal depends on p so that the formula holds for all primes. αK could also depend on algebraic extension of rationals to guarantee the independence of G on p. Note that preferred p-adic primes correspond to ramified primes of the extension so that extensions are labelled by collections of ramified primes, and the ramimified prime corresponding to gravitonic space-time sheets should appear in the formula for G/Lp2.
  3. Also the speculative scenario for coupling constant evolution could remain as such. Could the p-adic coupling constant evolution for the gauge coupling strengths be due to the breaking of number theoretical universality bringing in dependence on p? This would require mapping of p-adic coupling strength to their real counterparts and the variant of canonical identification used is not unique.
  4. A more attractive possibility is that coupling constants are algebraically universal (no dependence on number field). Even the value of αK, although number theoretically universal, could depend on the algebraic extension of rationals defining the adele. In this case coupling constant evolution would reflect the evolution assignable to the increasing complexity of algebraic extension of rationals. The dependence of coupling constants on p-adic prime would be induced by the fact that so called ramified primes are physically favored and characterize the algebraic extension of rationals used.
  5. One must also remember that the running coupling constants are associated with QFT limit of TGD obtained by lumping the sheets of many-sheeted space-time to single region of Minkowski space. Coupling constant evolution would emerge at this limit. Whether this evolution reflects number theoretical evolution as function of algebraic extension of rationals, is an interesting question.

See the chapter Coupling Constant Evolution in Quantum TGD and the chapter Unified Number Theoretic Vision of "Physics as Generalized Number Theory" or the article Could one realize number theoretical universality for functional integral?.

Field equations as conservation laws, Frobenius integrability conditions, and a connection with quaternion analyticity

The following represents qualitative picture of field equations of TGD trying to emphasize the physical aspects. What is new is the discussion of the possibility that Frobenius integrability conditions are satisfied and correspond to quaternion analyticity.

  1. Kähler action is Maxwell action for induced Kähler form and metric expressible in terms of imbedding space coordinates and their gradients. Field equations reduce to those for imbedding space coordinates defining the primary dynamical variables. By GCI only four of them are independent dynamical variables analogous to classical fields.
  2. The solution of field equations can be interpreted as a section in fiber bundle. In TGD the fiber bundle is just the Cartesian product X4× CD× CP2 of space-time surface X4 and causal diamond CD× CP2. CD is the intersection of future and past directed light-cones having two light-like boundaries, which are cone-like pieces of light-boundary δ M4+/-× CP2. Space-time surface serves as base space and CD× CP2 as fiber. Bundle projection Π is the projection to the factor X4. Section corresponds to the map x→ hk(x) giving imbedding space coordinates as functions of space-time coordinates. Bundle structure is now trivial and rather formal.

    By GCI one could also take suitably chosen 4 coordinates of CD× CP2 as space-time coordinates, and identify CD× CP2 as the fiber bundle. The choice of the base space depends on the character of space-time surface. For instance CD, CP2 or M2× S2 (S2 a geodesic sphere of CP2), could define the base space. The bundle projection would be projection from CD× CP2 to the base space. Now the fiber bundle structure can be non-trivial and make sense only in some space-time region with same base space.

  3. The field equations derived from Kähler action must be satisfied. Even more: one must have a preferred extremal of Kähler action. One poses boundary conditions at the 3-D ends of space-time surfaces and at the light-like boundaries of CD× CP2.

    One can fix the values of conserved Noether charges at the ends of CD (total charges are same) and require that the Noether charges associated with a sub-algebra of super-symplectic algebra isomorphic to it and having conformal weights coming as n-ples of those for the entire algebra, vanish. This would realize the effective 2-dimensionality required by SH. One must pose boundary conditions also at the light-like partonic orbits. So called weak form of electric-magnetic duality is at least part of these boundary conditions.

    It seems that one must restrict the conformal weights of the entire algebra to be non-negative r≥ 0 and those of subalgebra to be positive: mn>0. The condition that also the commutators of sub-algebra generators with those of the entire algebra give rise to vanishing Noether charges implies that all algebra generators with conformal weight m≥ n vanish so the dynamical algebra becomes effectively finite-dimensional. This condition generalizes to the action of super-symplectic algebra generators to physical states.

    M4 time coordinate cannot have vanishing time derivative dm0/dt so that four-momentum is non-vanishing for non-vacuum extremals. For CP2 coordinates time derivatives dsk/dt can vanish and for space-like Minkowski coordinates dmi/dt can be assumed to be non-vanishing if M4 projection is 4-dimensional. For CP2 coordinates dsk/dt=0 implies the vanishing of electric parts of induced gauge fields. The non-vacuum extremals with the largest conformal gauge symmetry (very small n) would correspond to cosmic string solutions for which induced gauge fields have only magnetic parts. As n increases, also electric parts are generated. Situation becomes increasingly dynamical as conformal gauge symmetry is reduced and dynamical conformal symmetry increases.

  4. The field equations involve besides imbedding space coordinates hk also their partial derivatives up to second order. Induced Kähler form and metric involve first partial derivatives ∂αhk and second fundamental form appearing in field equations involves second order partial derivatives ∂αβhk.

    Field equations are hydrodynamical, in other worlds represent conservation laws for the Noether currents associated with the isometries of M4× CP2. By GCI there are only 4 independent dynamical variables so that the conservation of m≤ 4 isometry currents is enough if chosen to be independent. The dimension m of the tangent space spanned by the conserved currents can be smaller than 4. For vacuum extremals one has m= 0 and for massless extremals (MEs) m= 1! The conservation of these currents can be also interpreted as an existence of m≤ 4 closed 3-forms defined by the duals of these currents.

  5. The hydrodynamical picture suggests that in some situations it might be possible to assign to the conserved currents flow lines of currents even globally. They would define m≤ 4 global coordinates for some subset of conserved currents (4+8 for four-momentum and color quantum numbers). Without additional conditions the individual flow lines are well-defined but do not organize to a coherent hydrodynamic flow but are more like orbits of randomly moving gas particles. To achieve global flow the flow lines must satisfy the condition dφA/dxμ= kABJBμ or dφA= kABJB so that one can special of 3-D family of flow lines parallel to kABJB at each point - I have considered this kind of possibility in detail earlier but the treatment is not so general as in the recent case.

    Frobenius integrability conditions follow from the condition d2φA=0= dkAB∧ JB+ kABdJB=0 and implies that dJB is in the ideal of exterior algebra generated by the JA appearing in kABJB. If Frobenius conditions are satisfied, the field equations can define coordinates for which the coordinate lines are along the basis elements for a sub-space of at most 4-D space defined by conserved currents. Of course, the possibility that for preferred extremals there exists m≤ 4 conserved currents satisfying integrability conditions is only a conjecture.

    It is quite possible to have m<4. For instance for vacuum extremals the currents vanish identically For MEs various currents are parallel and light-like so that only single light-like coordinate can be defined globally as flow lines. For cosmic strings (cartesian products of minimal surfaces X2 in M4 and geodesic spheres S2 in CP2 4 independent currents exist). This is expected to be true also for the deformations of cosmic strings defining magnetic flux tubes.

  6. Cauchy-Riemann conditions in 2-D situation represent a special case of Frobenius conditions. Now the gradients of real and imaginary parts of complex function w=w(z)= u+iv define two conserved currents by Laplace equations. In TGD isometry currents would be gradients apart from scalar function multipliers and one would have generalization of C-R conditions. In citeallb/prefextremals,twistorstory I have considered the possibility that the generalization of Cauchy-Riemann-Fuerter conditions could define quaternion analyticity - having many non-equivalent variants - as a defining property of preferred extremals. The integrability conditions for the isometry currents would be the natural physical formulation of CRF conditions. Different variants of CRF conditions would correspond to varying number of independent conserved isometry currents.
  7. The problem caused by GCI is that there is infinite number of coordinate choices. How to pick a physically preferred coordinate system? One possible manner to do this is to use coordinates for the projection of space-time surface to some preferred sub-space of imbedding - geodesic manifold is an excellent choice. Only M1× X3 geodesic manifolds are not possible but these correspond to vacuum extremals.

    One could also consider a philosophical principle behind integrability. The variational principle itself could give rise to at least some preferred space-time coordinates in the same manner as TGD based quantum physics would realize finite measurement resolution in terms of inclusions of HFFs in terms of hierarchy of quantum criticalities and fermionic strings connecting partonic 2-surfaces. Frobenius integrability of the isometry currents would define some preferred coordinates. Their number need not be the maximal four however.

    For instance, for massless extremals only light-like coordinate corresponding to the light-like momentum is obtained. To this one can however assign another local light-like coordinate uniquely to obtain integrable distribution of planes M2. The solution ansatz however defines directly an integrable choice of two pairs of coordinates at imbedding space level usable also as space-time coordinates - light-like local direction defining local plane M2 and polarization direction defining a local plane E2. These choices define integrable distributions of orthogonal planes and local hypercomplex and complex coordinates. Pair of analogs of C-R equations is the outcome. I have called these coordinates Hamilton-Jacobi coordinates for M4.

  8. This picture allows to consider a generalization of the notion of solution of field equation to that of integral manifold. If the number of independent isometry currents is smaller than 4 (possibly locally) and the integrability conditions hold true, lower-dimensional sub-manifolds of space-time surface define integral manifolds as kind of lower-dimensional effective solutions. Genuinely lower-dimensional solutions would of course have vanishing (g41/2) and vanishing Kähler action.

    String world sheets can be regarded as 2-D integral surfaces. Charged (possibly all) weak boson gauge fields vanish at them since otherwise the electromagnetic charge for spinors would not be well-defined. These conditions force string world sheets to be 2-D in the generic case. In special case 4-D space-time region as a whole can satisfy these conditions. Well-definedness of Kähler-Dirac equation demands that the isometry currents of Kähler action flow along these string world sheets so that one has integral manifold. The integrability conditions would allow 2<m≤ n integrable flows outside the string world sheets, and at string world sheets one or two isometry currents would vanish so that the flows would give rise 2-D independent sub-flow.

  9. The method of characteristics is used to solve hyperbolic partial differential equations by reducing them to ordinary differential equations. The (say 4-D) surface representing the solution in the field space has a foliation using 1-D characteristics. The method is especially simple for linear equations but can work also in the non-linear case. For instance, the expansion of wave front can be described in terms of characteristics representing light rays. It can happen that two characteristics intersect and a singularity results. This gives rise to physical phenomena like caustics and shock waves.

    In TGD framework the flow lines for a given isometry current in the case of an integrable flow would be analogous to characteristics, and one could also have purely geometric counterparts of shockwaves and caustics. The light-like orbits of partonic 2-surface at which the signature of the induced metric changes from Minkowskian to Euclidian might be seen as an example about the analog of wave front in induced geometry. These surfaces serve as carriers of fermion lines in generalized Feynman diagrams. Could one see the particle vertices at which the 4-D space-time surfaces intersect along their ends as analogs of intersections of characteristics - kind of caustics? At these 3-surfaces the isometry currents should be continuous although the space-time surface has "edge".

For details see the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" or the article Could One Define Dynamical Homotopy Groups in WCW?.

Could one define dynamical homotopy groups in WCW?

I learned that Agostino Prastaro has done highly interesting work with partial differential equations, also those assignable to geometric variational principles such as Kähler action in TGD. I do not understand the mathematical details but the key idea is a simple and elegant generalization of Thom's cobordism theory, and it is difficult to avoid the idea that the application of Prastaro's idea might provide insights about the preferred extremals, whose identification is now on rather firm basis.

One could also consider a definition of what one might call dynamical homotopy groups as a genuine characteristics of WCW topology. The first prediction is that the values of conserved classical Noether charges correspond to disjoint components of WCW. Could the natural topology in the parameter space of Noether charges zero modes of WCW metric) be p-adic and realize adelic physics at the level of WCW? An analogous conjecture was made on basis of spin glass analogy long time ago. Second surprise is that the only the 6 lowest dynamical homotopy/homology groups of WCW would be non-trivial. The Kähler structure of WCW suggets that only Π0, Π2, and Π4 are non-trivial.

The interpretation of the analog of Π1 as deformations of generalized Feynman diagrams with elementary cobordism snipping away a loop as a move leaving scattering amplitude invariant conforms with the number theoretic vision about scattering amplitude as a representation for a sequence of algebraic operation can be always reduced to a tree diagram. TGD would be indeed topological QFT: only the dynamical topology would matter.

See the chapter Recent View about Kähler Geometry and Spin Structure of WCW or the article Could one define dynamical homotopy groups in WCW?.

The relation between U- and M-matrices

U- and M-matrices are key objects in zero energy ontology (ZEO). M-matrix for large causal diamonds (CDs) is the counterpart of thermal S-matrix and proportional to scale dependent S-matrix: the dependence on the scale of CD characterized by integer is S(n)=Sn in accordance with the idea that S corresponds to the counterpart of ordinary unitary time evolution operator. U-matrix characterizes the time evolution as dispersion in the moduli space of CDs tending to increase the size of CD and giving rise to the experience arrow of geometric time and also to the notion of self in TGD insprired theory of consciousness.

The original view about the relationship between U- and M-matrices was a purely formal guess: M-matrices would define the orthonormal rows of U-matrix. This guess is not correct physically and one must consider in detail what U-matrix really means.

  1. First about the geometry of CD. The boundaries of CD will be called passive and active: passive boundary correspond to the boundary at which repeated state function reductions take place and give rise to a sequence of unitary time evolutions U followed by localization in the moduli of CD each. Active boundary corresponds to the boundary for which U induces delocalization and modifies the states at it.

    The moduli space for the CDs consists of a discrete subgroup of scalings for the size of CD characterized by the proper time distance between the tips and the sub-group of Lorentz boosts leaving passive boundary and its tip invariant and acting on the active boundary only. This group is assumed to be represented unitarily by matrices Λ forming the same group for all values of n.

    The proper time distance between the tips of CDs is quantized as integer multiples of the minimal distance defined by CP2 time: T= nT0. Also in quantum jump in which the size scale n of CD increases the increase corresponds to integer multiple of T0. Using the logarithm of proper time, one can interpret this in terms of a scaling parametrized by an integer. The possibility to interpret proper time translation as a scaling is essential for having a manifest Lorentz invariance: the ordinary definition of S-matrix introduces preferred rest system.

  2. The physical interpretation would be roughly as follows. M-matrix for a given CD codes for the physics as we usually understand it. M-matrix is product of square root of density matrix and S-matrix depending on the size scale of CD and is the analog of thermal S-matrix. State function at the opposite boundary of CD corresponds to what happens in the state function reduction in particle physics experiments. The repeated state function reductions at same boundary of CD correspond to TGD version of Zeno effect crucial for understanding consciousness. Unitary U-matrix describes the time evolution zero energy states due to the increase of the size scale of CD (at least in statistical sense). This process is dispersion in the moduli space of CDs: all possible scalings are allowed and localization in the space of moduli of CD localizes the active boundary of CD after each unitary evolution.
In the following I will proceed by making questions. One ends up to formulas allowing to understand the architecture of U-matrix and to reduce its construction to that for S-matrix having interpretation as exponential of the generator L-1 of the Virasoro algebra associated with the super-symplectic algebra.

What one can say about M-matrices?

  1. The first thing to be kept in mind is that M-matrices act in the space of zero energy states rather than in the space of positive or negative energy states. For a given CD M-matrices are products of hermitian square roots of hermitian density matrices acting in the space of zero energy states and universal unitary S-matrix S(CD) acting on states at the active end of CD (this is also very important to notice) depending on the scale of CD:

    Mi=HiS(CD) .

    Hi is hermitian square root of density matrix and the matrices Hi must be orthogonal for given CD from the orthonormality of zero energy states associated with the same CD. The zero energy states associated with different CDs are not orthogonal and this makes the unitary time evolution operator U non-trivial.

  2. Could quantum measurement be seen as a measurement of the observables defined by the Hermitian generators Hi? This is not quite clear since their action is on zero energy states. One might actually argue that the action of this kind of observables on zero energy states does not affect their vanishing net quantum numbers. This suggests that Hi carry no net quantum numbers and belong to the Cartan algebra. The action of S is restricted at the active boundary of CD and therefore it does not commute with Hi unless the action is in a separate tensor factor. Therefore the idea that S would be an exponential of generators Hi and thus commute with them so that Hi would correspond to sub-spaces remaining invariant under S acting unitarily inside them does not make sense.
  3. In TGD framework symplectic algebra actings as isometries of WCW is analogous to a Kac-Moody algebra with finite-dimensional Lie-algebra replaced with the infinite-dimensional symplectic algebra with elements characterized by conformal weights. There is a temptation to think that the Hi could be seen as a representation for this algebra or its sub-algebra. This algebra allows an infinite fractal hierarchy of sub-algebras of the super-symplectic algebra isomorphic to the full algebra and with conformal weights coming as n-ples of those for the full algebra. In the proposed realization of quantum criticality the elements of the sub-algebra characterized by n act as a gauge algebra. An interesting question is whether this sub-algebra is involved with the realization of M-matrices for CD with size scale n. The natural expectation is that n defines a cutoff for conformal weights relating to finite measurement resolution.
How does the size scale of CD affect M-matrices?

  1. In standard quantum field theory S-matrix represents time translation. The obvious generalization is that now scaling characterized by integer n is represented by a unitary S-matrix that is as n:th power of some unitary matrix S assignable to a CD with minimal size: S(CD)= Sn. S(CD) is a discrete analog of the ordinary unitary time evolution operator with n replacing the continuous time parameter.
  2. One can see M-matrices also as a generalization of Kac-Moody type algebra. Also this suggests S(CD)= Sn, where S is the S-matrix associated with the minimal CD. S becomes representative of phase exp(iφ). The inner product between CDs of different size scales can n1 and n2 can be defined as

    ⟨ Mi(m), Mj(n)⟩ =Tr(S-m• HiHj• Sn) × θ(n-m) ,

    θ(n)=1 for n≥ 0 , θ (n)=0 for n<0 .

    Here I have denoted the action of S-matrix at the active end of CD by "•" in order to distinguish it from the action of matrices on zero energy states which could be seen as belonging to the tensor product of states at active and passive boundary.

    It turns out that unitarity conditions for U-matrix are invariant under the translations of n if one assumes that the transitions obey strict arrow of time expressed by nj-ni≥ 0. This simplifies dramatically unitarity conditions. This gives orthonormality for M-matrices associated with identical CDs. This inner product could be used to identify U-matrix.

  3. How do the discrete Lorentz boosts affecting the moduli for CD with a fixed passive boundary affect the M-matrices? The natural assumption is that the discrete Lorentz group is represented by unitary matrices λ: the matrices Mi are transformed to Mi•λ for a given Lorentz boost acting on states at active boundary only.

    One cannot completely exclude the possibility that S acts unitarily on zero energy states. In this case the scaling would be interpreted as acting on zero energy states rather than those at active boundary only. The zero energy state basis defined by Mi would depend on the size scale of CD in more complex manner. This would not affect the above formulas except by dropping away the "•".

Unitary U must characterize the transitions in which the moduli of the active boundary of causal diamond (CD) change and also states at the active boundary (paired with unchanging states at the passive boundary) change. The arrow of the experienced flow of time emerges during the period as state function reductions take place to the fixed ("passive") boundary of CD and do not affect the states at it. Note that these states form correlated pairs with the changing states at the active boundary. The physically motivated question is whether the arrow of time emerges statistically from the fact that the size of CD tends to increase in average sense in repeated state function reductions or whether the arrow of geometric time is strict. It turns out that unitarity conditions simplify dramatically if the arrow of time is strict.

What can one say about U-matrix?

  1. Just from the basic definitions the elements of a unitary matrix, the elements of U are between zero energy states (M-matrices) between two CDs with possibly different moduli of the active boundary. Given matrix element of U should be proportional to an inner product of two M-matrices associated with these CDs. The obvious guess is as the inner product between M-matrices

    Uijm,n= ⟨Mi(m,λ1), Mj(n,λ2)⟩

    =Tr(λ1 S-m• HiHj• Sn λ2)

    =Tr(S-m• HiHj • Sn λ2λ1-1)θ(n-m) .

    Here the usual properties of the trace are assumed. The justification is that the operators acting at the active boundary of CD are special case of operators acting non-trivially at both boundaries.

  2. Unitarity conditions must be satisfied. These conditions relate S and the hermitian generators Hi serving as square roots of density matrices. Unitarity conditions UU=UU=1 is defined in the space of zero energy states and read as

    j1n1 Uij1mn1(U)j1jn1n = δi,jδm,nδλ12

    To simplify the situation let us make the plausible hypothesis contribution of Lorentz boosts in unitary conditions is trivial by the unitarity of the representation of discrete boosts and the independence on n.

  3. In the remaining degrees of freedom one would have

    j1,k≥ Max(0,n-m) Tr(Sk• HiHj1) Tr(Hj1Hj• Sn-m-k)= δi,jδm,n .

    The condition k≥ Max(0,n-m) reflects the assumption about a strict arrow of time and implies that unitarity conditions are invariant under the proper time translation (n,m)→ (n+r,m+r). Without this condition n back-wards translations (or rather scalings) to the direction of geometric past would be possible for CDs of size scale n and this would break the translational invariance and it would be very difficult to see how unitarity could be achieved. Stating it in a general manner: time translations act as semigroup rather than group.

  4. Irreversibility reduces dramatically the number of the conditions. Despite this their number is infinite and correlates the Hermitian basis and the unitary matrix S. There is an obvious analogy with a Kac-Moody algebra at circle with S replacing the phase factor exp(inφ) and Hi replacing the finite-dimensional Lie-algebra. The conditions could be seen as analogs for the orthogonality conditions for the inner product. The unitarity condition for the analog situation would involve phases exp(ikφ1)↔ Sk and exp(i(n-m-k)φ2)↔ Sn-m-k and trace would correspond to integration ∫ dφ1 over φ1 in accordance with the basic idea of non-commutative geometry that trace corresponds to integral. The integration of φi would give δk,0 and δm,n. Hence there are hopes that the conditions might be satisfied. There is however a clear distinction to the Kac-Moody case since Sn does not in general act in the orthogonal complement of the space spanned by Hi.
  5. The idea about reduction of the action of S to a phase multiplication is highly attractive and one could consider the possibility that the basis of Hi can be chosen in such a manner that Hi are eigenstates of of S. This would reduce the unitarity constraint to a form in which the summation over k can be separated from the summation over j1.

    k≥ Max(0,n-m) exp(iksi-(n-m-k)sj)∑j1Tr(HiHj1) Tr(Hj1Hj)= δi,jδm,n .

    The summation over k should gives a factor proportional to δsi,sj. If the correspondence between Hi and eigenvalues si is one-to-one, one obtains something proportional to δ (i,j) apart from a normalization factor. Using the orthonormality Tr(HiHj)=δi,j one obtains for the left hand side of the unitarity condition

    exp(isi(n-m)) ∑j1Tr(HiHj1) Tr(Hj1Hj)= exp(isi(n-m)) δi,j .

    Clearly, the phase factor exp(isi(n-m)) is the problem. One should have Kronecker delta δm,n instead. One should obtain behavior resembling Kac-Moody generators. Hi should be analogs of Kac-Moody generators and include the analog of a phase factor coming visible by the action of S.

It seems that the simple picture is not quite correct yet. One should obtain somehow an integration over angle in order to obtain Kronecker delta.

  1. A generalization based on replacement of real numbers with function field on circle suggests itself. The idea is to the identify eigenvalues of generalized Hermitian/unitary operators as Hermitian/unitary operators with a spectrum of eigenvalues, which can be continuous. In the recent case S would have as eigenvalues functions λi(φ) = exp(isiφ). For a discretized version φ would have has discrete spectrum φ(n)= 2π k/n. The spectrum of λi would have n as cutoff. Trace operation would include integration over φ and one would have analogs of Kac-Moody generators on circle.
  2. One possible interpretation for φ is as an angle parameter associated with a fermionic string connecting partonic 2-surface. For the super-symplectic generators suitable normalized radial light-like coordinate rM of the light-cone boundary (containing boundary of CD) would be the counterpart of angle variable if periodic boundary conditions are assumed.

    The eigenvalues could have interpretation as analogs of conformal weights. Usually conformal weights are real and integer valued and in this case it is necessary to have generalization of the notion of eigenvalues since otherwise the exponentials exp(isi) would be trivial. In the case of super-symplectic algebra I have proposed that the generating elements of the algebra have conformal weights given by the zeros of Riemann zeta. The spectrum of conformal weights for the generators would consist of linear combinations of the zeros of zeta with integer coefficients. The imaginary parts of the conformal weights could appear as eigenvalues of S.

  3. It is best to return to the definition of the U-matrix element to check whether the trace operation appearing in it can already contain the angle integration. If one includes to the trace operation appearing the integration over φ it gives δm,n factor and U-matrix has elements only between states assignable to the same causal diamond. Hence one must interpret U-matrix elements as functions of φ realized factors exp(i(sn-sm)φ). This brings strongly in mind operators defined as distributions of operators on line encountered in the theory of representations of non-compact groups such as Lorentz group. In fact, the unitary representations of discrete Lorentz groups are involved now.
  4. The unitarity condition contains besides the trace also the integrations over the two angle parameters φi associated with the two U-matrix elements involved. The left hand side of the unitarity condition reads as

    k≥ Max(0,n-m) =I(ksi)I((n-m-k)sj) × ∑ j1Tr(HiHj1) Tr(Hj1Hj) = δi,jδm,n ,

    I(s)=(1/2π)× ∫ dφ exp(isφ) =δs,0 .

    Integrations give the factor δk,0 eliminating the infinite sum obtained otherwise plus the factor δn,m. Traces give Kronecker deltas since the projectors are orthonormal. The left hand side equals to the right hand side and one achieves unitarity. It seems that the proposed ansatz works and the U-matrix can be reduced by a general ansatz to S-matrix.

What about the identification of S?

  1. S should be exponential of time the scaling operator whose action reduces to a time translation operator along the time axis connecting the tips of CD and realized as scaling. In other words, the shift t/T0=m→ m+n corresponds to a scaling t/T0=m→ km giving m+n=km in turn giving k= 1+ n/m. At the limit of large shifts one obtains k≈ n/m→ ∞, which corresponds to QFT limit. nS corresponds to (nT0)× (S/T0)= TH and one can ask whether QFT Hamiltonian could corresponds to H=S/T0.
  2. It is natural to assume that the operators Hi are eigenstates of radial scaling generator L0=irMd/drM at both boundaries of CD and have thus well-defined conformal weights. As noticed the spectrum for super-symplectic algebra could also be given in terms of zeros of Riemann zeta.
  3. The boundaries of CD are given by the equations rM=m0 and rM= T-m0, m0 is Minkowski time coordinate along the line between the tips of CD and T is the distance between the tips. From the relationship between rM and m0 the action of the infinitesimal translation H== i∂/∂m0 can be expressed as conformal generator L-1= i∂/∂rM = rM-1 L0 . Hence the action is non-diagonal in the eigenbasis of L0 and multiplies with the conformal weights and reduces the conformal weight by one unit. Hence the action of U can change the projection operator. For large values of conformal weight the action is classically near to that of L0: multiplication by L0 plus small relative change of conformal weight.
  4. Could the spectrum of H be identified as energy spectrum expressible in terms of zeros of zeta defining a good candidate for the super-symplectic radial conformal weights. This certainly means maximal complexity since the number of generators of the conformal algebra would be infinite. This identification might make sense in chaotic or critical systems. The functions (rM/r0)1/2+iy and (rM/r0)-2n, n>0, are eigenmodes of rM/drM with eigenvalues (1/2+iy) and -2n corresponding to non-trivial and trivial zeros of zeta.

    There are two options to consider. Either L0 or iL0 could be realized as a hermitian operator. These options would correspond to the identification of mass squared operator as L0 and approximation identification of Hamiltonian as iL1 as iL0 making sense for large conformal weights.

    1. Suppose that L0= rMd/drM realized as a hermitian operator would give harmonic oscillator spectrum for conformal confinement. In p-adic mass calculations the string model mass formula implies that L0 acts essentially as mass squared operator with integer spectrum. I have proposed conformal confinent for the physical states net conformal weight is real and integer valued and corresponds to the sum over negative integer valued conformal weights corresponding to the trivial zeros and sum over real parts of non-trivial zeros with conformal weight equal to 1/2. Imaginary parts of zeta would sum up to zero.
    2. The counterpart of Hamiltonian as a time translation is represented by H=iL0= irM d/drM. Conformal confinement is now realized as the vanishing of the sum for the real parts of the zeros of zeta: this can be achieved. As a matter fact the integration measure drM/rM brings implies that the net conformal weight must be 1/2. This is achieved if the number of non-trivial zeros is odd with a judicious choice of trivial zeros. The eigenvalues of Hamiltonian acting as time translation operator could correspond to the linear combination of imaginary part of zeros of zeta with integer coefficients. This is an attractive hypothesis in critical systems and TGD Universe is indeed quantum critical.

What about quantum classical correspondence and zero modes?

The one-one correspondence between the basis of quantum states and zero modes realizes quantum classical correspondence.

  1. M-matrices would act in the tensor product of quantum fluctuating degrees of freedom and zero modes. The assumption that zero energy states form an orthogonal basis implies that the hermitian square roots of the density matrices form an orthonormal basis. This condition generalizes the usual orthonormality condition.
  2. The dependence on zero modes at given boundary of CD would be trivial and induced by 1-1 correspondence |m⟩ → z(m) between states and zero modes assignable to the state basis |m+/-⟩ at the boundaries of CD, and would mean the presence of factors δz+,f(m+) × δz-,f(n-) multiplying M-matrix Mim,n.
To sum up, it seems that the architecture of the U-matrix and its relationship to the S-matrix is now understood and in accordance with the intuitive expectations the construction of U-matrix reduces to that for S-matrix and one can see S-matrix as discretized counterpart of ordinary unitary time evolution operator with time translation represented as scaling: this allows to circumvent problems with loss of manifest Poincare symmetry encountered in quantum field theories and allows Lorentz invariance although CD has finite size. What came as surprise was the connection with stringy picture: strings are necessary in order to satisfy the unitary conditions for U-matrix. Second outcome was that the connection with super-symplectic algebra suggests itself strongly. The identification of hermitian square roots of density matrices with Hermitian symmetry algebra is very elegant aspect discovered already earlier. A further unexpected result was that U-matrix is unitary only for strict arrow of time (which changes in the state function reduction to opposite boundary of CD).

See the chapter Zero Energy Ontology and Matrices or the article The relation between U-matrix and M-matrices.

Could the Universe be doing Yangian quantum arithmetics?

One of the old TGD inspired really crazy ideas about scattering amplitudes is that Universe is doing some sort of arithmetics so that scattering amplitude are representations for computational sequences of minimum length. The idea is so crazy that I have even given up its original form, which led to an attempt to assimilate the basic ideas about bi-algebras, quantum groups, Yangians and related exotic things. The work with twistor Grassmannian approach inspired a reconsideration of the original idea seriously with the idea that super-symplectic Yangian could define the arithmetics. I try to describe the background, motivation, and the ensuing reckless speculations in the following.

Do scattering amplitudes represent quantal algebraic manipulations?

  1. I seems that tensor product ⊗ and direct sum ⊕ - very much analogous to product and sum but defined between Hilbert spaces rather than numbers - are naturally associated with the basic vertices of TGD. I have written about this a highly speculative chapter - both mathematically and physically.
    1. In ⊗ vertex 3-surface splits to two 3-surfaces meaning that the 2 "incoming" 4-surfaces meet at single common 3-surface and become the outgoing 3-surface: 3 lines of Feynman diagram meeting at their ends. This has a lower-dimensional shadow realized for partonic 2-surfaces. This topological 3-particle vertex would be higher-D variant of 3-vertex for Feynman diagrams.
    2. The second vertex is trouser vertex for strings generalized so that it applies to 3-surfaces. It does not represent particle decay as in string models but the branching of the particle wave function so that particle can be said to propagate along two different paths simultaneously. In double slit experiment this would occur for the photon space-time sheets.
  2. The idea is that Universe is doing arithmetics of some kind in the sense that particle 3-vertex in the above topological sense represents either multiplication or its time-reversal co-multiplication.
The product, call it •, can be something very general, say algebraic operation assignable to some algebraic structure. The algebraic structure could be almost anything: a random list of structures popping into mind consists of group, Lie-algebra, super-conformal algebra quantum algebra, Yangian, etc.... The algebraic operation • can be group multiplication, Lie-bracket, its generalization to super-algebra level, etc...). Tensor product and thus linear (Hilbert) spaces are involved always, and in product operation tensor product ⊗ is replaced with •.
  1. The product Ak⊗ Al→ C= Ak• Al is analogous to a particle reaction in which particles Ak and Al fuse to particle Ak⊗ Al→ C=Ak• Al. One can say that ⊗ between reactants is transformed to • in the particle reaction: kind of bound state is formed.
  2. There are very many pairs Ak, Al giving the same product C just as given integer can be divided in many manners to a product of two integers if it is not prime. This of course suggests that elementary particles are primes of the algebra if this notion is defined for it! One can use some basis for the algebra and in this basis one has C=Ak• Al= fklmAm, fklm are the structure constants of the algebra and satisfy constraints. For instance, associativity A(BC)=(AB)C is a constraint making the life of algebraist more tolerable and is almost routinely assumed.

    For instance, in the number theoretic approach to TGD associativity is proposed to serve as fundamental law of physics and allows to identify space-time surfaces as 4-surfaces with associative (quaternionic) tangent space or normal space at each point of octonionic imbedding space M4× CP2. Lie algebras are not associative but Jacobi-identities following from the associativity of Lie group product replace associativity.

  3. Co-product can be said to be time reversal of the algebraic operation •. Co-product can be defined as C=Ak→ ∑lm fklmAl⊗ Bm is co-product in which one has quantum superposition of final states which can fuse to C (Ak⊗ Bkl→ C=Ak• Bl is possible). One can say that • is replaced with ⊗: bound state decays to a superposition of all pairs, which can form the bound states by product vertex.
There are motivations for representing scattering amplitudes as sequences of algebraic operations performed for the incoming set of particles leading to an outgoing set of particles with particles identified as algebraic objects acting on vacuum state. The outcome would be analogous to Feynman diagrams but only the diagram with minimal length to which a preferred extremal can be assigned is needed. Larger ones must be equivalent with it.

The question is whether it could be indeed possible to characterize particle reactions as computations involving transformation of tensor products to products in vertices and co-products to tensor products in co-vertices (time reversals of the vertices). A couple of examples gives some idea about what is involved.

  1. The simplest operations would preserve particle number and to just permute the particles: the permutation generalizes to a braiding and the scattering matrix would be basically unitary braiding matrix utilized in topological quantum computation.
  2. A more complex situation occurs, when the number of particles is preserved but quantum numbers for the final state are not same as for the initial state so that particles must interact. This requires both product and co-product vertices. For instance, Ak⊗ Al→ fklmAm followed by Am→ fmrsAr⊗ As giving Ak→ fklmfmrsAr⊗ As representing 2-particle scattering. State function reduction in the final state can select any pair Ar⊗ As in the final state. This reaction is characterized by the ordinary tree diagram in which two lines fuse to single line and defuse back to two lines. Note also that there is a non-deterministic element involved. A given final state can be achieved from a given initial state after large enough number of trials. The analogy with problem solving and mathematical theorem proving is obvious. If the interpretation is correct, Universe would be problem solver and theorem prover!
  3. More complex reactions affect also the particle number. 3-vertex and its co-vertex are the simplest examples and generate more complex particle number changing vertices. For instance, on twistor Grassmann approach on can construct all diagrams using two 3-vertices. This encourages the restriction to 3-vertice (recall that fermions have only 2-vertices)
  4. Intuitively it is clear that the final collection of algebraic objects can be reached by a large - maybe infinite - number of ways. It seems also clear that there is the shortest manner to end up to the final state from a given initial state. Of course, it can happen that there is no way to achieve it! For instance, if • corresponds to group multiplication the co-vertex can lead only to a pair of particles for which the product of final state group elements equals to the initial state group element.
  5. Quantum theorists of course worry about unitarity. How can avoid the situation in which the product gives zero if the outcome is element of linear space. Somehow the product should be such that this can be avoided. For instance, if product is Lie-algebra commutator, Cartan algebra would give zero as outcome.

Generalized Feynman diagram as shortest possible algebraic manipulation connecting initial and final algebraic objects

There is a strong motivation for the interpretation of generalized Feynman diagrams as shortest possible algebraic operations connecting initial and final states. The reason is that in TGD one does not have path integral over all possible space-time surfaces connecting the 3-surfaces at the ends of CD. Rather, one has in the optimal situation a space-time surface unique apart from conformal gauge degeneracy connecting the 3-surfaces at the ends of CD (they can have disjoint components).

Path integral is replaced with integral over 3-surfaces. There is therefore only single minimal generalized Feynman diagram (or twistor diagram, or whatever is the appropriate term). It would be nice if this diagram had interpretation as the shortest possible computation leading from the initial state to the final state specified by 3-surfaces and basically fermionic states at them. This would of course simplify enormously the theory and the connection to the twistor Grassmann approach is very suggestive. A further motivation comes from the observation that the state basis created by the fermionic Clifford algebra has an interpretation in terms of Boolean quantum logic and that in ZEO the fermionic states would have interpretation as analogs of Boolean statements A→ B.

To see whether and how this idea could be realized in TGD framework, let us try to find counterparts for the basic operations ⊗ and • and identify the algebra involved. Consider first the basic geometric objects.

  1. Tensor product could correspond geometrically to two disjoint 3-surfaces representing 3-particles. Partonic 2-surfaces associated with a given 3-surface represent second possibility. The splitting of a partonic 2-surface to two could be the geometric counterpart for co-product.
  2. Partonic 2-surfaces are however connected to each other and possibly even to themselves by strings. It seems that partonic 2-surface cannot be the basic unit. Indeed, elementary particles are identified as pairs of wormhole throats (partonic 2-surfaces) with magnetic monopole flux flowing from throat to another at first space-time sheet, then through throat to another sheet, then back along second sheet to the lower throat of the first contact and then back to the thirst throat. This unit seems to be the natural basic object to consider. The flux tubes at both sheets are accompanied by fermionic strings. Whether also wormhole throats contain strings so that one would have single closed string rather than two open ones, is an open question.
  3. The connecting strings give rise to the formation of gravitationally bound states and the hierarchy of Planck constants is crucially involved. For elementary particle there are just two wormhole contacts each involving two wormhole throats connected by wormhole contact. Wormhole throats are connected by one or more strings, which define space-like boundaries of corresponding string world sheets at the boundaries of CD. These strings are responsible for the formation of bound states, even macroscopic gravitational bound states.
Super-symplectic Yangian would be a reasonable guess for the algebra involved.
  1. The 2-local generators of Yangian would be of form TA1= fABCTB⊗ TC, where fABC are the structure constants of the super-symplectic algebra. n-local generators would be obtained by iterating this rule. Note that the generator TA1 creates an entangled state of TB and TC with fABC the entanglement coefficients. TAn is entangled state of TB and TCn-1 with the same coefficients. A kind replication of TAn-1 is clearly involved, and the fundamental replication is that of TA. Note that one can start from any irreducible representation with well defined symplectic quantum numbers and form similar hierarchy by using TA and the representation as a starting point.

    That the hierarchy TAn and hierarchies irreducible representations would define a hierarchy of states associated with the partonic 2-surface is a highly non-trivial and powerful hypothesis about the formation of many-fermion bound states inside partonic 2-surfaces.

  2. The charges TA correspond to fermionic and bosonic super-symplectic generators. The geometric counterpart for the replication at the lowest level could correspond to a fermionic/bosonic string carrying super-symplectic generator splitting to fermionic/bosonic string and a string carrying bosonic symplectic generator TA. This splitting of string brings in mind the basic gauge boson-gauge boson or gauge boson-fermion vertex.

    The vision about emission of virtual particle suggests that the entire wormhole contact pair replicates. Second wormhole throat would carry the string corresponding to TA assignable to gauge boson naturally. TA should involve pairs of fermionic creation and annihilation operators as well as fermionic and anti-fermionic creation operator (and annihilation operators) as in quantum field theory.

  3. Bosonic emergence suggests that bosonic generators are constructed from fermion pairs with fermion and anti-fermion at opposite wormhole throats: this would allow to avoid the problems with the singular character of purely local fermion current. Fermionic and anti-fermionic string would reside at opposite space-time sheets and the whole structure would correspond to a closed magnetic tube carrying monopole flux. Fermions would correspond to superpositions of states in which string is located at either half of the closed flux tube.
  4. The basic arithmetic operation in co-vertex would be co-multiplication transforming TAn to TAn+1 = fABCTBn ⊗ TC. In vertex the transformation of TAn+1 to TAn would take place. The interpretations would be as emission/absorption of gauge boson. One must include also emission of fermion and this means replacement of TA with corresponding fermionic generators FA, so that the fermion number of the second part of the state is reduced by one unit. Particle reactions would be more than mere braidings and re-grouping of fermions and anti-fermions inside partonic 2-surfaces, which can split.
  5. Inside the light-like orbits of the partonic 2-surfaces there is also a braiding affecting the M-matrix. The arithmetics involved would be therefore essentially that of measuring and "co-measuring" symplectic charges.

    Generalized Feynman diagrams (preferred extremals) connecting given 3-surfaces and many-fermion states (bosons are counted as fermion-anti-fermion states) would have a minimum number of vertices and co-vertices. The splitting of string lines implies creation of pairs of fermion lines. Whether regroupings are part of the story is not quite clear. In any case, without the replication of 3-surfaces it would not be possible to understand processes like e-e scattering by photon exchange in the proposed picture.

This was not the whole story yet

The proposed amplitude represents only the value of WCW spinor field for single pair of 3-surfaces at the opposite boundaries of given CD. Hence Yangian construction does not tell the whole story.

  1. Yangian algebra would give only the vertices of the scattering amplitudes. On basis of previous considerations, one expects that each fermion line carries propagator defined by 8-momentum. The structure would resemble that of super-symmetric YM theory. Fermionic propagators should emerge from summing over intermediate fermion states in various vertices and one would have integrations over virtual momenta which are carried as residue integrations in twistor Grassmann approach. 8-D counterpart of twistorialization would apply.
  2. Super-symplectic Yangian would give the scattering amplitudes for single space-time surface and the purely group theoretical form of these amplitudes gives hopes about the independence of the scattering amplitude on the pair of 3-surfaces at the ends of CD near the maximum of Kähler function. This is perhaps too much to hope except approximately but if true, the integration over WCW would give only exponent of Kähler action since metric and poorly defined Gaussian and determinants would cancel by the basic properties of Kähler metric. Exponent would give a non-analytic dependence on αK.

    The Yangian supercharges are proportional to 1/αK since covariant Kähler-Dirac gamma matrices are proportional to canonical momentum currents of Kähler action and thus to 1/αK. Perturbation theory in powers of αK= gK2/4πhbareff is possible after factorizing out the exponent of vacuum functional at the maximum of Kähler function and the factors 1/αK multiplying super-symplectic charges.

    The additional complication is that the characteristics of preferred extremals contributing significantly to the scattering amplitudes are expected to depend on the value of αK by quantum interference effects. Kähler action is proportional to 1/αK. The analogy of AdS/CFT correspondence states the expressibility of Kähler function in terms of string area in the effective metric defined by the anti-commutators of K-D matrices. Interference effects eliminate string length for which the area action has a value considerably larger than one so that the string length and thus also the minimal size of CD containing it scales as heff. Quantum interference effects therefore give an additional dependence of Yangian super-charges on heff leading to a perturbative expansion in powers of αK although the basic expression for scattering amplitude would not suggest this.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Is the formation of gravitational bound states impossible in superstring models?

I decided to take here from a previous posting an argument allowing to conclude that super string models are unable to describe macroscopic gravitation involving formation of gravitationally bound states. Therefore superstrings models cannot have desired macroscopic limit and are simply wrong. This is of course reflected also by the landscape catastrophe meaning that the theory ceases to be a theory in macroscopic scales. The failure is not only at the level of superstring models: it is at the level of quantum theory itself. Instead of single value of Planck constant one must allow a hierarchy of Planck constants predicted by TGD. My sincere hope is that this message could gradually leak through the iron curtain to the ears of the super string gurus.

Superstring action has bosonic part proportional to string area. The proportionality constant is string tension proportional to 1/hbar G and is gigantic. One expects only strings of length of order Planck length be of significance.

It is now clear that also in TGD the action in Minkowskian regions contains a string area. In Minkowskian regions of space-time strings dominate the dynamics in an excellent approximation and the naive expectation is that string theory should give an excellent description of the situation.

String tension would be proportional to 1/hbar G and this however raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required non-perturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.

In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.

I have however proposed that gravitons - at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has heff= hgr=GMm/v0, where v0/c<1 (v0 has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v0/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v0/G2Mm. For v0= GMm/hbar, which remains below unity for Mm/mPl2 one would have hgr/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water) - probably not an accident - would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. hgr=heff hypothesis is indeed central in TGD based view about living matter.

To conclude, it seems that superstring theory with single value of Planck constant cannot give rise to macroscopic gravitationally bound matter and would be therefore simply wrong much better than to be not-even-wrong.

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

Updated view about Kähler geometry of WCW

TGD differs in several respects from quantum field theories and string models. The basic mathematical difference is that the mathematically poorly defined notion of path integral is replaced with the mathematically well-defined notion of functional integral defined by the Kähler function defining Kähler metric for WCW ("world of classical worlds"). Apart from quantum jump, quantum TGD is essentially theory of classical WCW spinor fields with WCW spinors represented as fermionic Fock states. One can say that Einstein's geometrization of physics program is generalized to the level of quantum theory.

It has been clear from the beginning that the gigantic super-conformal symmetries generalizing ordinary super-conformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and space-time degrees of freedom (or between Minkowskian and Euclidian space-time regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anti-commutators of WCW gamma matrices identifiable as super-conformal Noether super-charges for the symplectic algebra assignable to δ M4+/-× CP2. The string model description of gravitation emerges and also the TGD based view about dark matter becomes more precise.

Kähler function, Kähler action, and connection with string models

The definition of Kähler function in terms of Kähler action is possible because space-time regions can have also Euclidian signature of induced metric. Euclidian regions with 4-D CP2 projection - wormhole contacts - are identified as lines of generalized Feynman diagrams - space-time correlates for basic building bricks of elementary particles. Kähler action from Minkowskian regions is imaginary and gives to the functional integrand a phase factor crucial for quantum field theoretic interpretation. The basic challenges are the precise specification of Kähler function of "world of classical worlds" (WCW) and Kähler metric.

There are two approaches concerning the definition of Kähler metric: the conjecture analogous to AdS/CFT duality is that these approaches are mathematically equivalent.

  1. The Kähler function defining Kähler metric can be identified as Kähler action for space-time regions with Euclidian signature for a preferred extremal containing 3-surface as the ends of the space-time surfaces inside causal diamond (CD). Minkowskian space-time regions give to Kähler action an imaginary contribution interpreted as the counterpart of quantum field theoretic action. The exponent of Kähler function defines functional integral in WCW. WCW metric is dictated by the Euclidian regions of space-time with 4-D CP2 projection.

    The basic question concerns the attribute "preferred". Physically the preferred extremal is analogous to Bohr orbit. What is the mathematical meaning of preferred extremal of Kähler action? The latest step of progress is the realization that the vanishing of generalized conformal charges for the ends of the space-time surface fixes the preferred extremals to high extent and is nothing but classical counterpart for generalized Virasoro and Kac-Moody conditions.

  2. Fermions are also needed. The well-definedness of electromagnetic charge led to the hypothesis that spinors are restricted to string world sheets. It has become also clear that string world sheets are most naturally minimal surfaces with 1-D CP2 projection (this brings in gravitational constant) and that Kähler action in Minkowskian regions involves also the string area (, which does not contribute to Kähler function) giving the entire action in the case of M4 type vacuum extremals with vanishing Kähler form. Hence vacuum extremals might serve as an excellent approximation for the sheets of the many-sheeted space-time in Minkowskian space-time regions.
  3. Second manner to define Kähler metric is as anticommutators of WCW gamma matrices identified as super-symplectic Noether charges for the Dirac action for induced spinors with string tension proportional to the inverse of Newton's constant. These charges are associated with the 1-D space-like ends of string world sheets connecting the wormhole throats. WCW metric contains contributions from the spinor modes associated with various string world sheets connecting the partonic 2-surfaces associated with the 3-surface.

    It is clear that the information carried by WCW metric about 3-surface is rather limited and that the larger the number of string world sheets, the larger the information. This conforms with strong form of holography and the notion of measurement resolution as a property of quantums state. Clearly. Duality means that Kähler function is determined either by space-time dynamics inside Euclidian wormhole contacts or by the dynamics of fermionic strings in Minkowskian regions outside wormhole contacts. This duality brings strongly in mind AdS/CFT duality. One could also speak about fermionic emergence since Kähler function is dictated by the Kähler metric part from a real part of gradient of holomorphic function: a possible identification of the exponent of Kähler function is as Dirac determinant.

Realization of super-conformal symmetries

The detailed realization of various super-conformal symmetries has been also a long standing problem but recent progress leads to very beautiful overall view.

  1. Super-conformal symmetry requires that Dirac action for string world sheets is accompanied by string world sheet area as part of bosonic action. String world sheets are implied and can be present only in Minkowskian regions if one demands that octonionic and ordinary representations of induced spinor structure are equivalent (this requires vanishing of induced spinor curvature to achieve associativity in turn implying that CP2 projection is 1-D). Note that 1-dimensionality of CP2 projection is symplectically invariant property. Neither string world sheet area nor Kähler action is invariant under symplectic transformations. This is necessary for having non-trivial Kähler metric. Whether WCW really possesses super-symplectic isometries remains an open problem.
  2. Super-conformal symmetry also demands that Kähler action is accompanied by what I call Kähler-Dirac action with gamma matrices defined by the contractions of the canonical momentum currents with imbedding space-gamma matrices. Hence also induced spinor fields in the space-time interior must be present. Indeed, inside wormhole contacts Kähler-Dirac equation reducing to CP2 Dirac equation for CP2 vacuum extremals dictates the fermionic dynamics.

    Strong form of holography implied by strong form of general coordinate invariance strongly suggests that super-conformal invariance in the interior of the space-time surface is a broken gauge invariance in the sense that the super-conformal charges for a sub-algebra with conformal weights vanishing modulo some integer n vanish. The proposal is that n corresponds to the effective Planck constant as heff/h=n. For string world sheets super-conformal symmetries are not gauge symmetries and strings dominate in good approximation the fermionic dynamics.

Interior dynamics for fermions, the role of vacuum extremals, dark matter, and SUSY

The key role of CP2-type and M4-type vacuum extremals has been rather obvious from the beginning but the detailed understanding has been lacking. Both kinds of extremals are invariant under symplectic transformations of δ M4× CP2, which inspires the idea that they give rise to isometries of WCW. The deformations CP2-type extremals correspond to lines of generalized Feynman diagrams. M4 type vacuum extremals in turn are excellent candidates for the building bricks of many-sheeted space-time giving rise to GRT space-time as approximation. For M4 type vacuum extremals CP2 projection is (at most 2-D) Lagrangian manifold so that the induced Kähler form vanishes and the action is fourth-order in small deformations. This implies the breakdown of the path integral approach and of canonical quantization, which led to the notion of WCW.

If the action in Minkowskian regions contains also string area, the situation changes dramatically since strings dominate the dynamics in excellent approximation and string theory should give an excellent description of the situation: this of course conforms with the dominance of gravitation.

String tension would be proportional to 1/hbar G and this raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required non-perturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.

In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.

I have however proposed that gravitons - at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has heff= hgr=GMm/v0, where v0/c<1 (v0 has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v0/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v0/G2Mm. For v0= GMm/hbar, which remains below unity for Mm/mPl2 one would have hgr/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water) - probably not an accident - would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. hgr=heff hypothesis is indeed central in TGD based view about living matter. In this framework superstring theory with single value of Planck constant would not give rise to macroscopic gravitationally bound matter and would be thus simply wrong.

If one assumes that for non-standard values of Planck constant only n-multiples of super-conformal algebra in interior annihilate the physical states, interior conformal gauge degrees of freedom become partly dynamical. The identification of dark matter as macroscopic quantum phases labeled by heff/h=n conforms with this.

The emergence of dark matter corresponds to the emergence of interior dynamics via breaking of super-conformal symmetry. The induced spinor fields in the interior of flux tubes obeying Kähler Dirac action should be highly relevant for the understanding of dark matter. The assumption that dark particles have essentially same masses as ordinary particles suggests that dark fermions correspond to induced spinor fields at both string world sheets and in the space-time interior: the spinor fields in the interior would be responsible for the long range correlations characterizing heff/h=n. Magnetic flux tubes carrying dark matter are key entities in TGD inspired quantum biology. Massless extremals represent second class of M4 type non-vacuum extremals.

This view forces once again to ask whether space-time SUSY is present in TGD and how it is realized. With a motivation coming from the observation that the mass scales of particles and sparticles most naturally have the same p-adic mass scale as particles in TGD Universe I have proposed that sparticles might be dark in TGD sense. The above argument leads to ask whether the dark variants of particles correspond to states in which one has ordinary fermion at string world sheet and 4-D fermion in the space-time interior so that dark matter in TGD sense would almost by definition correspond to sparticles!

See the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" .

Permutations, braidings, and amplitudes

Nima Arkani-Hamed et al demonstrate that various twistorially represented on-mass-shell amplitudes (allowing light-like complex momenta) constructible by taking products of the 3-particle on mass-shell amplitude and its conjugate can be assigned with unique permutations of the incoming lines. The article describes the graphical representation of the amplitudes and its generalization. For 3-particle amplitudes, which correspond to ++- and +-- twistor amplitudes, the corresponding permutations are cyclic permutations, which are inverses of each other. One actually introduces double cover for the labels of the particles and speaks of decorated permutations meaning that permutation is always a right shift in the integer and in the range [1,2× n].

Amplitudes as representation of permutations

It is shown that for on mass shell twistor amplitudes the definition using on-mass-shell 3-vertices as building bricks is highly reducible: there are two moves for squares defining 4-particle sub-amplitudes allowing to reduce the graph to a simpler one. The first move is topologically like the s-t duality of the old-fashioned string models and second one corresponds to the transformation black ↔ white for a square sub-diagram with lines of same color at the ends of the two diagonals and built from 3-vertices.

One can define the permutation characterizing the general on mass shell amplitude by a simple rule. Start from an external particle a and go through the graph turning in in white (black) vertex to left (right). Eventually this leads to a vertex containing an external particle and identified as the image P(a) of the a in the permutation. If permutations are taken as right shifts, one ends up with double covering of permutation group with 2× n! elements - decorated permutations. In this manner one can assign to any any line of the diagram two lines. This brings in mind 2-D integrable theories where scattering reduces to braiding and also topological QFTs where braiding defines the unitary S-matrix. In TGD parton lines involve braidings of the fermion lines so that an assignment of permutation also to vertex would be rather nice.

BCFW bridge has an interpretation as a transposition of two neighboring vertices affecting the lines of the permutation defining the diagram. One can construct all permutations as products of transpositions and therefore by building BCFW bridges. BCFW bridge can be constructed also between disjoint diagrams as done in the BCFW recursion formula.

Can one generalize this picture in TGD framework? There are several questions to be answered.

  1. What should one assume about the states at the light-like boundaries of string world sheets? What is the precise meaning of the supersymmetry: is it dynamical or gauge symmetry or both?
  2. What does one mean with particle: partonic 2-surface or boundary line of string world sheet? How the fundamental vertices are identified: 4 incoming boundaries of string world sheets or 3 incoming partonic orbits or are both aspects involved?
  3. How the 8-D generalization of twistors bringing in second helicity and doubling the M4 helicity states assignable to fermions does affect the situation?
  4. Does the crucial right-left rule relying heavily on the possibility of only 2 3-particle vertices generalize? Does M4 massivation imply more than 2 3-particle vertices implying many-to-one correspondence between on-mass-shell diagrams and permutations? Or should one generalize the right-left rule in TGD framework?

Fermion lines for fermions massless in 8-D sense

What does one mean with particle line at the level of fermions?

  1. How the addition of CP2 helicity and complete correlation between M4 and CP2 chiralities does affect the rules of N=4 SUSY? Chiral invariance in 8-D sense guarantees fermion number conservation for quarks and leptons separately and means conservation of the product of M4 and CP2 chiralities for 2-fermion vertices. Hence only M4 chirality need to be considered. M4 massivation allows more 4-fermion vertices than N=4 SUSY.
  2. One can assign to a given partonic orbit several lines as boundaries of string world sheets connecting the orbit to other partonic orbits. Supersymmetry could be understoond in two manners.
    1. The fermions generating the state of super-multiplet correspond to boundaries of different string world sheets which need not connect the string world sheet to same partonic orbit. This SUSY is dynamical and broken. The breaking is mildest breaking for line groups connected by string world sheets to same partonic orbit. Right handed neutrinos generated the least broken N=2 SUSY.
    2. Also single line carrying several fermions would provide realization of generalized SUSY since the multi-fermion state would be characterized by single 8-momentum and helicity. One would have N =4 SUSY for quarks and leptons separately and N =8 if both quarks and leptons are allowed. Conserved total for quark and antiquarks and leptons and antileptons characterize the lines as well.

      What would be the propagator associated with many-fermion line? The first guess is that it is just a tensor power of single fermion propagator applied to the tensor power of single fermion states at the end of the line. This gives power of 1/p2n to the denominator, which suggests that residue integral in momentum space gives zero unless one as just single fermion state unless the vertices give compensating powers of p. The reduction of fermion number to 0 or 1 would simplify the diagrammatics enormously and one would have only 0 or 1 fermions per given string boundary line. Multi-fermion lines would represent gauge degrees of freedom and SUSY would be realized as gauge invariance. This view about SUSY clearly gives the simplest picture, which is also consistent with the earlier one, and will be assumed in the sequel

  3. The multiline containing n fermion oscillator operators can transform by chirality mixing in 2n manners at 4-fermion vertex so that there is quite a large number of options for incoming lines with ni fermions.
  4. In 4-D Dirac equation light-likeness implies a complete correlation between fermion number and chirality. In 8-D case light-likeness should imply the same: now chirality correspond to fermion number. Does this mean that one must assume just superposition of different M4 chiralities at the fermion lines as 8-D Dirac equation requires. Or should one assume that virtual fermions at the end of the line have wrong chirality so that massless Dirac operator does not annihilate them?

Fundamental vertices

One can consider two candidates for fundamental vertices depending on whether one identifies the lines of Feynman diagram as fermion lines or as light-like orbits of partonic 2-surfaces. The latter vertices reduces microscopically to the fermionic 4-vertices.

  1. If many-fermion lines are identified as fundamental lines, 4-fermion vertex is the fundamental vertex assignable to single wormhole contact in the topological vertex defined by common partonic 2-surface at the ends of incoming light-like 3-surfaces. The discontinuity is what makes the vertex non-trivial.
  2. In the vertices generalization of OZI rule applies for many-fermion lines since there are no higher vertices at this level and interactions are mediated by classical induced gauge fields and chirality mixing. Classical induced gauge fields vanish if CP2 projection is 1-dimensional for string world sheets and even gauge potentials vanish if the projection is to geodesic circle. Hence only the chirality mixing due to the mixing of M4 and CP2 gamma matrices is possible and changes the fermionic M4 chiralities. This would dictate what vertices are possible.
  3. The possibility of two helicity states for fermions suggests that the number of amplitudes is considerably larger than in N=4 SUSY. One would have 5 independent fermion amplitudes and at each 4-fermion vertex one should be able to choose between 3 options if the right-left rule generalizes. Hence the number of amplitudes is larger than the number of permutations possibly obtained using a generalization of right-left rule to right-middle-left rule.
  4. Note however that for massless particles in M4 sense the reduction of helicity combinations for the fermion and antifermion making virtual gauge boson happens. The fermion and antifermion at the opposite wormhole throats have parallel four-momenta in good approximation. In M4 they would have opposite chiralities and opposite helicities so that the boson would be M4 scalar. No vector bosons would be obtained in this manner.

    In 8-D context it is possible to have also vector bosons since the M4 chiralities can be same for fermion and anti-fermion. The bosons are however massive, and even photon is predicted to have small mass given by p-adic thermodynamics. Massivation brings in also the M4 helicity 0 state. Only if zero helicity state is absent, the fundamental four-fermion vertex vanishes for ++++ and ---- combinations and one extend the right-left rule to right-middle-left rule. There is however no good reason for he reduction in the number of 4-fermion amplitudes to take place.

Partonic surfaces as 3-vertices

At space-time level one could identify vertices as partonic 2-surfaces.

  1. At space-time level the fundamental vertices are 3-particle vertices with particle identified as wormhole contact carrying many-fermion states at both wormhole throats. Each line of BCFW diagram would be doubled. This brings in mind the representation of permutations and leads to ask whether this representation could be re-interpreted in TGD framework. For this option the generalization of the decomposition of diagram to 3-particle vertices is very natural. If the states at throats consist of bound states of fermions as SUSY suggests, one could characterize them by total 8-momentum and helicity in good approximation. Both helicities would be however possible also for fermions by chirality mixing.
  2. A genuine decomposition to 3-vertices and lines connecting them takes place if two of the fermions reside at opposite throats of wormhole contact identified as fundamental gauge boson (physical elementary particles involve two wormhole contacts).

    The 3-vertex can be seen as fundamental and 4-fermion vertex becomes its microscopic representation. Since the 3-vertices are at fermion level 4-vertices their number is greater than two and there is no hope about the generalization of right-left rule.

OZI rule implies correspondence between permutations and amplitudes

The realization of the permutation in the same manner as for N=4 amplitudes does not work in TGD. OZI rule following from the absence of 4-fermion vertices however implies much simpler and physically quite a concrete manner to define the permutation for external fermion lines and also generalizes it to include braidings along partonic orbits.

  1. Already N=4 approach assumes decorated permutations meaning that each external fermion has effectively two states corresponding to labels k and k+n (permutations are shifts to the right). For decorated permutations the number of external states is effectively 2n and the number of decorated permutations is 2× n!. The number of different helicity configurations in TGD framework is 2n for incoming fermions at the vertex defined by the partonic 2-surface. By looking the values of these numbers for lowest integers one finds 2n≥ 2n: for n=2 the equation is saturated. The inequality log(n!) >nlog(n)/e)+1 (see Wikipedia) gives

    log(2n!)/log(2n)geq; log(2)+ 1+nlog(n/e)/nlog(2)= log(n/e)/log(2) +O(1/n)

    so that the desired inequality holds for all interesting values of n.

  2. If OZI rule holds true, the permutation has very natural physical definition. One just follows the fermion line which must eventually end up to some external fermion since the only fermion vertex is 2-fermion vertex. The helicity flip would map k→ k+n or vice versa.
  3. The labelling of diagrams by permutations generalizes to the case of diagrams involving partonic surfaces at the boundaries of causal diamond containing the external fermions and the partonic 2-surfaces in the interior of CD identified as vertices. Permutations generalize to braidings since also the braidings along the light-like partonic 2-surfaces are allowed. A quite concrete generalization of the analogs of braid diagrams in integrable 2-D theories emerges.
  4. BCFW bridge would be completely analogous to the fundamental braiding operation permuting two neighboring braid strands. The almost reduction to braid theory - apart from the presence of vertices conforms with the vision about reduction of TGD to almost topological QFT.
To sum up, the simplest option assumes SUSY as both gauge symmetry and broken dynamical symmetry. The gauge symmetry relates string boundaries with different fermion numbers and only fermion number 0 or 1 gives rise to a non-vanishing outcome in the residue integration and one obtains the picture used hitherto. If OZI rule applies, the decorated permutation symmetry generalizes to include braidings at the parton orbits and k→ k +/- n corresponds to a helicity flip for a fermion going through the 4-vertex. OZI rule follows from the absence of non-linearities in Dirac action and means that 4-fermion vertices in the usual sense making theory non-renormalizable are absent. Theory is essentially free field theory in fermionic degrees of freedom and interactions in the sense of QFT are transformed to non-trivial topology of space-time surfaces.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

What about non-planar diagrams?

Non-planar Feynman diagrams have remained a challenge for the twistor approach. The problem is simple: there is no canonical ordering of the extrenal particles and the loop integrand involving tricky shifts in integrations to get finite outcome is not unique and well-defined so that twistor Grasmann approach encounters difficulties.

Recently Nima Arkani-Hamed et al have have also considered non-planar MHV diagrams (having minimal number of "wrong" helicities) of N=4 SUSY, and shown that they can be reduced to non-planar diagrams for different permutations of vertices of planar diagrams ordered naturally. There are several integration regions identified as positive Grassmannians corresponding to different orderings of the external lines inducing non-planarity. This does not however hold true generally.

At the QFT limit the crossings of lines emerges purely combinatorially since Feynman diagrams are purely combinatorial objects with the ordering of vertices determining the topological properties of the diagram. Non-planar diagrams correspond to diagrams, which do not allow crossing-free imbedding to plane but require higher genus surface to get rid of crossings.

  1. The number of the vertices of the diagram and identification of lines connecting them determines the diagram as a graph. This defines also in TGD framework Feynman diagram like structure as a graph for the fermion lines and should be behind non-planarity in QFT sense.
  2. Could 2-D Feynman graphs exists also at geometric rather than only combinatorial level? Octonionization at imbedding space level requires identification of preferred M2⊂ M4 defining a preferred hyper-complex sub-space. Could the projection of the Fermion lines defined concrete geometric representation of Feynman diagrams?
  3. Despite their purely combinatorial character Feynman diagrams are analogous to knots and braids. For years ago I proposed the generalization of the construction of knot invariants in which one gradually eliminates the crossings of the knot projection to end up with a trivial knot is highly suggestive as a procedure for constructing the amplitudes associated with the non-planar diagrams. The outcome should be a collection of planar diagrams calculable using twistor Grassmannian methods. Scattering amplitudes could be seen as analogs of knot invariants. The reduction of MHV diagrams to planar diagrams could be an example of this procedure.
One can imagine also analogs of non-planarity, which are geometric and topological rather than combinatorial and not visible at the QFT limit of TGD.
  1. The fermion lines representing boundaries of string world sheets at the light-like orbits of partonic 2-surfaces can get braided. The same can happen also for the string boundaries at space-like 3-surfaces at the ends of the space-time surface. The projections of these braids to partonic 2-surfaces are analogs of non-planar diagrams. If the fermion lines at single wormhole throat are regarded effectively as a line representing one member of super-multiplet, this kind of braiding remains below the resolution used and cannot correspond to the braiding at QFT limit.
  2. 2-knotting and 2-braiding are possible for partonic 2-surfaces and string world sheets as 2-surfaces in 4-D space-time surfaces and have no counterpart at QFT limit.
  3. If one can approximate space-time sheets by maps from M4 to CP2, one expects General Relativity and QFT description to be good approximations. GRT space-time is obtained by replacing space-time sheets with single sheet - a piece of slightly deformed Minkowski space but without assupmtion about imbedding to H. Induced classical gravitational field and gauge fields are sums of those associated with the sheets. The generalized Feynman diagrams with lines at various sheets and going also between sheets are projected to single piece of M4. Many-sheetedness makes 1-homology non-trivial and implies analog of braiding, which should be however invisible at QFT limit.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

About the twistorial description of light-likeness in 8-D sense using octonionic spinors

The twistor approach to TGD require that the expression of light-likeness of M4 momenta in terms of twistors generalizes to 8-D case. The light-likeness condition for twistors states that the 2× 2 matrix representing M4 momentum annihilates a 2-spinor defining the second half of the twistor. The determinant of the matrix reduces to momentum squared and its vanishing implies the light-likeness. This should be generalized to a situation in one has M4 and CP2 twistor, which are not light-like separately but light-likeness in 8-D sense holds true (allowing massive particles in M4 sense and thus generalization of twistor approach for massive particles).

The case of M8=M4× E4

M8-H duality suggests that it might be useful to consider first the twistorialiation of 8-D light-likeness first the simpler case of M8 for which CP2 corresponds to E4. It turns out that octonionic representation of gamma matrices provide the most promising formulation.

In order to obtain quadratic dispersion relation, one must have 2× 2 matrix unless the determinant for the 4× 4 matrix reduces to the square of the generalized light-likeness condition.

  1. The first approach relies on the observation that the 2× 2 matrices characterizing four-momenta can be regarded as hyper-quaternions with imaginary units multiplied by a commuting imaginary unit. Why not identify space-like sigma matrices with hyper-octonion units?
  2. The square of hyper-octonionic norm would be defined as the determinant of 4× 4 matrix and reduce to the square of hyper-octonionic momentum. The light-likeness for pairs formed by M4 and E4 momenta would make sense.
  3. One can generalize the sigma matrices representing hyper-quaternion units so that they become the 8 hyper-octonion units. Hyper-octonionic representation of gamma matrices exists (γ0z× 1, γk= σy× Ik) but the octonionic sigma matrices represented by octonions span the Lie algebra of G2 rather than that of SO(1, 7). This dramatically modifies the physical picture and brings in also an additional source of non-associativity. Fortunately, the flatness of M8 saves the situation.
  4. One obtains the square of p2=0 condition from the massless octonionic Dirac equation as vanishing of the determinant much like in the 4-D case. Since the spinor connection is flat for M8 the hyper-octonionic generalization indeed works.
This is not the only possibility that I have considered.
  1. Is it enough to allow the four-momentum to be complex? One would still have 2× 2 matrix and vanishing of complex momentum squared meaning that the squares of real and imaginary parts are same (light-likeness in 8-D sense) and that real and imaginary parts are orthogonal to each other. Could E4 momentum correspond to the imaginary part of four-momentum?
  2. The signature causes the first problem: M8 must be replaced with complexified Minkowski space Mc4 for to make sense but this is not an attractive idea although Mc4 appears as sub-space of complexified octonions.
  3. For the extremals of Kähler action Euclidian regions (wormhole contacts identifiable as deformations of CP2 type vacuum extremals) give imaginary contribution to the four-momentum. Massless complex momenta and also color quantum numbers appear also in the standard twistor approach. Also this suggest that complexification occurs also in 8-D situation and is not the solution of the problem.

The case of M8=M4× CP2

What about twistorialization in the case of M4× CP2? The introduction of wave functions in the twistor space of CP2 seems to be enough to generalize Witten's construction to TGD framework and that algebraic variant of twistors might be needed only to realize quantum classical correspondence. It should correspond to tangent space counterpart of the induced twistor structure of space-time surface, which should reduce effectively to 4-D one by quaternionicity of the space-time surface.

  1. For H=M4× CP2 the spinor connection of CP2 is not trivial and the G2 sigma matrices are proportional to M4 sigma matrices and act in the normal space of CP2 and to M4 parts of octonionic imbedding space spinors, which brings in mind co-associativity. The octonionic charge matrices are also an additional potential source of non-associativity even when one has associativity for gamma matrices.

    Therefore the octonionic representation of gamma matrices in entire H cannot be physical. It is however equivalent with ordinary one at the boundaries of string world sheets, where induced gauge fields vanish. Gauge potentials are in general non-vanishing but can be gauge transformed away. Here one must be of course cautious since it can happen that gauge fields vanish but gauge potentials cannot be gauge transformed to zero globally: topological quantum field theories represent basic example of this.

  2. Clearly, the vanishing of the induced gauge fields is needed to obtain equivalence with ordinary induced Dirac equation. Therefore also string world sheets in Minkowskian regions should have 1-D CP2 projection rather than only having vanishing W fields if one requires that octonionic representation is equivalent with the ordinary one. For CP2 type vacuum extremals electroweak charge matrices correspond to quaternions, and one might hope that one can avoid problems due to non-associativity in the octonionic Dirac equation. Unless this is the case, one must assume that string world sheets are restricted to Minkowskian regions. Also imbedding space spinors can be regarded as octonionic (possibly quaternionic or co-quaternionic at space-time surfaces): this might force vanishing 1-D CP2 projection.
    1. Induced spinor fields would be localized at 2-surfaces at which they have no interaction with weak gauge fields: of course, also this is an interaction albeit very implicit one! This would not prevent the construction of non-trivial electroweak scattering amplitudes since boson emission vertices are essentially due to re-groupings of fermions and based on topology change.
    2. One could even consider the possibility that the projection of string world sheet to CP2 corresponds to CP2 geodesic circle at which also the induced gauge potentials vanish so that one could assign light-like 8-momentum to entire string world sheet, which would be minimal surface in M4× S1. This would mean enormous technical simplification in the structure of the theory. Whether the spinor harmonics of imbedding space with well-defined M4 and color quantum numbers can co-incide with the solutions of the induced Dirac operator at string world sheets defined by minimal surfaces remains an open problem.
    3. String world sheets cannot be present inside wormhole contacts, which have 4-D CP2 projection so that string world sheets cannot carry vanishing induced gauge fields. Therefore the strings in TGD are open.

To sum up, the generalization of the notion of twistor to 8-D context allows description of massive particles using twistors but requires that octonionic Dirac equation is introduced. If one requires that octonionic and ordinary description of Dirac equation are equivalent, the description is possible only at surfaces having at most 1-D CP2 projection - geodesic circle for the most stringent option. The boundaries of string world sheets are such surfaces and also string world sheets themselves if they have 1-D CP2 projection, which must be geodesic circle if also induce gauge potentials are required to vanish. In spirit with M8-H duality, string boundaries give rise to classical M8 twistorizalization analogous to the standard M4 twistorialization and generalize 4-momentum to massless 8-momentum whereas imbedding space spinor harmonics give description in terms of four-momentum and color quantum numbers. One has SO(4)-SU(3) duality: a wave function in the space of 8-momenta corresponds to SO(4) description of hadrons at low energies as opposed to that for quarks at high energies in terms of color. The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle (EP) in the most concrete form: gravitational four-momentum equals to inertial four-momentum. EP for internal quantum numbers is clearly more delicate. In twistorialization also helicity is brought and for CP2 degrees of freedom M8 helicity means that electroweak spin is described in terms of helicity.

Biologists have a principle known as "ontogeny recapitulates phylogeny" (ORP) stating that the morphogenesis of the individual reflects evolution of the species. The principle seems to be realized also in theoretical physics - at least in TGD Universe. ORP would now say that the evolution of theoretical physics via the emergence of increasingly complex notion of particle reflects the structure physics itself. Point like particles are really there as points at partonic 2-surfaces carrying fermion number: their 1-D orbits correspond to the boundaries of string world sheets; 2-D hyper-complex string world sheets in flat space (M4× S1) are there and carry induced spinors; also complex (or co-complex) partonic 2-surfaces (Euclidian string world sheets) and carry particle numbers; 3-D space-like surfaces at the ends of causal diamonds (CDs) and the 3-D light-like orbits of partonic 2-surfaces are there; 4-D space-time surfaces are there as quaternionic or co-quaternionic sub-manifolds of 8-D octonionic imbedding space: there the hierarchy ends since there are no higher-dimensional classical number fields. ORP would thus also realize evolution of mathematics at the level of physics.

The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle in the most concrete form: gravitational four-momentum equals to inertial four-momentum.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Witten's twistor string approach and TGD

The twistor Grassmann approach has led to a phenomenal progress in the understanding of the scattering amplitudes of gauge theories, in particular the N=4 SUSY.

As a non-specialist I have been frustrated about the lack of concrete picture, which would help to see how twistorial amplitudes might generalize to TGD framework. A pleasant surprise in this respect was the proposal of a particle interpretation for the twistor amplitudes by Nima Arkani Hamed et al in the article "Unification of Residues and Grassmannian Dualities" .

In this interpretation incoming particles correspond to spheres CP1 so that n-particle state corresponds to (CP1)n/Gl(2) (the modding by Gl(2) might be seen as a kind of formal generalization of particle identity by replacing permutation group S2 with Gl(2) of 2× 2 matrices). If the number of "wrong" helicities in twistor diagram is k, this space is imbedded to CPk-1n/Gl(k) as a surface having degree k-1 using Veronese map to achieve the imbedding. The imbedding space can be identified as Grassmannian G(k,n) . This surface defines the locus of the multiple residue integral defining the twistorial amplitude.

The particle interpretation brings in mind the extension of single particle configuration space E3 to its Cartesian power E3n/Sn for n-particle system in wave mechanics. This description could make sense when point-like particle is replaced with 3-surface or partonic 2-surface: one would have Cartesian product of WCWs divided my Sn. The generalization might be an excellent idea as far calculations are considered but is not in spirit with the very idea of string models and TGD that many-particle states correspond to unions of 3-surfaces in H (or light-like boundaries of causal diamond (CD) in Zero Energy Ontology (ZEO).

Witten's twistor string theory

Witten's twistor string theory is more in spirit with TGD at fundamental level since it is based on the identification of generalization of vertices as 2-surfaces in twistor space.

  1. There are several kinds of twistors involved. For massless external particles in eigenstates of momentum and helicity null twistors code the momentum and helicity and are pairs of 2-spinor and its conjugate. More general momenta correspond to two independent 2-spinors.

    One can perform twistor Fourier transform for the conjugate 2-spinor to obtain twistors coding for the points of compactified Minkowski space. Wave functions in this twistor space characterized by massless momentum and helicity appear in the construction of twistor amplitudes. BCFW recursion relation allows to construct more complex amplitudes assuming that intermediate states are on mass shells massless states with complex momenta.

    One can perform twistor Fourier transformation (there are some technical problems in Minkowski signature) also for the second 2-spinor to get what are called momentum twistors providing in some aspects simpler description of twistor amplitudes. These code for the four-momenta propagating between vertices at which the incoming particles arrive and the differences if two subsequent momenta are equal to massless external momenta.

  2. In Witten's theory the interactions of incoming particles correspond to amplitudes in which the twistors appearing as arguments of the twistor space wave functions characterized by momentum and helicity are localized to complex curves X2 of twistor space CP3 or its Minkowskian counterpart. This can be seen as a non-local twistor space variant of local interactions in Minkowski space.

    The surfaces X2 are characterized by their degree d (of the polynomial of complex coordinates defining the algebraic 2-surface) the genus g of the algebraic surface, by the number k of "wrong" (helicity violating) helicities, and by the number of loops of corresponding diagram of SUSY amplitude: one has d= k-1+ l, g≤ l. The interaction vertex in twistor space is not anymore completely local but the n particles are at points of the twistorial surface X2.

Generalization of the notion of twistor to 8-D context

In the addition to the article Classical part of the twistor story a proposal generalizing Witten's approach to TGD is discussed. The following gives a very concise summary of the article.

Consider first various aspects of twistorialization.

  1. The fundamental challenge is the generalization of the notion of twistor associated with massless particle to 8-D context, first for M4=M4× E4 and then for H= M4 × CP2. The notion of twistor space solves this question at geometric level. As far as construction of the TGD variant of Witten's twistor string is considered, this might be quite enough.

    What is especially nice, that twistorialization extends to from spin to include also electroweak spin. These two spins correspond correspond to M4 and CP2helicities for the twistor space amplitude, and are non-local properties of these amplitudes. In TGD framework only twistor amplitudes for which helicities correspond to that for massless fermion and antifermion are possible and by fermion number conservation the numbers of positive and negative helicities are identical and equal to the fermion number (or antifermion number). Separate lepton and baryon number conservation realizing 8-D chiral symmetry implies that M4 and CP2 helicities are completely correlated. At twistorial level this means restriction to amplitudes with k=2(n(F)-n(Fbar)) if no mixing of M4and CP2 occurs (massivation in M4 sense) . This in quark and lepton sector separately.

  2. M8-H duality and quantum-classical correspondence however suggest that M8 twistors might allow tangent space description of four-momentum, spin, color quantum numbers and electroweak numbers and that this is needed. What comes in mind is the identification of fermion lines as light-like geodesics possessing M8 valued 8-momentum, which would define the long sought gravitational counterparts of four-momentum and color quantum numbers at classical point-particle level. The M8 part of this four-momentum would be equal to that associated with imbedding space spinor mode characterizing the ground state of super-conformal representation for fundamental fermion.

    Hence one might also think of starting from the 4-D condition relating Minkowski coordinates to twistors and looking what it could mean in the case of M8. The generalization is indeed possible in M8 =M4× E4 by its flatness if one replaces gamma matrices with octonionic gamma matrices.

    In the case of M4× CP2 situation is different since for octonionic gamma matrices SO(1,7) is replaced with G2, and the induced gauge fields have different holonomy structure than for ordinary gamma matrices and octonionic sigma matrides appearing as charge matrices bring in also an additional source of non-associativity. Certainly the notion of the twistor Fourier transform fails since CP2 Dirac operator cannot be algebraized.

    Algebraic twistorialization however works for the light-like fermion lines at which the ordinary and octonionic representations for the induced Dirac operator are equivalent. One can indeed assign 8-D counterpart of twistor to the particle classically as a representation of light-like hyper-octonionic four-momentum having massive M4 and CP2 projections and CP2 part perhaps having interpretation in terms of classical tangent space representation for color and electroweak quantum numbers at fermionic lines.

    If all induced electroweak gauge fields - rather than only charged ones as assumed hitherto - vanish at string world sheets, the octonionic representation is equivalent with the ordinary one. The CP2 projection of string world sheet should be 1-dimensional: inside CP2 type vacuum extremals this is impossible, and one could even consider the possibility that the projection corresponds to CP2 geodesic circle. This would be enormous technical simplification. What is important that this would not prevent obtaining non-trivial scattering amplitudes at elementary particle level since vertices would correspond to re-arrangement of fermion lines between the generalized lines of Feynman diagram meeting at the vertices (partonic 2-surfaces).

  3. In the fermionic sector one is forced to reconsider the notion of the induced spinor field. The modes of the imbedding space spinor field should co-incide in some region of the space-time surface with those of the induced spinor fields. The light-like fermionic lines defined by the boundaries of string world sheets or their ends are the obvious candidates in this respect. String world sheets is perhaps too much to require. Number theoretically string world sheets would be commutative surfaces analogous to space-time surfaces as quaternionic surfaces of octonionic imbedding space.

    The only reasonable identification of string world sheet gamma matrices is as induced gamma matrices and super-conformal symmetry requires that the action contains string world sheet area as an additional term just as in string models. String tension would correspond to gravitational constant and its value - that is ratio to the CP2 radius squared, would be fixed by quantum criticality.

Generalization of Witten's construction in TGD framework

The generalization of the Witten's geometric construction of scattering amplitudes relying on the induction of the twistor structure of the imbedding space to that associated with space-time surface looks surprisingly straight-forward and would provide more precise formulation of the notion of generalized Feynman diagrams forcing to correct some wrong details.

  1. All generalized Feynman graphs defined in terms of Euclidian regions of space-time surface are lifted to twistor spaces. Incoming particles correspond quantum mechanically to twistor space amplitudes defined by their momenta and helicities and and classically to the entire twistor space of space-time surface as a surface in the twistor space of H. Of course, also the Minkowskian regions have this lift. The vertices of Feynman diagrams correspond to regions of twistor space in which the incoming twistor spaces meet along their 5-D ends having also S2 bundle structure over space-like 3-surfaces. These space-like 3-surfaces correspond to ends of Euclidian and Minkowskian space-time regions separated from each other by light-like 3-surfaces at which the signature of the metric changes from Minkowskian to Euclidian. These "partonic orbits" have as their ends 2-D partonic surfaces. By strong form of General Coordinate Invariance implying strong of holography, these 2-D partonic surfaces and their 4-D tangent space data should code for quantum physics. Their lifts to twistor space are 4-D S2 bundles having partonic 2-surface X2 as base.

    One of the nice outcomes is that the genus appearing in Witten's formulation naturally corresponds to family replication in TGD framework.

  2. All elementary particles are many particle bound states of massless fundamental fermions: the non-collinearity of massless momenta explains massivation. The fundamental fermions are localized at wormhole throats defining the light-like orbits of partonic 2-surfaces. Throats are associated with wormhole contacts connecting two space-time sheets. Stability of the contact is guaranteed by non-vanishing monopole magnetic flux through it and this requires the presence of second wormhole contact so that a closed magnetic flux tube carrying monopole flux and involving the two space-time sheets is formed. The net fermionic quantum numbers of the second throat correspond to particle's quantum numbers and above weak scale the weak isospins of the throats sum up to zero.
  3. Fermionic 2-vertex is the only local many-fermion vertex being analogous to a mass insertion. The non-triviality of the theory could be seen to follow from the analog of OZI mechanism: the fermionic lines inside partonic orbits are redistributed in vertices. Lines can also turn around in time direction which corresponds to creation or annihilation of a pair. 3-particle vertices are obtained only in topological sense as 3 space-time surfaces are glued together at their ends. The interaction between fermions at different wormhole throats is described in terms of string world sheets.

  4. The earlier proposal was that the fermions in the internal fermion lines are massless in M4 sense but have non-physical helicity so that the algebraic M4 Dirac operator emerging from the residue integration over internal four-momentum does not annihilate the state at the end of the propagator line. Now the algebraic induced Dirac operator defines the propagator at fermion lines. Should one assume generalization of non-physical helicity also now?
  5. All this stuff must be lifted to twistorial level and one expects that the lift to S2 bundle allows an alternative description of fermions and spinor structure so that one can speak of induced twistor structure instead of induced spinor structure. This approach allows also a realization of M4 conformal symmetries in terms of globally well-defined linear transformations so that it might be that twistorialization is not a mere reformulation but provides a profound unification of bosonic and fermionic degrees of freedom.

The emergence of fundamental 4-fermion vertex and of boson exchanges

The emergence of the fundamental 4-fermion vertex and of boson exchanges deserves a more detailed discussion.

  1. I have proposed that the discontinuity of the Dirac operator at partonic two-surface (corner of fermion line) defines both the fermion boson vertex and TGD analog of mass insertion (not scalar but imbedding space vector) giving rise to mass parameter having interpretation as Higgs vacuum expectation and various fermionic mixing parameters at QFT limit of TGD obtained by approximating many-sheeted space-time of TGD with the single sheeted region of M4 such that gravitational field and gauge potentials are obtained as sums of those associated with the sheets.
  2. Non-trivial scattering requires also correlations between fermions at different partonic 2-surfaces. Both partonic 2-surfaces and string world sheets are needed to describe these correlations. Therefore the string world sheets and partonic 2-surfaces cannot be dual: both are needed and this means deviation from Witten's theory. Fermion vertex corresponds to a "corner" of a fermion line at partonic 2-surface at which generalized 4-D lines of Feynman diagram meet and light-like fermion line changes to space-like one. String world sheet with its corners at partonic 2-surfaces (wormhole throats) describes the momentum exchange between fermions. The space-like string curve connecting two wormhole throats serves as the analog of the exchanged gauge boson.
  3. Two kinds of 4-fermion amplitudes can be considered depending on whether the string connects throats of single wormhole contact (CP2 scale) or of two wormhole contacts (p-adic length scale - typically of order elementary particle Compton length). If string worlds sheets have 1-D CP2 projection, only Minkowskian string world sheets are possible. The exchange in Compton scale should be assignable to the TGD counterpart of gauge boson exchange and the fundamental 4-fermion amplitude should correspond to single wormhole contact: string need not to be involved now. Interaction is basically classical interaction assignable to single wormhole contact generalizing the point like vertex.
  4. The possible TGD counterparts of BCFW recursion relations should use the twistorial representations of fundamental 4-fermion scattering amplitude as seeds. Yangian invariance poses very strong conditions on the form of these amplitudes and the exchange of massless bosons is suggestive for the general form of amplitude.

    The 4-fermion amplitude assignable to two wormhole throats defines the analog of gauge boson exchange and is expressible as fusion of two fundamental 4-fermion amplitudes such that the 8-momenta assignable to the fermion and anti-fermion at the opposite throats of exchanged wormhole contact are complex by BCFW shift acting on them to make the exchanged momenta massless but complex. This entity could be called fundamental boson (not elementary particle).

  5. Can one assume that the fundamental 4-fermion amplitude allows a purely formal composition to a product of BFF amplitudes? Two 8-momenta at both FFB vertices must be complex so that at least two external fermionic momenta must be complex. These external momenta are naturally associated with the throats of the a wormhole contact defining virtual fundamental boson. Rather remarkably, without the assumption about product representation one would have general four-fermion vertex rather than boson exchange. Hence gauge theory structure is not put in by hand but emerges.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Could quaternion analyticity make sense for the preferred extremals?

The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. The obvious ideas coming in mind are appropriately defined quaternionic and octonion analyticity. I have used a considerable amount of time to consider these possibilities but had to give up the idea about octonion analyticity could somehow allow to preferred extemals.

Basic idea

One can argue that quaternion analyticity is the more natural option in the sense that the local octonionic imbedding space coordinate (or at least M8 or E8 coordinate, which is enough if M8-H duality holds true) would for preferred extremals be expressible in the form

o(q)= u(q) + v(q)× I .

Here q is quaternion serving as a coordinate of a quaternionic sub-space of octonions, and I is octonion unit belonging to the complement of the quaternionic sub-space, and multiplies v(q) from right so that quaternions and qiaternionic differential operators acting from left do not notice these coefficients at all. A stronger condition would be that the coefficients are real. u(q) and v(q) would be quaternionic Taylor- of even Laurent series with coefficients multiplying powers of q from right for the same reason.

I ended up to this idea after finding two very interesting articles discussing the generalization of Cauchy-Riemann equations. The first article was about so called triholomorphic maps between 4-D almost quaternionic manifolds. The article gave as a reference an article about quaternionic analogs of Cauchy-Riemann conditions discussed by Fueter long ago (somehow I have managed to miss Fueter's work just like I missed Hitchin's work about twistorial uniqueness of M4 and CP2), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.

The so called Cauhy-Riemann-Fueter conditions generalize Cauchy-Riemann conditions. These conditions are however not unique.

  1. The translationally invariant form of CRF conditions is ∂q* f=0 or explicitly

    (∂t+ ∂x I+∂yJ +∂zK)f=0 .

    This form does not allow quaternionic Taylor series although it allows functions depending on complex coordinate z of some complex-plane only.

    Note that the Taylor coefficients multiplying powers of the coordinate from right are arbitrary quaternions. What looks pathological is that even linear functions of q fail be solve this condition. What is however interesting that in flat space the equation is equivalent with Dirac equation for a pair of Majorana spinors.

  2. Second form of CRF conditions is

    (∂t+ (xI+yJ+zK)/r2) r∂r)f=0 .

    Here r denotes the imaginary part of q. This form allows both the desired quaternionic Taylor series and ordinary holomorphic functions of complex variable in one of the 3 complex coordinate planes as general solutions.

    This form of CRF is neither Lorentz invariant nor translationally invariant but remains invariant under simultaneous scalings of t and r and under time translations. Under rotations of either coordinates or of imaginary units the spatial part transforms like vector so that quaternionic automorphism group SO(3) serves as a moduli space for these operators.

  3. The interpretation of the latter solutions inspired by ZEO would be that in Minkowskian regions r corresponds to the light-like radial coordinate of the either boundary of CD, which is part of δ M4+/-. The radial scaling operator is that assigned with the light-like radial coordinate of the light-cone boundary. A slicing of CD by surfaces parallel to the δ M4+/- is assumed and implies that the line r=0 connecting the tips of CD is in a special role. The line connecting the tips of CDa defines coordinate line of time coordinate. The breaking of rotational invariance corresponds to the selection of a preferred quaternion unit defining the twistor structure and preferred complex sub-space.

    In regions of Euclidian signature r could corresponds to the the radial Eguchi-Hanson coordinates and r=0 corresponds to a fixed point of U(2) subgroup under which CP2 complex coordinates transform linearly.

  4. In TGD framework one must have also ordinary holomorphic functions mapping 4-D quaterionic space to 2-D complex space associated with real and/or imaginary part of octonion: they would be associated with extremals for which M4 and/or CP2 projection is 2-dimensional (cosmic strings and massless extremals).

    Since the conditions are linear one can also have superpositions of different types of solutions and they could describe perturbations of say cosmic strings and massless extremals.

  5. It is of course possible that an entire moduli space of tri-holomorphic operators exists and would be interesting to know the most general form of tri-holomorphic operator. For instance, if one performs a quaternion analytic map of the space-time surface the form of the operator defining C-R-F conditions changes and becomes rather complex. This suggests that the operator as it is defined above indeed refers to geometrically preferred coordinates as already suggested. One must be however very cautious. It might well be that quaternion conformal transforms of this operator are possible but that they are equivalent.
  6. Both generalizations of the C-R-F conditions generalize to the octonionic situation and right multiplication of powers of octonion by Taylor coefficients plus linearity imply that there are no problems with associativity. This inspires several questions.

    Could octonion analytic maps of imbedding space allow to construct new solutions from the existing ones? Could quaternion analytic maps applied at space-time level act as analogs of holomorphic maps and generalize conformal invariance to 4-D context?

The first form of Cauchy-Rieman-Fueter conditions

Cauhy-Riemann-Fueter (CRF) conditions generalize Cauchy-Riemann conditions. These conditions are however not unique. Consider first the translationally invariant form of CRF conditions.

  1. The translationally invariant form of CRF conditions is ∂q*f=0 or explicitly

    qf=(∂t- ∂x I-∂y J -∂zK)f=0 .

    This form does not allow quaternionic Taylor series. Note that the Taylor coefficients multiplying powers of the coordinate from right are arbitrary quaternions. What looks pathological is that even linear functions of q fail be solve this condition. What is however interesting that in flat space the equation is equivalent with Dirac equation for a pair of Majorana spinors.

  2. The condition allows functions depending on complex coordinate z of some complex-plane only. It also allows functions satisfying two separate analyticity conditions, say

    ∂u*f= (∂t- ∂x I)f=0 ,

    v*f=-(∂y J+∂z K)f=-J(∂y -∂ zI) f=0 .

    In the latter formula J multiplies from left! One has good hopes of obtaining holomorphic functions of two complex coordinates. This might be enough to understand the preferred extremals of Kähler action as quaternion analytic mops.

    There are potential problems due to non-commutativity of u=t+/- xI and v=yJ+/- zK= (y+/- zI) J (note that J ~ multiplies from right!) and ∂u and ∂v. A prescription for the ordering of the powers u and v in the polynomials of u and v appearing in the double Taylor series seems to be needed. For instance, powers of u can be taken to be at left and v or of a related variable at right.

    By the linearity of ∂v* one can leave J to the left and commute only (∂y -∂ zI) through the u-dependent part of the series: this operation is trivial. The condition ∂vf=0 is satisfied if the polynomials of y and z are polynomials of y+iz multiplied by J from right. The solution ansatz is thus product of Taylor series of monomials fmn= (x+iy)m (y+iz)nJ with Taylor coefficients amn, which multiply the monomials from right and are arbitrary quaternions. Note that the monomials (y+iz)n do not reduce to polynomials of v and that the ordering of these powers is arbitrary. If the coefficients amn are real f maps 4-D quaternionic region to 2-D region spanned by J and K. Otherwise the image is 4-D.

  3. By linearity the solutions obey linear superposition. They can be also multiplied if product is defined as ordered product in such a manner that only the powers t+ix and y+iz are multiplied together at left and coefficients amn are multiplied together at right. The analogy with quantum non-commutativity is obvious.

  4. In Minkowskian signature one must multiply imaginary units I,J,K with an additional commuting imaginary unit i. This would give solutions as powers of (say) t+ex, e=iI with e2=1 representing imaginary unit of hyper-complex numbers. The natural interpretation would be as algebraic extension which is analogous to the extension of rational number by adding algebraic number, say 21/2 to get algebraically 2-dimensional structure but as real numbers 1-D structure. Only the non-commutativity with J and K distinguishes e from e=+/- 1 and if J and K do not appear in the function, one can replace e by +/- 1 in t+ex to get just t+/- x appearing as argument for waves propagating with light velocity.

Second form of CRF conditions

Second form of CRF conditions proposed in the second reference is tailored in order to realize the almost obvious manner to realize quaternion analyticity.

  1. The ingenious idea is to replace preferred quaternionic imaginary unit by a imaginary unit which is in radial direction: er= (xI+yJ+zK)/r and require analyticity with respect to the coordinate t+e r. The solution to the condition is power series in t+rer= q so that one obtains quaternion analyticity.
  2. The excplicit form of the conditions is

    l (∂t- err)f= (∂t-er/r r∂r)f=0 .

    This form allows both the desired quaternionic Taylor series and ordinary holomorphic functions of complex variable in one of the 3 complex coordinate planes as general solutions.

  3. This form of CRF is neither Lorentz invariant nor translationally invariant but remains invariant under simultaneous scalings of t and r and under time translations. Under rotations of either coordinates or of imaginary units the spatial part transforms like vector so that quaternionic automorphism group SO(3) serves as a moduli space for these operators.
  4. The interpretation of the latter solutions inspired by ZEO would be that in Minkowskian regions r corresponds to the light-like radial coordinate of the either boundary of CD, which is part of δ M4+/-. The radial scaling operator is that assigned with the light-like radial coordinate of the light-cone boundary. A slicing of CD by surfaces parallel to the δ M4+/- is assumed and implies that the line r=0 connecting the tips of CD is in a special role. The line connecting the tips of CD defines coordinate line of time coordinate. The breaking of rotational invariance corresponds to the selection of a preferred quaternion unit defining the twistor structure and preferred complex sub-space.

    In regions of Euclidian signature r could correspond to the radial Eguchi-Hanson coordinate of CP2 and r=0 corresponds to a fixed point of U(2) subgroup under which CP2 complex coordinates transform linearly.

  5. Also in this case one can ask whether solutions depending on two complex local coordinates analogous to those for translationally invariant CRF condition are possible. The remain imaginary units would be associated with the surface of sphere allowing complex structure.

Generalization of CRF conditions?

Could the proposed forms of CRF conditions be special cases of much more general CRF conditions as CR conditions are?

  1. Ordinary complex analysis suggests that there is an infinite number of choices of the quaternionic coordinates related by the above described quaternion-analytic maps with 4-D images. The form of of the CRF conditions would be different in each of these coordinate systems and would be obtained in a straightforward manner by chain rule.
  2. One expects the existence of large number of different quaternion-conformal structures not related by quaternion-analytic transformations analogous to those allowed by higher genus Riemann surfaces and that these conformal equivalence classes of four-manifolds are characterized by a moduli space and the analogs of Teichmueller parameters depending on 3-topology. In TGD framework strong form of holography suggests that these conformal equivalence classes for preferred extremals could reduce to ordinary conformal classes for the partonic 2-surfaces. An attractive possibility is that by conformal gauge symmetries the functional integral over WCW reduces to the integral over the conformal equivalence classes.
  3. The quaternion-conformal structures could be characterized by a standard choice of quaternionic coordinates reducing to the choice of a pair of complex coordinates. In these coordinates the general solution to quaternion-analyticity conditions would be of form described for the linear ansatz. The moduli space corresponds to that for complex or hyper-complex structures defined in the space-time region.

Geometric formulation of the CRF conditions

The previous naive generalization of CRF conditions treats imaginary units without trying to understand their geometric content. This leads to difficulties when when tries to formulate these conditions for maps between quaternionic and hyper-quaternionic spaces using purely algebraic representation of imaginary units since it is not clear how these units relate to each other.

In the first article the CRF conditions are formulated in terms of the antisymmetric (1,1) type tensors representing the imaginary units: they exist for almost quaternionic structure and presumably also for almost hyper-quaternionic structure needed in Minkowskian signature.

The generalization of CRF conditions is proposed in terms of the Jacobian J of the map mapping tangent space TM to TN and antisymmetric tensors Ju and Ju representing the quaternionic imaginary units of N and M. The generalization of CRF conditions reads as

J- ∑u Ju J ju=0 .

For N=M it reduces to the translationally invariant algebraic form of the conditions discussed above. These conditions seem to be well-defined also when one maps quaternionic to hyper-quaternionic space or vice versa. These conditions are not unique. One can perform an SO(3) rotation (quaternion automorphism) of the imaginary units mediated by matrix Λuv to obtain

J- Λuv JuJ jv=0 .

The matrix Λ can depend on point so that one has a kind of gauge symmetry. The most general triholomorphic map allows the presence of Λ Note that these conditions make sense on any coordinates and complex analytic maps generate new forms of these conditions.

Covariant forms of structure constant tensors reduce to octonionic structure constants and this allows to write the conditions explicitly. The index raising of the second index of the structure constants is however needed using the metrics of M and N. This complicates the situation and spoils linearity: in particular, for surfaces induced metric is needed. Whether local SO(3) rotation can eliminate the dependence on induced metric is an interesting question. Minkowskian imaginary units differ only by multiplication by i from Euclidian so that Minkowskian structure constants differ only by sign from those for Euclidian ones.

In the octonionic case the geometric generalization of CRF conditions does not seem to make sense. By non-associativity of octonion product it is not possible to have a matrix representation for the matrices so that a faithful representation of octonionic imaginary units as antisymmetric 1-1 forms does not make sense. If this representation exists it it must map octonionic associators to zero. Note however that CRF conditions do not involve products of three octonion units so that they make sense as algebraic conditions at least.

Does residue calculus generalize?

CRF conditions allow to generalize Cauchy formula allowing to express value of analytic function in terms of its boundary values. This would give a concrete realization of the holography in the sense that the physical variables in the interior could be expressed in terms of the data at the light-like partonic orbits and and the ends of the space-time surface. Triholomorphic function satisfies d'Alembert/Laplace equations - in induced metric in TGD framework- so that the maximum modulus principle holds true. The general ansatz for a preferred extremals involving Hamilton-Jacobi structure leads to d'Alembert type equations for preferred extremals.

Could the analog of residue calculus exist? Line integral would become 3-D integral reducing to a sum over poles and possible cuts inside the 3-D contour. The space-like 3-surfaces at the ends of CDs could define natural integration contours, and the freedom to choose contour rather freely would reflect General Coordinate Invariance. A possible choice for the integration contour would be the closed 3-surface defined by the union of space-like surfaces at the ends of CD and by the light-like partonic orbits.

Poles and cuts would be in the interior of the space-time surface. Poles have co-dimension 2 and cuts co-dimension 1. Strong form of holography suggests that partonic 2-surfaces and perhaps also string world sheets serve as candidates for poles. Light-like 3-surfaces (partonic orbits) defining the boundaries between Euclidian and Minkowskian regions are singular objects and could serve as cuts. The discontinuity would be due to the change of the signature of the induced metric. There are CDs inside CDs and one can also consider the possibility that sub-CDs define cuts, which in turn reduce to cuts associated with sub-CDs.

Could one understand the preferred extremals in terms of quaternion-analyticity?

Could one understand the preferred extremals in terms of quaternion-analyticity or its possible generalization to an analytic representation for co-quaternionicity expected in space-time regions with Euclidian signature? What is the generalization of the CRF conditions for the counterparts of quaternion-analytic maps from hyper-quaternionic X4 to quaternionic CP2 and from quaternionic X4 to hyper-quaternionic M4? It has already become clear that this problem can be probably solved by using the the geometric representation for quaternionic imaginary units.

The best thing to do is to look whether this is possible for the known extremals: CP2 type vacuum extremals, vacuum extremals expressible as graph of map from M4 to a Lagrangian sub-manifold of CP2, cosmic strings of form X2× Y2⊂ M4 × CP2 such that X2 is string world sheet (minimal surface) and Y2 complex sub-manifold of CP2. One can also check whether Hamilton-Jacobi structure of M4 and of Minkowskian space-time regions and complex structure of CP2 have natural counterparts in the quaternion-analytic framework.

  1. Consider first cosmic strings. In this case the quaternionic-analytic map from X4 = X2× Y2 to M4× CP2 with octonion structure would be map X4 to 2-D string world sheet in M2 and Y2 to 2-D complex manifold of CP2. This could be achieved by using the linear variant of CRF condition. The map from X4 to M4 would reduce to ordinary hyper-analytic map from X2 with hyper-complex coordinate to M4 with hyper-complex coordinates just as in string models. The map from X4 to CP2 would reduce to an ordinary analytic map from X2 with complex coordinates. One would not leave the realm of string models.
  2. For the simplest massless extremals (MEs) CP2 coordinates are arbitrary functions of light-like coordinate u=k•.m, k constant light-like vector, and of v=ε • m, ε a polarization vector orthogonal to k. The interpretation as classical counterpart of photon or Bose-Einstein condense of photons is obvious. There are good reasons to expect that this ansatz generalizes by replacing the variables u and v with coordinate along the light-like and space-like coordinate lines of Hamilton-Jacobi structure. The non-geodesic motion of photons with light-velocity and variation of the polarization direction would be due to interactions with the space-time sheet to which it is topologically condensed. Note that light-likeness condition for the coordinate curve gives rise to Virasoro conditions. This observation led long time ago to the idea that 2-D conformal invariance must have a non-trivial generalization to 4-D case.

    Now space-time surface would have naturally M4 coordinates and the map M4→ M4 would be just identity map satisfying the radial CRF condition. Can one understand CP2 coordinates in terms of quaternion- analyticity? The dependence of CP2 coordinates on u=t-x only can be formulated as CFR condition ∂u* sk=0 and this could is expected to generalize in the formulation using the geometric representation of quaternionic imaginary units at both sides. The dependence on light-light coordinate u follows from the translationally invariant CRF condition.

    The dependence on the real coordinate v is however problematic since the dependence is naturally on complex coordinate w assignable to the polarization plane of form z= f(w). This would give dependence on 2 transversal coordinates and CP2 projection would be 3-D rather than 2-D. One can of course ask whether this dependence is actually present for preferred extremals? Could the polarization vector be complex local polarization vector orthogonal to the light-like vector? In quantum theory complex polarization vectors are used of routinely and become oscillator operators in second quantization and in TGD Universe MEs indeed serve as space-time correlates for photons or their BE condensates.

  3. Vacuum extremals with Lagrangian manifold as (in the generic case 2-D) CP2 projection are not expected to be preferred extremals for obvious reasons. One one can however try similar approach. Hyper-quaternionic structure for space-time surface using Hamilton-Jacobi structure is the first guess. CP2 should allow a quaternionic coordinate decomposing to a pair of complex coordinates such that second complex coordinate is constant for 2-D Lagrangian manifold and second parameterizes it. Any 2-D surface allows complex structure defined by the induced metric so that there are good hopes that these coordinates exist. The quaternion-analytic map would map in the most general case is trivial for both hypercomplex and complex coordinate of M4 but the quaternionic Taylor coefficients reduce to real numbers to that the image is 2-D.
  4. For CP2 type vacuum extremals the M4 projection is random light-like curve. Now one expects co-quaternionicity and that quaternion-analyticity is not the correct manner to formulate the situation. "Co-" suggests that instead of expressing surface as graph one should perhaps express it in terms of conditions stating that some quaternionic analytic functions in H are vanish.

    One can fix the coordinates of X4 to be complex coordinates of CP2 so that one gets rid of the degeneracy due to the choice of coordinates. M4 allows hyper-quaternionic coordinates and Hamilton-Jacobi structures define different choices of hyper-quaternionic coordinates. Now the second light- like coordinate would vary along random light-like curves providing slicing of M4 by 3-D surfaces. Hamilton-Jacobi structure defines at each point a plane M2(x) fixed by the light-like vector at the point and the 2-D orthogonal plane. In fact 4-D coordinate grid is defined. This local choice must be integrable, which means that one has slicing by 2-D string world sheets and polarization planes orthogonal to them.

    The problem is that the mapping of quaternionic CP2 coordinate to hyper-quaternionic coordinates of M4 (say v=0, w=0) in terms of quaternionic analyticity is not easy. "Co-" suggets that , one could formulate light-likeness condition using Hamilton-Jacobi structure as conditions w*-constant=0 and v-constant=0. Note that one has u*=v.

  5. In the naive generalization CRF conditions are linear. Whether this is the case in the formulation using the geometric representation of the imaginary units is not clear since the quaternionic imaginary units depend on the vielbein of the induced 3-metric (note however that the SO(3) gauge rotation appearing in the conditions could allow to compensate for the change of the tensors in small deformations of the spaced-time surface). If linearity is real and not true only for small perturbations, one could have linear superpositions of different types of solutions, which looks strange. Could the superpositions describe perturbations of say cosmic strings and massless extremals?
  6. Both forms of algebraic C-R-F conditions generalize to the octonionic situation and right multiplication of powers of octonion by Taylor coefficients plus linearity imply that there are no problems with associativity. This inspires several questions.

    Could octonion analytic maps of imbedding space allow to construct new solutions from the existing ones? Could quaternion analytic maps applied at space-time level act as analogs of holomorphic maps and generalize conformal gauge invariance to 4-D context?


To sum up, connections between different conjectures related to the preferred extremals - M8-H duality, Hamilton-Jacobi structure, induced twistor space structure, quaternion-Kähler property and its Minkowskian counterpart, and even quaternion analyticity, are clearly emerging. The underlying reason is strong form of GCI forced by the construction of WCW geometry and implying strong from of holography posing extremely powerful quantization conditions on the extremals of Kähler action in ZEO. Without the conformal gauge conditions the mutual inconsistency of these conjectures looks rather infeasible.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Are Euclidian regions of preferred extremals quaternion-Kähler manifolds?

In blog comments Anonymous gave a link to an article about construction of 4-D quaternion-Kähler metrics with an isometry: they are determined by so called SU(∞) Toda equation. I tried to see whether quaternion-Kähler manifolds could be relevant for TGD.

From Wikipedia one can learn that QK is characterized by its holonomy, which is a subgroup of Sp(n)×Sp(1): Sp(n) acts as linear symplectic transformations of 2n-dimensional space (now real). In 4-D case tangent space contains 3-D sub-manifold identifiable as imaginary quaternions. CP2 is one example of QK manifold for which the subgroup in question is SU(2)× U(1) and which has non-vanishing constant curvature: the components of Weyl tensor represent the quaternionic imaginary units. QKs are Einstein manifolds: Einstein tensor is proportional to metric.

What is really interesting from TGD point of view is that twistorial considerations show that one can assign to QK a special kind of twistor space (twistor space in the mildest sense requires only orientability). Wiki tells that if Ricci curvature is positive, this (6-D) twistor space is what is known as projective Fano manifold with a holomorphic contact structure. Fano variety has the nice property that as (complex) line bundle it has enough sections to define the imbedding of its base space to a projective variety. Fano variety is also complete: this is algebraic geometric analogy of topological property known as compactness.

QK manifolds and twistorial formulation of TGD

How the QKs could relate to the twistorial formulation of TGD?

  1. In the twistor formulation of TGD the space-time surfaces are 4-D base spaces of 6-D twistor spaces in the Cartesian product of 6-D twistor spaces of M4 and CP2 - the only twistor spaces with Kähler structure. In TGD framework space-time regions can have either Euclidian or Minkowskian signature of induced metric. The lines of generalized Feynman diagrams have Euclidian signature.
  2. Could the twistor spaces associated with the lines of generalized Feynman diagrams be projective Fano manifolds? Could QK structure characterize Euclidian regions of preferred extremals of Kähler action. Could a generalization to Minkowskian regions exist. I have proposed that so called Hamilton-Jacobi structure characterizes preferred extremals in Minkowskian regions. It could be the natural Minkowskian counterpart for the quaternion Kähler structure, which involves only imaginary quaternions and could make sense also in Minkowski signature. Note that unit sphere of imaginary quaternions defines the sphere serving as fiber of the twistor bundle.
  3. Why it would be natural to have QK that is corresponding twistor space which is projective contact Fano manifold?
    1. Fano property implies that the 4-D Euclidian space-time region representing line of the Feynman diagram can be imbedded as a sub-manifold to complex projective space CPn. This would allow to use the powerful machinery of projective geometry in TGD framework. This could also be a space-time correlate for the fact that CPns emerge in twistor Grassmann approach expected to generalize to TGD framework.
    2. CP2 allows both projective (trivially) and contact (even symplectic) structures. δ M4+ × CP2 allows contact structure - I call it loosely symplectic structure. Also 3-D light-like orbits of partonic 2-surfaces allow contact structure. Therefore holomorphic contact structure for the twistor space is natural.
    3. Both the holomorphic contact structure and projectivity of CP2 would be inherited if QK property is true. Contact structures at orbits of partonic 2-surfaces would extend to holomorphic contact structures in the Euclidian regions of space-time surface representing lines of generalized Feynman diagrams. Projectivity of Fano space would be also inherited from CP2 or its twistor space SU(3)/U(1)× U(1) (flag manifold identifiable as the space of choices for quantization axes of color isospin and hypercharge).
  4. Could the isometry (or possibly isometries) for QK be seen as a remnant of color symmetry or rotational symmetries of M4 factor of imbedding space? The only remnant of color symmetry at the level of imbedding space spinors is anomalous color hyper charge (color is like orbital angular momentum and associated with spinor harmonic in CP2 center of mass degrees of freedom). Could the isometry correspond to anomalous hypercharge?
How to choose the quaternionic imaginary units for the space-time surface?

Parallellizability is a very special property of 3-manifolds allowing to choose quaternionic imaginary units: global choice of one of them gives rise to twistor structure.

  1. The selection of time coordinate defines a slicing of space-time surface by 3-surfaces. GCI would suggest that a generic slicing gives rise to 3 quaternionic units at each point each 3-surface? The parallelizability of 3-manifolds - a unique property of 3-manifolds - means the possibility to select global coordinate frame as section of the frame bundle: one has 3 sections of tangent bundle whose inner products give rose to the components of the metric (now induced metric) guarantees this. The tri-bein or its dual defined by two-forms obtained by contracting tri-bein vectors with permutation tensor gives the quanternionic imaginary units. The construction depends on 3-metric only and could be carried out also in GRT context. Note however that topology change for 3-manifold might cause some non-trivialities. The metric 2-dimensionality at the light-like orbits of partonic 2-surfaces should not be a problem for a slicing by space-like 3-surfaces. The construction makes sense also for the regions of Minkowskian signature.
  2. In zero energy ontology (ZEO)- a purely TGD based feature - there are very natural special slicings. The first one is by linear time-like Minkowski coordinate defined by the direction of the line connecting the tips of the causal diamond (CD). Second one is defined by the light-cone proper time associated with either light-cone in the intersection of future and past directed light-cones defining CD. Neither slicing is global as it is easy to see.

The relationship to quaternionicity conjecture and M8-H duality

One of the basic conjectures of TGD is that preferred extremals consist of quaternionic/ co-quaternionic (associative/co-associative) regions (see this). Second closely related conjecture is M8-H duality allowing to map quaternionic/co-quaternionic surfaces of M8 to those of M4× CP2. Are these conjectures consistent with QK in Euclidian regions and Hamilton-Jacobi property in Minkowskian regions? Consider first the definition of quaternionic and co-quaternionic space-time regions.

  1. Quaternionic/associative space-time region (with Minkowskian signature) is defined in terms of induced octonion structure obtained by projecting octonion units defined by vielbein of H= M4× CP2 to space-time surface and demanding that the 4 projections generate quaternionic sub-algebra at each point of space-time.

    If there is also unique complex sub-algebra associated with each point of space-time, one obtains one can assign to the tangent space-of space-time surface a point of CP2. This allows to realize M8-H duality (see this) as the number theoretic analog of spontaneous compactification (but involving no compactification) by assigning to a point of M4=M4× CP2 a point of M4× CP2. If the image surface is also quaternionic, this assignment makes sense also for space-time surfaces in H so that M8-H duality generalizes to H-H duality allowing to assign to given preferred extremal a hierarchy of extremals by iterating this assignment. One obtains a category with morphisms identifiable as these duality maps.

  2. Co-quaternionic/co-associative structure is conjectured for space-time regions of Euclidian signature and 4-D CP2 projection. In this case normal space of space-time surface is quaternionic/associative. A multiplication of the basis by preferred unit of basis gives rise to a quaternionic tangent space basis so that one can speak of quaternionic structure also in this case.
  3. Quaternionicity in this sense requires unique identification of a preferred time coordinate as imbedding space coordinate and corresponding slicing by 3-surfaces and is possible only in TGD context. The preferred time direction would correspond to real quaternionic unit. Preferred time coordinate implies that quaternionic structure in TGD sense is more specific than the QK structure in Euclidian regions.
  4. The basis of induced octonionic imaginary unit allows to identify quaternionic imaginary units linearly related to the corresponding units defined by tri-bein vectors. Note that the multiplication of octonionic units is replaced with multiplication of antisymetric tensors representing them when one assigns to the quaternionic structure potential QK structure. Quaternionic structure does not require Kähler structure and makes sense for both signatures of the induced metric. Hence a consistency with QK and its possible analog in Minkowskian regions is possible.
  5. The selection of the preferred imaginary quaternion unit is necessary for M8-H correspondence. This selection would also define the twistor structure. For quaternion-Kähler manifold this unit would be covariantly constant and define Kähler form - maybe as the induced Kähler form.
  6. Also in Minkowskian regions twistor structure requires a selection of a preferred imaginary quaternion unit. Could the induced Kähler form define the preferred imaginary unit also now? Is the Hamilton-Jacobi structure consistent with this?

    Hamilton-Jacobi structure involves a selection of 2-D complex plane at each point of space-time surface. Could induced Kähler magnetic form for each 3-slice define this plane? It is not necessary to require that 3-D Kähler form is covariantly constant for Minkowskian regions. Indeed, massless extremals representing analogs of photons are characterized by local polarization and momentum direction and carry time-dependent Kähler-electric and -magnetic fields. One can however ask whether monopole flux tubes carry covariantly constant Kähler magnetic field: they are indeed deformations of what I call cosmic strings (see this) for which this condition holds true?

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Surface area as geometric representation of entanglement entropy?

In Thinking Allowed Original there was a link to a talk by James Sully and having the title Geometry of Compression. I must admit that I understood very little about the talk. My not so educated guess is however that information is compressed: UV or IR cutoff eliminating entanglement in short length scales and describing its presence in terms of density matrix - that is thermodynamically - is another manner to say it. The TGD inspired proposal for the interpretation of the inclusions of hyper-finite factors of type II1 (HFFs) is in spirit with this.

The space-time counterpart for the compression would be in TGD framework discretization. Discretizations using rational points (or points in algebraic extensions of rationals) make sense also p-adically and thus satisfy number theoretic universality. Discretization would be defined in terms of intersection (rational or in algebraic extension of rationals) of real and p-adic surfaces. At the level of "world of classical worlds" the discretization would correspond to - say - surfaces defined in terms of polynomials, whose coefficients are rational or in some algebraic extension of rationals. Pinary UV and IR cutoffs are involved too. The notion of p-adic manifold allows to interpret rthe p-adic variants of space-time surfaces as cognitive representations of real space-time surfaces.

Finite measurement resolution does not allow state function reduction reducing entanglement totally. In TGD framework also negentropic entanglement stable under Negentropy Maximixation Principle (NMP) is possible. For HFFs the projection into single ray of Hilbert space is indeed impossible: the reduction takes always to infinite-D sub-space.

The visit to the URL was however not in vain. There was a link to an article discussing the geometrization of entanglement entropy inspired by the AdS/CFT hypothesis.

Quantum classical correspondence is basic guiding principle of TGD and suggests that entanglement entropy should indeed have space-time correlate, which would be the analog of Hawking-Bekenstein entropy.

Generalization of AdS/CFT to TGD context

AdS/CFT generalizes to TGD context in non-trivial manner. There are two alternative interpretations, which both could make sense. These interpretations are not mutually exclusive. The first interpretation makes sense at the level of "world of classical worlds" (WCW) with symplectic algebra and extended conformal algebra associated with δ M4+/- replacing ordinary conformal and Kac-Moody algebras. Second interpretation at the level of space-time surface with the extended conformal algebras of the light-likes orbits of partonic 2-surfaces replacing the conformal algebra of boundary of AdSn.

1. First interpretation

For the first interpretation 2-D conformal invariance is generalised to 4-D conformal invariance relying crucially on the 4-dimensionality of space-time surfaces and Minkowski space.

  1. One has an extension of the conformal invariance provided by the symplectic transformations of δ CD× CP2 for which Lie algebra has the structure of conformal algebra with radial light-like coordinate of δ M4+ replacing complex coordinate z.
  2. One could see the counterpart of AdSn as imbedding space H=M4 × CP2 completely unique by twistorial considerations and from the condition that standard model symmetries are obtained and its causal diamonds defined as sub-sets CD× CP2, where CD is an intersection of future and past directed light-cones. I will use the shorthand CD for CD× CP2. Strings in AdS5× S5 are replaced with space-time surfaces inside 8-D CD.
  3. For this interpretation 8-D CD replaces the 10-D space-time AdS5× S5. 7-D light-like boundaries of CD correspond to the boundary of say AdS5, which is 4-D Minkowski space so that zero energy ontology (ZEO) allows rather natural formulation of the generalization of AdS/CFT correspondence since the positive and negative energy parts of zero energy states are localized at the boundaries of CD.

2. Second interpretation

For the second interpretation relies on the observation that string world sheets as carriers of induced spinor fields emerge in TGD framework from the condition that electromagnetic charge is well-defined for the modes of induced spinor field.

  1. One could see the 4-D space-time surfaces X4 as counterparts of AdS4. The boundary of AdS4 is replaced in this picture with 3-surfaces at the ends of space-time surface at opposite boundaries of CD and by strong form of holography the union of partonic 2-surfaces defining the intersections of the 3-D boundaries between Euclidian and Minkowskian regions of space-time surface with the boundaries of CD. Strong form of holography in TGD is very much like ordinary holography.
  2. Note that one has a dimensional hierarchy: the ends of the boundaries of string world sheets at boundaries of CD as pointlike partices, boundaries as fermion number carrying lines, string world sheets, light-like orbits of partonic 2-surfaces, 4-surfaces, imbedding space M4× CP2. Clearly the situation is more complex than for AdS/CFT correspondence.
  3. One can restrict the consideration to 3-D sub-manifolds X3 at either boundary of causal diamond (CD): the ends of space-time surface. In fact, the position of the other boundary is not well-defined since one has superposition of CDs with only one boundary fixed to be piece of light-cone boundary. The delocalization of the other boundary is essential for the understanding of the arrow of time. The state function reductions at fixed boundary leave positive energy part (say) of the zero energy state at that boundary invariant (in positive energy ontology entire state would remain unchanged) but affect the states associated with opposite boundaries forming a superposition which also changes between reduction: this is analog for unitary time evolution. The average for the distance between tips of CDs in the superposition increases and gives rise to the flow of time.
  4. One wants an expression for the entanglement entropy between X3 and its partner. Bekenstein area law allows to guess the general expression for the entanglement entropy: for the proposal discussed in the article the entropy would be the area of the boundary of X3 divided by gravitational constant: S= A/4G. In TGD framework gravitational constant might be replaced by the square of CP2 radius apart from numerical constant. How gravitational constant emerges in TGD framework is not completely understood although one can deduce for it an estimate using dimensional analyses. In any case, gravitational constant is a parameter which characterizes GRT limit of TGD in which many-sheeted space-time is in long scales replaced with a piece of Minkowski space such that the classical gravitational fields and gauge potentials for sheets are summed. The physics behind this relies on the generalization of linear superposition of fields: the effects of different space-time sheets particle touching them sum up rather than fields.
  5. The counterpart for the boundary of X3 appearing in the proposal for the geometrization of the entanglement entropy naturally corresponds to partonic 2-surface or a collection of them if strong form of holography holds true.

With what kind of systems 3-surfaces can entangle?

With what system X3 is entangled/can entangle? There are several options to consider and they could correspond to the two TGD variants for the AdS/CFT correspondence.

  1. X3 could correspond to a wormhole contact with Euclidian signature of induced metric. The entanglement would be between it and the exterior region with Minkowskian signature of the induced metric.
  2. X3 could correspond to single sheet of space-time surface connected by wormhole contacts to a larger space-time sheet defining its environment. More precisely, X3 and its complement would be obtained by throwing away the wormhole contacts with Euclidian signature of induce metric. Entanglement would be between these regions. In the generalization of the formula

    S= A/4hbar G

    area A would be replaced by the total area of partonic 2-surfaces and G perhaps with CP2 length scale squared.

  3. In ZEO the entanglement could also correspond to time-like entanglement between the 3-D ends of the space-time surface at opposite light-like boundaries of CD. M-matrix, which can be seen as the analog of thermal S-matrix, decomposes to a product of hermitian square root of density matrix and unitary S-matrix and this hermitian matrix could also define p-adic thermodynamics. Note that in ZEO quantum theory can be regarded as square root of thermodynamics.

Minimal surface property is not favored in TGD framework

Minimal surface property for the 3-surfaces X3 at the ends of space-time surface looks at first glance strange but a proper generalization of this condition makes sense if one assumes strong form of holography. Strong form of holography realizes General Coordinate Invariance (GCI) in strong sense meaning that light-like parton orbits and space-like 3-surfaces at the ends of space-time surfaces are equivalent physically. As a consequence, partonic 2-surfaces and their 4-D tangent space data must code for the quantum dynamics.

The mathematical realization is in terms of conformal symmetries accompanying the symplectic symmetries of δ M4+/-× CP2 and conformal transformations of the light-like partonic orbit. The generalizations of ordinary conformal algebras correspond to conformal algebra, Kac-Moody algebra at the light-like parton orbits and to symplectic transformations δ M4× CP2 acting as isometries of WCW and having conformal structure with respect to the light-like radial coordinate plus conformal transformations of δ M4+/-, which is metrically 2-dimensional and allows extended conformal symmetries.

  1. If the conformal realization of the strong form of holography works, conformal transformations act at quantum level as gauge symmetries in the sense that generators with no-vanishing conformal weight are zero or generate zero norm states. Conformal degeneracy can be eliminated by fixing the gauge somehow. Classical conformal gauge conditions analogous to Virasoro and Kac-Moody conditions satisfied by the 3-surfaces at the ends of CD are natural in this respect. Similar conditions would hold true for the light-like partonic orbits at which the signature of the induced metric changes.
  2. What is also completely new is the hierarchy of conformal symmetry breakings associated with the hierarchy of Planck constants heff/h=n. The deformations of the 3-surfaces which correspond to non-vanishing conformal weight in algebra or any sub-algebra with conformal weights vanishing modulo n give rise to vanishing classical charges and thus do not affect the value of the Kähler action.

    The inclusion hierarchies of conformal sub-algebras are assumed to correspond to those for hyper-finite factors. There is obviously a precise analogy with quantal conformal invariance conditions for Virasoro algebra and Kac-Moody algebra. There is also hierarchy of inclusions which corresponds to hierarchy of measurement resolutions. An attractive interpretation is that singular conformal transformations relate to each other the states for broken conformal symmetry. Infinitesimal transformations for symmetry broken phase would carry fractional conformal weights coming as multiples of 1/n.

  3. Conformal gauge conditions need not reduce to minimal surface conditions holding true for all variations.
  4. Note that Kähler action reduces to Chern-Simons term at the ends of CD if weak form of electric magnetic duality holds true. The conformal charges at the ends of CD cannot however reduce to Chern-Simons charges by this condition since only the charges associated with CP2 degrees of freedom would be non-trivial.


The generalisation of the conjecture about surface area proportionality of entropy to TGD context looks rather straightforward but is physically highly non-trivial. There are however some technicalities involved.

  1. In TGD framework it is not quite clear whether
    1. G still appears in the formula or
    2. whether G should be replaced with the square R2 of CP2 radius to give

      S= A/4π R2

      apart from numerical constant.

    For option a) one must include Planck constant explicitly to the formula to give S= A/4heffG: the entropy would decrease as heff=n× h increases. The condition heff=hgr= GM2/v0 would give S= v0/c<1. The entropy using b) would be by a factor of order 10-5 smaller and would not depend on the value of heff at all. It will be found that p-adic mass calculations lead to entropy allowing to circumvent these problems.
  2. There is also the question about the identification of the area A. For blackhole A would be determined by Schwartschild radius rS= 2GM depending on mass only. In TGD framework one has several candidates.
    1. The area of partonic 2-surface is an obvious first guess. One cannot however expect that the area of partonic 2-surface is constant. Could conformal gauge fixing fixes the 3-surfaces highly uniquely. Ordinary conformal invariance for partonic 2-surface does not however seem to be consistent with the fixing of the area of partonic 2-surface since conformal transformations do not preserve area.
    2. Could the area of partonic 2-surface be replaced with the area of the boundary of space-time sheet at which particle is topologically condensed and has size scale of order Compton length? This option looks the most feasible one on basis of p-adic mass calculations as will be found.

p-Adic variant of Bekenstein-Hawking law

When the 3-surface corresponds to elementary particle, a direct connection with p-adic thermodynamics suggests itself and allows to answer the questions above. p-Adic thermodynamics could be interpreted as a description of the entanglement with environment. In ZEO the entanglement could also correspond to time-like entanglement between the 3-D ends of the space-time surface at opposite light-like boundaries of CD. M-matrix, which can be seen as the analog of thermal S-matrix, decomposes to a product of hermitian square root of density matrix and unitary S-matrix and this hermitian matrix could also define p-adic thermodynamics.

  1. p-Adic thermodynamics would not be for energy but for mass squared (or scaling generator L0) would describe the entanglement of the particle with environment defined by the larger space-time sheet. Conformal weights would comes as positive powers of integers (pL0 would replace exp(-H/T) to guarantee the number theoretical existence and convergence of the Boltzmann weight: note that conformal invariance that is integer spectrum of L0 is also essential).
  2. The interactions with environment would excite very massive CP2 mass scale excitations (mass scale is about 10-4 times Planck mass) of the particle and give it thermal mass squared identifiable as the observed mass squared. The Boltzmann weights would be extremely small having p-adic norm about 1/pn, p the p-adic prime: M127=2127-1 for electron.
  3. I have proposed earlier p-adic entropy as a p-adic counterpart of Bekenstein-Hawking entropy. S= (R2/hbar2)× M2 holds true identically apart from numerical constant. Note that one could interpret R2M/hbar as the counterpart of Schwartschild radius. Note that this radius is proportional to 1/p1/2 so that the area A would correspond to the area defined by Compton length. This is in accordance with the third option.

What is the space-time correlate for negentropic entanglement?

The new element brought in by TGD framework is that number theoretic entanglement entropy is negative for negentropic entanglement assignable to unitary entanglement and NMP states that this negentropy increases. Since entropy is essentially number of energy degenerate states, a good guess is that the number n=heff/h of space-time sheets associated with heff defines the negentropy. An attractive space-time correlate for the negentropic entanglement is braiding. Braiding defines unitary S-matrix between the states at the ends of braid and this entanglement is negentropic. This entanglement gives also rise to topological quantum computation.

See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article Surface area as geometric representation of entanglement entropy?.

A more detailed view about the construction of scattering amplitudes

The following represents an update view about construction of scattering amplitudes at the level of "world of classical worlds" (WCW).

Basic principles

In order to facilitate the challenge of the reader I summarize basic ideas behind the construction of scattering amplitudes.

1. Zero energy ontology

In Zero Energy Ontology (ZEO) quantum theory as hermitian square root of thermodynamics, which leads to a generalization of the notion of S-matrix to a unitary U-matrix between zero energy states having as rows M-matrices which are products of hermitian square roots of density matrices with a common unitary matrix S for given CD. Number theoretical considerations suggests that CD size comes as integer multiples of CP2 size so that one obtain a hierarchy U-matrices having interpretation in terms of length scale evolution. For given CD also sub-CDs contribute down to some minimal scale defining UV scale. The largest CD defines the IR cutoff.

Scattering amplitudes would characterize the modes of WCW spinor field as time-like entanglement coefficients between positive and negative energy states associated with zero energy states.

The construction of scattering amplitudes - or M-matrix elements in ZEO - reduces at basic level to the construction of the Feynman diagram like entities for fundamental fermions, which serve basic building bricks of elementary particles.

2. Construction of scattering amplitudes as functional integrals in WCW

The decomposition of space-time surface to Minkowskian and Eucldian regions is the basic distinction from ordinary quantum field theories since it replaces path integral with mathematically well-defined functional integral over WCW.

  1. Space-time surface decomposes to regions with Minkowskian or Euclidian signature of the induced metric. The regions with Euclidian metric are identified as lines of generalized Feynman diagrams. The boundaries between two kinds of regions - to be called parton orbits - can be regarded as carriers of elementary particle quantum numbers such as fermion number assignable to the boundaries of string world sheets at them. Induced spinor fields are localized at them from the well-definedness of electromagnetic charge requiring that induced W boson fields vanish. Hence strings emerge from TGD. Note that at boundary between Euclidian and Minkowskian regions the metric determinant vanishes.
  2. Weak form of electric magnetic duality together with the assumption that the term jαAα in Kähler action vanishes imply that Kähler action reduces to 3-D Chern-Simons term. This hypothesis is inspired by TGD as almost topological quantum field theory conjecture. In Minkowskian regions this conjecture is very natural. In the Euclidian region the contribution to Kähler action need not reduce to a mere Chern-Simons term associated with its boundary. This would be due to the non-triviality of the U(1) bundle defined by Kähler form giving also Chern-Simons terms inside the CP2 type vacuum extremal.
  3. Scattering amplitude is a functional integral over space-time surfaces: the data about these space-time surfaces are coded by their ends about the opposite light-like boundaries of causal diamond (CD) of given scale. The weight function in the functional integral is exponential of Kähler function of "world of classical worlds" coming from Euclidian regions of the space-time surface representing lines of generalized Feynman diagram and being deformation of CP2 type vacuum extremals representing wormhole contacts connecting two space-time sheets with Minkowskian signature of induced metric. Kähler function is the exponent of Kähler action from Euclidian regions. The real exponent takes care that the functional integral is obtained instead of path integral so that the outcome is mathematically well-defined.
  4. Euclidian region would give only the analog of thermodynamics but there is also an imaginary exponential coming from the exponential of the imaginary Kähler action from Minkowskian regions. Space-time surfaces are extremals of Kähler action and for very general ansatz Minkowskian contribution to Kähler action reduces to imaginary Chern-Simons term at the light-like 3-D boundary between regions at which the 4-D metric is degenerate. This term makes possible interference of different contributions to the functional integral which is absolutely essential in quantum field theory.

3. Why it might work?

There are many reasons encouraging the hopes about calculable theory.

  1. The theory has huge super-conformal symmetries dramatically reducing the dynamical degrees of freedom by the choice of conformal gauge. This implies that both the space-like 3-surfaces at the ends of space-time surface and partonic orbits satisfy classical Super conformal conditions for generalizations of ordinary super-conformal algebras perhaps extending to multilocal Yangian with locus identified as single partonic 2-surface at the light-like boundary of CD.

    Yangian symmetry in turn gives excellent hopes about twistorialization: in fact, M4× CP2 is completely unique choice for the imbedding space by twistorial considerations and the product of the twistor spaces of M4 and CP2 allows to construct the twistor spaces of space-time surfaces as liftings of the extremals of Kähler action to 6-D sphere bundles over space-time surface.

  2. The integrand in the functional integral represents the analog of ordinary Feynman diagrams involving only fermions and 1-D lines. Indeed, by bosonic emergence all bosons (in fact all elementary particles) can be regarded as composites of fundamental fermions. The only fermionic vertices are 2-fermion vertices since 3-vertices correspond to space-time surfaces meeting along common 3-surface and are thus purely topological. This is of course excellent news from the point of view of finiteness. The fermionic vertices are represented by the discontinuity of the modified Dirac operator associated with the string boundary line at partonic 2-surface so that there are no coupling constants involved. The only fundamental coupling parameter is Kähler strength whose value is dictated by quantum criticality as the analog of critical temperature.

One must have a view about what elementary particles - as opposed to fundamental fermions - are, how the ordinary view about scattering based on exchanges of elementary particles emerges from this picture and how say BFF vertex reduces to a diagram at for fundamental fermions involving only 2-fermion vertices.

Elementary particles in TGD framework

The notion of elementary particle involves two aspects: elementary particles as space-time surfaces and elementary particles as many-fermion states with fundamental fermions localized at the wormhole throats and defining elementary particles as their bound states (including physical fermions).

1. Elementary particles as space-time surfaces

Let us first summarize what kind of picture ZEO suggests about elementary particles.

  1. Kähler magnetically charged wormhole throats are the basic building bricks of elementary particles. The lines of generalized Feynman diagrams are identified as the Euclidian regions of space-time surface. The weak form of electric magnetic duality forces magnetic monopoles and gives classical quantization of the Kähler electric charge. Wormhole throat is a carrier of many-fermion state with parallel momenta and the fermionic oscillator algebra gives rise to a badly broken large N SUSY.
  2. The first guess would be that elementary fermions correspond to wormhole throats with unit fermion number and bosons to wormhole contacts carrying fermion and anti-fermion at opposite throats. The magnetic charges of wormhole throats do not however allow this option. The reason is that the field lines of Kähler magnetic monopole field must close. Both in the case of fermions and bosons one must have a pair of wormhole contacts (see figure ) connected by flux tubes. The most general option is that net quantum numbers are distributed amongst the four wormhole throats. A simpler option is that quantum numbers are carried by the second wormhole: fermion quantum numbers would be carried by its second throat and bosonic quantum numbers by fermion and anti-fermion at the opposite throats. All elementary particles would therefore be accompanied by parallel flux tubes and string world sheets.
  3. A cautious proposal in its original form was that the throats of the other wormhole contact could carry weak isospin represented in terms of neutrinos and neutralizing the weak isospin of the fermion at second end. This would imply weak neutrality and weak confinement above length scales longer than the length of the flux tube. This condition might be un-necessarily strong.

    The realization of the weak neutrality using pair of left handed neutrino and right handed antineutrino or a conjugate of this state is possible if one allows right-handed neutrino to have also unphysical helicity. The weak screening of a fermion at wormhole throat is possible if νR is a constant spinor since in this case Dirac equation trivializes and allows both helicities as solutions. The new element from the solution of the modified Dirac equation is that νR would be interior mode de-localized either to the other wormhole contact or to the Minkowskian flux tube. The state at the other end of the flux tube is sparticle of left-handed neutrino.

    It must be emphasized that weak confinement is just a proposal and looks somewhat complex: Nature is perhaps not so complex at the basic level. To understand this better, one can think about how M89 mesons having quark and antiquark at the ends of long flux tube returning back along second space-time sheet could decay to ordinary quark and antiquark.

Localization of the induced spinor fields at string world sheets and fermionic propagators

The localization of induced spinors to string world sheets emerges from the condition that electromagnetic charge is well-defined for the modes of induced spinor fields. There is however an exception: covariantly constant right handed neutrino spinor νR: it can be de-localized along entire space-time surface. Right handed neutrino has no couplings to electroweak fields. It couples however to left handed neutrino by induced gamma matrices except when it is covariantly constant. Note that standard model does not predict νR but its existence is necessary if neutrinos develop Dirac mass. νR is indeed something which must be considered carefully in any generalization of standard model.

It has turned out that covariantly constant right-handed neutrino very probably corresponds to a pure gauge degree of freedom. Non-covariantly constant right-handed neutrino however mixes with left handed neutrino since the modified gamma matrices involve both M4 and CP2 gamma matrices and latter mix M4 chiralities. These right-handed neutrinos localized to partonic 2-surfaces would generate broken SUSY. There are however good reasons to expect that the mass scale for SUSY breaking corresponds to that for the mixing of right and left handed neutrinos inducing neutrino massivation and that the p-adic mass scale of particle and sparticle are same. The only manner to avoid conflict with experimental facts is that sparticles are dark in TGD sense that is having Planck constant heff=n× h. This would conform with the idea that the hierarchy of Planck constants corresponds to a hierarchy of breakings of conformal invariance, which is indeed behind the massivation.

The localization has powerful consequences since it gives in fermionic degrees of freedom what looks like ordinary Feynman diagrams but with only 2-fermion vertex. Space-time topology describes the vertices, say BFF vertex.

Fermion lines correspond to boundaries of string world sheets at the parton orbits. At these lines one must pose a boundary condition and the boundary condition is that the action of the modified Dirac operator associated with the boundary equals to the action of massless Dirac operator in momentum space representation. This reduces the fermionic propagator to massless Dirac propagator and simplify the construction decisively. If residue integral applied in twistor approach makes sense for the fermionic virtual momenta, this in turn gives inverse of Dirac propagator contracted between massless spinors with non-physical helicities.

What is the correct choice for the modified Dirac operator?

  1. Chern-Simons-Dirac operator modified gamma matrix consists of CP2 gamma matrices only and has square which does not vanish identically. Hence the covariant derivative of the induced spinor field along the string boundary must vanish if one wants that that fermion four-momentum is light-like and it must annihilate the spinors at its ends. The helicity of fermion would be physical and one would have incoming or outgoing on mass-shell fermion. This is certainly highly undesirable.

    Covariant constancy of the spinor mode has however the nice implication that it gives the familiar non-integrable phase factor for the dependence of the induced spinor at the fermion line behaving like Wilson line. Wilson lines are known to lead in string model picture to twistor diagrams so that it seems that we are in correct track.

    This forces to ask whether fermion propagator are needed at all in the construction of fermionic Feynman diagrams. Dimensional arguments force them - at least if the scattering is just fermion scattering. The exchanged particles consist however of wormhole contacts: it is wormhole contact with propagates. Boson exchange corresponds to the exchange of wormhole contact with fermion and antifermion at throats. The four-momenta of fermion and anti-fermion making the virtual boson are tightly correlated reducing the pair of fermionic propagators to bosonic propagator and eliminating one virtual momentum integration. This conforms with the fact that in twistor approach the integration over bosonic virtual momenta over boson cancels the bosonic propagator.

  2. What about Kähler-Dirac operator, which is indeed the most natural first guess. For Kähler-Dirac operator one could have light-like K-D gamma matrix at the string boundary and formally one can have modes, which are not covariantly constant. Unfortunately, the definition of the projection of Kähler-Dirac operator to the line is problematic since the determinant of the induced metric vanishes at partonic orbit and the component of canonical momentum density along the string boundary can diverge.

    One could of course consider the possibility of defining the condition as a limit. The infinitesimal value of the covariant derivative would compensate for the divergence of Kähler-Dirac gamma matrix Γt at the limit and one would obtain light-like momenta and non-physical fermion helicities of TGD based variant of twistor approach. For Kähler-Dirac term Γt can be infinite at the limit. If gti=0 holds true then simple matrix algebra shows that gtt becomes infinite whereas gti are limiting values of form 0/0. If Jti=0 holds true for Kähler form, manifest infinity transforms to 0/0 type limit and one has hope of finite limiting value. Weak form of electric magnetic duality could in factor guarantee Jti=0 by forcing Jtn and Jxy to be the only non-vanishing components of induced Kähler form.

    Remarkably, also now one would obtain the non-integrable phase factor characterizing Wilson line. The graph would depend on space-time surface only through this phase factor appearing at 2-fermion vertices and by the discontinuity of the modified Dirac operator at partonic 2-surface defining the most natural candidate for the 2-fermion vertex.

  3. If the modified Dirac operator is defined by the induced metric at the string boundary and if its is light-like. This is trivial manner to solve the problem but now one loses the non-integrable phase factor and Wilson line picture.


Vertices can be considered at both space-time level and fermionic level.

  1. At space-time level vertices correspond to the fusion of space-time surfaces representing particles along common 3-surface defining the vertex. At the parton level 3-light-like parton orbits fuse together along partonic 2-surface. In these vertices particle number changes this change correspond the change of particle number for elementary particles.
  2. At fermion level vertices are localized at the partonic 2-surfaces and vertex corresponds to the discontinuity of the Kähler Dirac operator at the corner of the line representing the boundary of string world sheet. The creation of fermion pair from vacuum corresponds to an corner of string boundary at which the boundaries of string world sheets associated with two outgoing or incoming sheets meet. The creation/annihilation of a fermion pair is essential for the realization of say tree diagrams describing fermion scattering by virtual boson exchange.
The discontinuity of the modified Dirac operator is the key quantity. If one assumes continuity of the time derivative the discontinuity reduces to the discontinuity of D=ΓtDt acting on say incoming spinor. ∂t is continuous as operator. The continuity of Γt could be posed as a condition and would state that the canonical momentum density at the corner of string is continuous. This might well be achieved with the proposed conditions for J=0 and g=0. The outcome is non-trivial since D(out/in) need not annihilate Ψ(in/out) so that with the assumptions just listed one obtains ΓtΔ At, where Δ At is the discontinuity of the induced spinor connection and is gauge invariant quantity. Also the non-integrable phase factor associated with the line entering to the vertex appears in the vertex and comes from covariant constancy of the induced spinor field along the line.

As already noticed, the definition of Γt might be problematic. For Chern-Simons term Γt is finite but in this case one looses the justification for M4 propagator of fermion as coming from the boundary condition. For Kähler-Dirac action there are hopes of obtaining a finite limiting value for Γt and therefore of ΓtDt if the induced Kähler form satisfies condition Jti=0 in the case that one has g=0. Furthermore, the continuity of Γt could be posed as a condition for vertices. Note that one obtains also the non-integrable phase factor allowing Wilson line interpretation.

Nothing has been said about the discontinuity of the modified Dirac operator through the wormhole contact as one traverses from Euclidian to Minkowskian side. This discontinuity might be also relevant.One could consider also the difference of the above discussed discontinuity between Euclidian and Minkowskian regions.

What one should obtain at QFT limit?

After functional integration over WCW of one should obtain a scattering amplitude in which the fermionic 2- vertices defined as discontinuities of the modified Dirac operator at partonic 2-surfaces should boil down to a contraction of an M8 vector with gamma matrices of M8. This vector has dimension of mass. This basic parameter should characterize many different physical situations. Consider only the description of massivation of elementary particles regarded as bound states of fundamental massless fermions and the mixing of left and right-handed fermions. Also CKM mixing should involve this parameter. These vectors should also appear in Higgs couplings, which in QFT description contain Higgs vacuum expectation as a factor.

In twistor approach virtual particles have complex light-like momenta. Fundamental fermions have most naturally real and light-like momenta. N=4 SUSY describes gauge bosons which correspond to bound states of fundamental fermions in TGD. This suggests that the four-momenta of bound states of massless fermions - be they hadrons, leptons, or gauge bosons - can be taken to be complex.

There is an intriguing connection with TGD based notion of space-time. In TGD one obtains at space-time level complexified four-momenta since the four-momentum from Minkowskian/ Euclidian region is real/imaginary. In the case of physical particle necessary involving two wormhole contacts and two flux tubes connecting them the total complexifies four momentum would be sum of two real and two imaginary contributions. Every elementary particle should have also imaginary part in its four-momentum and would be massless in complexified sense allowing mass in real sense given by the length of the imaginary four-momentum.

TGD predicts Higgs field although Higgs expectation does not have any role in quantum TGD proper. Higgs vacuum expectation is however a necessary part of QFT limit (Higgs decays to WW pairs require that vacuum expectation is non-vanishing). Higgs vacuum expectation must correspond in TGD framework to a quantity with dimensions of mass. In TGD Higgs cannot be scalar but a vector in CP2 degrees of freedom. The problem is that CP2 does not allow covariantly constant vectors. The imaginary part of classical four-momentum gives a parameter which has interpretation as a vector in the tangent space of which is same as that of M4× CP2. Could M8-H duality be realized at the level of tangent space and for relate four-momentum and color quantum numbers to 8-momentum?

Elementary particles of course need not be eigenstates of the imaginary part of four-momentum. For a fixed mass one can have wave functions in the space of imaginary four-momentum analogous to S3 spherical harmonics at the sphere of E4 with radius defined by the length of imaginary four-momentum (mass). These harmonics are characterized by SO(4) quantum numbers. Could one interpret this complexification in terms of M8-H duality and say that SO(4) defines the symmetries for the low energy dual of WCW defining high energy description of QCD based on SU(3) symmetry. SO(4) would correspond to the symmetry group assigned to hadrons in the approach based on conserved vector currents and partially conserved axial currents. SO(4) would be much more general and associated also with leptons.

The anomalous color hyper-charge of leptonic spinors would imply that one can have also in the case of leptons a wave function in S3. Higher harmonics would correspond to color excitations of leptons and quarks. If one considers gamma matrices, complexification of M4 means introduction of gamma matrix algebra of complexified M4 requiring 8 gamma matrices. This suggests a connection with M8-H duality. All elementary particles have also imaginary part of four-momentum and the 8-momentum can be interpreted as M8-momentum combining the four-momentum and color quantum numbers together.

See the chapter A more detailed view about the construction of scattering amplitudes or the article A more detailed view about the construction of scattering amplitudes.

What could be the TGD counterpart of SUSY

Supersymmetry is very beautiful generalization of the ordinary symmetry concept by generalizing Lie-algebra by allowing grading such that ordinary Lie algebra generators are accompanied by super-generators transforming in some representation of the Lie algebra for which Lie-algebra commutators are replaced with anti-commutators. In the case of Poincare group the super-generators would transform like spinors. Clifford algebras are actually super-algebras. Gamma matrices anti-commute to metric tensor and transform like vectors under the vielbein group (SO(n) in Euclidian signature). In supersymmetric gauge theories one introduced super translations anti-commuting to ordinary translations.

Supersymmetry algebras defined in this manner are characterized by the number of super-generators and in the simplest situation their number is one: one speaks about N=1 SUSY and minimal super-symmetric extension of standard model (MSSM) in this case. These models are most studied because they are the simplest ones. They have however the strange property that the spinors generating SUSY are Majorana spinors- real in well-defined sense unlike Dirac spinors. This implies that fermion number is conserved only modulo two: this has not been observed experimentally. A second problem is that the proposed mechanisms for the breaking of SUSY do not look feasible.

LHC results suggest MSSM does not become visible at LHC energies. This does not exclude more complex scenarios hiding simplest N=1 to higher energies but the number of real believers is decreasing. Something is definitely wrong and one must be ready to consider more complex options or totally new view abot SUSY.

What is the situation in TGD? Here I must admit that I am still fighting to gain understanding of SUSY in TGD framework. That I can still imagine several scenarios shows that I have not yet completely understood the problem and am working hardly to avoid falling to the sin of sloppying myself. In the following I summarize the situation as it seems just now.

  1. In TGD framework N=1 SUSY is excluded since B and L and conserved separately and imbedding space spinors are not Majorana spinors. The possible analog of space-time SUSY should be a remnant of a much larger super-conformal symmetry in which the Clifford algebra generated by fermionic oscillator operators giving also rise to the Clifford algebra generated by the gamma matrices of the "world of classical worlds" (WCW) and assignable with string world sheets. This algebra is indeed part of infinite-D super-conformal algebra behind quantum TGD. One can construct explicitly the conserved super conformal charges accompanying ordinary charges and one obtains something analogous to N=∞ super algebra. This SUSY is however badly broken by electroweak interactions.
  2. The localization of induced spinors to string world sheets emerges from the condition that electromagnetic charge is well-defined for the modes of induced spinor fields. There is however an exception: covariantly constant right handed neutrino spinor νR: it can be de-localized along entire space-time surface. Right-handed neutrino has no couplings to electroweak fields. It couples however to the left handed neutrino by induced gamma matrices except when it is covariantly constant. Note that standard model does not predict νR but its existence is necessary if neutrinos develop Dirac mass. νR is indeed something which must be considered carefully in any generalization of standard model.

Could covariantly constant right handed neutrinos generate SUSY?

Could covariantly constant right-handed spinors generate exact N=2 SUSY? There are two spin directions for them meaning the analog N=2 Poincare SUSY. Could these spin directions correspond to right-handed neutrino and antineutrino. This SUSY would not look like Poincare SUSY for which anticommutator of super generators would be proportional to four-momentum. The problem is that four-momentum vanishes for covariantly constant spinors! Does this mean that the sparticles generated by covariantly constant νR are zero norm states and represent super gauge degrees of freedom? This might well be the case although I have considered also alternative scenarios.

What about non-covariantly constant right-handed neutrinos?

Both imbedding space spinor harmonics and the modified Dirac equation have also right-handed neutrino spinor modes not constant in M4. If these are responsible for SUSY then SUSY is broken.

  1. Consider first the situation at space-time level. Both induced gamma matrices and their generalizations to modified gamma matrices defined as contractions of imbedding space gamma matrices with the canonical momentum currents for Kähler action are superpositions of M4 and CP2 parts. This gives rise to the mixing of right-handed and left-handed neutrinos. Note that non-covariantly constant right-handed neutrinos must be localized at string world sheets.

    This in turn leads neutrino massivation and SUSY breaking. Given particle would be accompanied by sparticles containing varying number of right-handed neutrinos and antineutrinos localized at partonic 2-surfaces.

  2. One an consider also the SUSY breaking at imbedding space level. The ground states of the representations of extended conformal algebras are constructed in terms of spinor harmonics of the imbedding space and form the addition of right handed neutrino with non-vanishing four-momentum would make sense. But the non-vanishing four-momentum means that the members of the super-multiplet cannot have same masses. This is one manner to state what SUSY breaking is.

What one can say about the masses of sparticles?

The simplest form of massivation would be that all members of the super- multiplet obey the same mass formula but that the p-adic length scales associated with them are different. This could allow very heavy sparticles. What fixes the p-adic mass scales of sparticles? If this scale is CP2 mass scale SUSY would be experimentally unreachable. The estimate below does not support this option.

One can even consider the possibility that SUSY breaking makes sparticles unstable against phase transition to their dark variants with heff =n× h. Sparticles could have same mass but be non-observable as dark matter not appearing in same vertices as ordinary matter! Geometrically the addition of right-handed neutrino to the state would induce many-sheeted covering in this case with right handed neutrino perhaps associated with different space-time sheet of the covering.

This idea need not be so outlandish at it looks first.

  1. The generation of many-sheeted covering has interpretation in terms of breaking of conformal invariance. The sub-algebra for which conformal weights are n-tuples of integers becomes the algebra of conformal transformations and the remaining conformal generators do note represent gauge degrees of freedom anymore. They could however represent conserved conformal charges still.
  2. This generalization of conformal symmetry breaking gives rise to infinite number of fractal hierarchies formed by sub-algebras of conformal algebra and is also something new and a fruit of an attempt to avoid sloppy thinking. The breaking of conformal symmetry is indeed expected in massivation related to the SUSY breaking.
The following poor man's estimate supports the idea about dark sfermions and the view that sfermions cannot be very heavy.
  1. Neutrino mixing rate should correspond to the mass scale of neutrinos known to be in eV range for ordinary value of Planck constant. For heff/h=n it is reduced by factor 1/n, when mass kept constant. Hence sfermions could be stabilized by making them dark.
  2. A very rough order of magnitude estimate for sfermion mass scale is obtained from Uncertainty Principle: particle mass should be higher than its decay rate. Therefore an estimate for the decay rate of sfermion could give a lower bound for its mass scale.
  3. Assume the transformation νR→ νL makes sfermion unstable against the decay to fermion and ordinary neutrino. If so, the decay rate would be dictated by the mixing rate and therefore to neutrino mass scale for the ordinary value of Planck constant. Particles and sparticles would have the same p-adic mass scale. Large heff could however make sfermion dark, stable, and non-observable.
A rough model for the neutrino mixing in TGD framework

The mixing of right- and left handed neutrinos would be the basic mechanism in the decays of sfermions. The mixing mechanism is mystery in standard model framework but in TGD it is implied by both induced and modified gamma matrices. The following argument tries to capture what is essential in this process.

  1. Conformal invariance requires that the string ends at which fermions are localized at wormhole throats are light-like curves. In fact, light-likeness gives rise to Virasosoro conditions.
  2. Mixing is described by a vertex residing at partonic surface at which two partonic orbits join. Localization of fermions to string boundaries reduces the problem to a problem completely analogous to the coupling of point particle coupled to external gauge field. What is new that orbit of the particle has corner at partonic 2-surface. Corner breaks conformal invariance since one cannot say that curve is light-like at the corner. At the corner neutrino transforms from right-handed to left handed one.
  3. In complete analogy with Ψbar;γtAtΨ vertex for the point-like particle with spin in external field, the amplitude describing nuRL transition involves matrix elements of form νbarRΓt(CP2)ZtνL at the vertex of the CP2 part of the modified gamma matrix and classical Z0 field.

    How Γt is identified? The modified gamma matrices associated with the interior need not be well-defined at the light-like surface and light-like curve. One basis of weak form of electric magnetic duality the modified gamma matrix corresponds to the canonical momentum density associated with the Chern-Simons term for Kähler action. This gamma matrix contains only the CP2 part.

The following provides as more detailed view.
  1. Let us denote by ΓtCP2(in/out) the CP2 part of the modified gamma matrix at string at at partonic 2-surface and by Z0t the value of Z0 gauge potential along boundary of string world sheet. The direction of string line in imbedding space changes at the partonic 2-surface. The question is what happens to the modified Dirac action at the vertex.
  2. For incoming and outgoing lines the equation

    D(in/out)Ψ(in/out)= pk(in,out)γk Ψ(in/out) ,

    where the modified Dirac operator is D(in/out)=Γt(in/out)Dt, is assumed. νR corresponds to "in" and νR to "out". It implies that lines corresponds to massless M4 Dirac propagator and one obtains something resembling ordinary perturbation theory.

    It also implies that the residue integration over fermionic internal momenta gives as a residue massless fermion lines with non-physical helicities as one can expect in twistor approach. For physical particles the four-momenta are massless but in complex sense and the imaginary part comes classical from four-momenta assignable to the lines of generalized Feynman diagram possessing Euclidian signature of induced metric so that the square root of the metric determinant differs by imaginary unit from that in Minkowskian regions.

  3. In the vertex D(in/out) could act in Ψ(out/in) and the natural idea is that νRL mixing is due to this so that it would be described the classical weak current couplings νbarR ΓtCP2(out)Z0t(in)νL and νbarR ΓtCP2(out)Z0t(in)νL.
To get some idea about orders of magnitude assume that the CP2 projection of string boundary is geodesic circle thus describable as Φ= ω t, where Φ is angle coordinate for the circle and t is Minkowski time coordinate. The contribution of CP2 to the induced metric gtt is Δ gtt =-R2ω2.
  1. In the first approximation string end is a light-like curve in Minkowski space meaning that CP2 contribution to the induced metric vanishes. Neutrino mixing vanishes at this limit.
  2. For a non-vanishing value of ω R the mixing and the order of magnitude for mixing rate and neutrino mass is expected to be R∼ ω and m∼ ω/h. p-Adic length scale hypothesis and the experimental value of neutrino mass allows to estimate m to correspond to p-adic mass to be of order eV so that the corresponding p-adic prime p could be p≈ 2167. Note that k=127 defines largest of the four Gaussian Mersennes MG,k= (1+i)k-1 appearing in the length scale range 10 nm - 2.5 μm. Hence the decay rate for ordinary Planck constant would be of order R∼ 1014/s but large value of Planck constant could reduced it dramatically. In living matter reductions by a factor 10-12 can be considered.

See the chapter Does the QFT Limit of TGD Have Space-Time Super-Symmetry? or the article What went wrong with symmetries?.

The vanishing of conformal charges as a gauge conditions selecting preferred extremals of Kähler action

Classical TGD involves several key questions waiting for clearcut answers.

  1. The notion of preferred extremal emerges naturally in positive energy ontology, where Kähler metric assigns a unique (apart from gauge symmetries) preferred extremal to given 3-surface at M4 time= constant section of imbedding space H=M4× CP2. This would quantize the initial values of the time derivatives of imbedding coordinates and this could correspond to the Bohr orbitology in quantum mechanics.
  2. In zero energy ontology (ZEO) initial conditions are replaced by boundary conditions. One fixes only the 3-surfaces at the opposite boundaries of CD and in an ideal situation there would exist a unique space-time surface connecting them. One must however notice that the existence of light-like wormhole throat orbits at which the signature of the induced metric changes (det(g4)=0) its signature might change the situation. Does the attribute "preferred" become obsolete and does one lose the beautiful Bohr orbitology which looks intuitively compelling and would realize quantum classical correspondence?
  3. Intuitively it has become clear that the generalization of super-conformal symmetries by replacing 2-D manifold with metrically 2-D but topologically 3-D light-like boundary of causal diamond makes sense. Generalized super-conformal symmetries should apply also to the wormhole throat orbits which are also metrically 2-D and for which conformal symmetries respect detg(g4)=0 condition. Quantum classical correspondence demands that the generalized super-confornal invariance has classical counterpart. How could this classical counterpart be realized?
  4. Holography is one key aspect of TGD and mean that 3-surfaces dictate everything. In positive energy ontology the content w of this statement would be rather obvious and reduce to Bohr orbitology but in ZEO situation is different. On the other hand, TGD strongly suggests strong form of holography based stating that partonic 2-surfaces (the ends of wormhole throat orbits at boundaries of CD) and tangent space data at them code for quantum physics of TGD. General coordinate invariance would be realied in strong sense: one could formulate the theory either in terms of space-like 3-surfaces at the ends of CD or in terms of light-like wormhole throat orbits. This would realize Bohr orbitology also in ZEO by reducing the boundary conditions to those at partonic 2-surfaces. How to realize this explicitly at the level of field equations? This has been the challenge.
Answering questions is extremely useful activity. During last years Hamed has posed continually questions related to the basic TGD. At this time Hamed asked about the derivation of field equations of TGD. In "simple" field theories involving some polynomial non-linearities the deduction of field equations is of course totally trivial process but in the extremely non-linear geometric framework of TGD situation is quite different.

While answering the questions I made what I immediately dare to call a breakthrough discovery in the mathematical understanding of TGD. To put it concisely: one can assume that the variations at the light-like boundaries of CD vanish for all conformal variations which are not isometries. For isometries the contributions from the ends of CD cancel each other so that the corresponding variations need not vanish separately at boundaries of CD! This is extremely simple and profound fact. This would be nothing but the realisation of the analogs of conformal symmetries classically and give precise content for the notion of preferred external, Bohr orbitology, and strong form of holography. And the condition makes sense only in ZEO!

I attach below the answers to the questions of Hamed almost as such apart from slight editing and little additions, re-organization, and correction of typos.

The physical interpretation of the canonical momentum current

Hamed asked about the physical meaning of Tnk== ∂ L/∂(∂n hk) - normal components of canonical momentum labelled by the label k of imbedding space coordinates - it is good to start from the physical meaning of a more general vector field

Tαk == ∂ L/∂(∂α hk)

with both imbedding space indices k and space-time indices α - canonical momentum currents. L refers to Kähler action.

  1. One can start from the analogy with Newton's equations derived from action principle (Lagrangian). Now the analogs are the partial derivatives ∂ L/∂(dxk/dt). For a particle in potential one obtains just the momentum. Therefore the term canonical momentum current/density: one has kind of momentum current for each imbedding space coordinate.
  2. By contracting with generators of imbedding space isometries (Poincare and color) one indeed obtains conserved currents associated with isometries by Noether's theorem:

    jA α= TαkjAk .

    By field equations the divergences of these currents vanish and one obtains conserved charged- classical four-momentum and color charges:

    Dα TA α=0 .

  3. The normal component of conserved current must vanish at space-like boundaries if one has such


    if one has boundaries with Minkowskian signature of induced metric. Now one has wormhole throat orbits which are not genuine boundaries albeit analogous to them and one must be very careful. The quantity Tnk determines the values of normal components of currents and must vanish at possible space-like boundaries.

Note that in TGD field equations reduce to the conservation of isometry currents as in hydrodynamics where basic equations are just conservation laws.

The basic steps in the derivation of field equations

First a general recipe for deriving field equations from Kähler action - or any action as a matter of fact.

  1. At the first step one writes an expression of the variation of the Kähler action as sum of variations with respect to the induced metric g and induced Kähler form J. The partial derivatives in question are energy momentum tensor and contravariant Kähler form.
  2. After this the variations of g and J are expressed in terms of variations of imbedding space coordinates, which are the primary dynamical variables.
  3. The integral defining the variation can be decomposed to a total divergence plus a term vanishing for extremals for all variations: this gives the field equations. Total divergence term gives a boundary term and it vanishes by boundary conditions if the boundaries in question have time-like direction.

    If the boundary is space-like, the situation is more delicate in TGD framework: this will be considered in the sequel. In TGD situation is also delicate also because the light-like 3-surfaces which are common boundaries of regions with Minkowskian or Euclidian signature of the induced metric are not ordinary topological boundaries. Therefore a careful treatment of both cases is required in order to not to miss important physics.

Expressing this summary more explicitly, the variation of the Kahler action with respect to the gradients of the imbedding space coordinates reduces to an integral of

Tαkαδ hk + (∂ L/∂ hk) δ hk .

The latter term comes only from the dependence of the imbedding space metric and Kähler form on imbedding space coordinates. One can use a simple trick. Assume that they do not depend at all on imbedding space coordinates, derive field equations, and replaced partial derivatives by covariant derivatives at the end. Covariant derivative means covariance with respect to both space-time and imbedding space vector indices for the tensorial quantities involved. The trick works because imbedding space metric and Kähler form are covariantly constant quantities.

The integral of Tαkαδ hk decomposes to two parts.

  1. The first term, whose vanishing gives rise to field equations, is integral of

    Dα Tαk δ hk .

  2. The second term is integral of

    α (Tαk δ hk) .

    This term reduces as a total divergence to a 3-D surface integral over the boundary of the region of fixed signature of the induced metric consisting of the ends of CD and wormhole throat orbits (boundary of region with fixed signature of induced metric). This term vanishes if the normal components Tnk of canonical momentum currents vanishes at the boundary like region.

In the sequel the boundary terms are discussed explicitly and it will be found that their treatment indeed involves highly non-trivial physics.

Boundary conditions at boundaries of CD

In positive energy ontology one would formulate boundary conditions as initial conditions by fixing both the 3-surface and associated canonical momentum densities at either end of CD (positions and momenta of particles in mechanics). This would bring asymmetry between boundaries of CD.

In TGD framework one must carefully consider the boundary conditions at the boundaries of CDs. What is clear that the time-like boundary contributions from the boundaries of CD to the variation must vanish.

  1. This is true if the variations are assumed to vanish at the ends of CD. This might be however too strong a condition.
  2. One cannot demand vanishing of Ttk (t refers to time coordinate as normal coordinate) since this would give only vacuum extremals. One could however require quantum classical correspondence for any Cartan sub-algebra of isometries, whose elements define maximal set of isometry generators. The eigenvalues of quantal variants of isometry charge assignable to second quantized induced spinors at the ends of space-time surface are equal to the classical charges. Is this actually formulation of Equivalence Principle, is not quite clear to me.
While writing this a completely new idea popped to my mind. What if one poses the vanishing of the boundary terms at boundaries of CDs as additional boundary conditions for all variations except isometries? Or perhaps for all conformal variations (conformal in TGD sense)? This would not imply vanishing of isometry charges since the variations coming from the opposite ends of CD cancel each other! It soon became clear that this would allow to meet all the challenges listed in the beginning!
  1. These conditions would realize Bohr orbitology also to ZEO approach and define what "preferred extremal" means.
  2. The conditions would be very much like super-Virasoro conditions stating that super conformal generators with non-vanishing conformal weight annihilate states or create zero norm states but no conditions are posed on generators with vanishing conformal weight (now isometries). One could indeed assume only deformations, which are local isometries assignable to the generalised conformal algebra of the δ M4+/-× CP2. For arbitrary variations one would not require the vanishing. This could be the long sought for precise formulation of super-conformal invariance at the level of classical field equations!

    It is enough co consider the weaker conditions that the conformal charges defined as integrals of corresponding Noether currents vanish. These conditions would be direct equivalents of quantal conditions.

  3. The natural interpretation would be as a fixing of conformal gauge. This fixing would be motivated by the fact that WCW Kähler metric must possess isometries associated with the conformal algebra and can depend only on the tangent data at partonic 2-surfaces as became clear already for more than two decades ago. An alternative, non-practical option would be to allow all 3-surfaces at the ends of CD: this would lead to the problem of eliminating the analog of the volume of gauge group from the functional integral.
  4. The conditions would also define precisely the notion of holography and its reduction to strong form of holography in which partonic 2-surfaces and their tangent space data code for the dynamics.

Needless to say, the modification of this approach could make sense also at partonic orbits.

Isometry charges are complex

One must be careful also at the light-like 3-surfaces (orbits of wormhole throats) at which the induced metric changes its signature.

  1. Should one assume that det(g4)1/2 is imaginary in Minkowskian and real in Euclidian region? For Kähler action this is sensible and Euclidian region would give a real negative contribution giving rise to exponent of Kähler function of WCW ("world of classical worlds") making the functional integral convergent. Minkowskian regions would give imaginary contribution to the exponent causing interference effects absolutely essential in quantum field theory. This contribution would correspond to Morse function for WCW.

    The implication would be that the classical four-momenta in Euclidian/Minkowskian regions are imaginary/real. What could the interpretation be? Should one accept as a fact that four-momenta are complex.

  2. Twistor approach to TGD is now in quite good shape. M4× CP2 is the unique choice is one requires that the Cartesian factors allow twistor space with Kähler structure and classical TGD allows twistor formulation.

    In the recent formulation the fundamental fermions are assumed to propagate with light-like momenta along wormhole throats. At gauge theory limit particles must have massless or massive four-momenta. One can however also consider the possibility of complex massless momenta and in the standard twistor approach on mass shell massless particles appearing in graphs indeed have complex momenta. These complex momenta should by quantum classical correspondence correspond directly to classical complex momenta.

  3. A funny question popping in mind is whether the massivation of particles could be such that the momenta remain massless in complex sense! The complex variant of light-likeness condition would be

    p2Re= p2Im , pRe• pIm=0 .

    Could one interpret p2Im as the mass squared of the particle? Or could p2Im code for the decay width of an unstable particle?

Boundary conditions at the wormhole throat orbits and connection with quantum criticality and hierarchy of Planck constants defining dark matter hierarchy

The contributions from the orbits of wormhole throats are singular since the contravariant form of the induced metric develops components which are infinite (det(g4)=0). The contributions are real at Euclidian side of throat orbit and imaginary at the Minkowskian side so that they must be treated as independently.

  1. One can consider the possibility that under rather general conditions the normal components Tnkdet(g4) 1/2 approach to zero at partonic orbits since det(g4) is vanishing. Note however the appearance of contravariant appearing twice as index raising operator in Kähler action. If so, the vanishing of Tnkdet(g4) 1/2 need not fix completely the "boundary" conditions. In fact, I assign to the wormhole throat orbits conformal gauge symmetries so that just this is expected on physical grounds.
  2. Generalized conformal invariance would suggest that the variations defined as integrals of Tnkdet(g4) 1/2δ hk vanish in a non-trivial manner for the conformal algebra associated with the light-like wormhole throats with deformations respecting det(g4)=0 condition. Also the variations defined by infinitesimal isometries (zero conformal weight sector) should vanish since otherwise one would lose the conservation laws for isometry charges. The conditions for isometries might reduce to Tnkdet(g4) 1/2→ 0 at partonic orbits. Also now the interpretaton would be in terms of fixing of conformal gauge.
  3. Even Tnkdet(g4) 1/2=0 condition need not fix the partonic orbit completely. The Gribov ambiguity meaning that gauge conditions do not fix uniquely the gauge potential could have counterpart in TGD framework. It could be that there are several conformally non-equivalent space-time surfaces connecting 3-surfaces at the opposite ends of CD.

    If so, the boundary values at wormhole throats orbits could matter to some degree: very natural in boundary value problem thinking but new in initial value thinking. This would conform with the non-determinism of Kähler action implying criticality and the possibility that the 3-surfaces at the ends of CD are connected by several space-time surfaces which are physically non-equivalent.

    The hierarchy of Planck constants assigned to dark matter, quantum criticality and even criticality indeed relies on the assumption that heff=n× h corresponds to n-fold coverings having n space-time sheets which coincide at the ends of CD and that conformal symmetries act on the sheets as gauge symmetries. One would have as Gribov copies n conformal equivalence classes of wormhole throat orbits and corresponding space-time surfaces. Depending on whether one fixes the conformal gauge one has n equivalence classes of space-time surfaces or just one representative from each conformal equivalent class.

  4. There is also the question about the correspondence with the weak form of electric magnetic duality. This duality plus the condition that jαAα=0 in the interior of space-time surface imply the reduction of Kähler action to Chern-Simons terms. This would suggest that the boundary variation of the Kähler action reduces to that for Chern-Simons action which is indeed well-defined for light-like 3-surfaces.

    If so, the gauge fixing would reduce to variational equations for Chern-Simons action! A weaker condition is that classical conformal charges vanish. This would give a nice connection to the vision about TGD as almost topological QFT. In TGD framework these conditions do not imply the vanishing of Kähler form at boundaries. The conditions are satisfied if the CP2 projection of the partonic orbit is 2-D: the reason is that Chern-Simons term vanishes identically in this case.

  5. A further intuitively natural hypothesis is that there is a breaking of conformal symmetry: only the generators of conformal sub-algebra with conformal weight multiple of n act as gauge symmetries. This would give infinite hierarchies of breakings of conformal symmetry interpreted in terms of criticality: in the hierarchy the integers ni would satisfy ni divides ni+1.

    Similar degeneracy would be associated with the space-like ends at CD boundaries and I have considered the possibility that the integer n appearing in heff has decomposition n=n1n2 corresponding to the degeneracies associated with the two kinds of boundaries. Alternatively, one could have just n=n1=n2 from the condition that the two conformal symmetries are 3-dimensional manifestations of single 4-D analog of conformal symmetry.

As should have become clear, the derivation of field equations in TGD framework is not just an application of a formal recipe as in field theories and a lot of non-trivial physics is involved!

See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article The vanishing of conformal charges as a gauge conditions selecting preferred extremals of Kähler action.

Positivity of N=4 scattering amplitudes from number theoretical universality?

Lubos wrote a commentary about a new article of Nima Arkani Hamed et al about the positivity of the amplitudes in in the amplituhedron. Nima et al argue using explicit calculations that positivity could be a quite general property: as if the amplitudes were analogous to generalized volumes of higher-dimensional polyhedra.

The Grassmannian twistor approach considers also positive Grassmannians. This is not the same thing but most be closely related to the positivity of amplitudes. I recall that the core idea is however that one considers higher-D analogs of polygons. Simple representative for positive space polygon would be a triangle bounded by positive x- and y-axis and the line y=-x. x and y coordinates are positive.

What is of course certain is that the entire scattering amplitude cannot be positive since it is complex number. Rather, it must decompose into a product of "trivial" part determined by symmetries and non-trivial part which for some reason must be positive. What does this mean? Lubos considers this question in his posting: the idea is roughly that the real amplitude in question is an exponential and this guarantees positivity. Bundle theorist might speak about global everywhere non-vanishing section of vector bundle having geometric and topological meaning in algebraic geometry. Below I try to interpret positivity in terms of number theoretic arguments.

Does number theoretical universality require positivity?

Number theoretical universality of physics is one of the key principles of TGD and states that real physics must allow algebraic continuation to p-adic physics and vice versa. I suggest that number theoretical universality requires the positivity.

  1. The p-adicization of the amplitudes requires positivity. The "non-trivial" factor of the amplitude is mapped to its p-adic counterpart by a variant of canonical identification mapping p-adics to non-negative reals and vice versa and is well-defined only for non-negative real numbers. Hence positivity. This condition could also explain why positive Grassmannians are needed: the preferred coordinates must be positive in order to have well-defined p-adic counterparts and vice versa.

    The basic reason for positivity is that p-adic numbers are not well-ordered. When one has two p-adic numbers with the same p-adic norm, one cannot tell which of them is the larger one. This makes it impossible to tell whether a given p-adic number is positive or negative and one cannot talk about p-adic boundaries. p-Adic line segment has no ends. As a consequence, definite integral is very difficult notion p-adically although the notion of integral function can be defined. p-Adic counterparts of differential forms are also far from trivial to define.

  2. What about the symmetry determined parts of amplitudes, which are typically analogs of partial waves depending on angular coordinates and are complex and cannot be positive? Also here one encounters a technical problem related to another conceptual challenge of p-adicization program: the notion of angle is not well-defined in p-adic context and one can talk only about discrete phases. More precisely, one can define exponential function exp(x) if x has p-adic norm smaller than one but it does not have the properties of the ordinary exponential function (it has p-adic norm 1 for all values of x). One can define also trigonometric functions with the help of exp(ix) for p mod 4=3 formally but the trigonometric functions are not periodic. Something is missing.

    In the case of ordinary trigonometry the only way out is to perform algebraic extension of p-adic numbers by adding roots of unity representing phases associated with corresponding angles: phases replace angles. For instance, Un=exp(i2π/n) exists in an algebraic extension of p-adic numbers. Angle degrees of freedom are discretized.

    In the case of hyperbolic geometry, one can add roots of e to obtain p-adic counterparts of e(1/n) (note that ep is ordinary p-adic number so that these extensions are finite-dimensional algebraically and e is completely unique real transcendental in that it is algebraic number p-adically!): this extension allows to get the counterparts of ordinary exponential functions in extension.

    One can also multiply the points of discretization by p-adic numbers of norm smaller than 1 (or some negative power of p) to get something as near as possible to continuum. Hence discretization in both hyperbolic and trigonometric degrees of freedom by algebraic extension solves the problems quite generally for amplitudes defined in highly symmetric spaces. In Grassmann twistor approach one indeed considers projective spaces.

  3. Consider as an example the p-adic counterpart of Euclidian 2-space with coordinates (ρ,φ). ρ is non-negative radial coordinate and has p-adic counterpart obtained by canonical identification or its variant. The values of φ are replaced with discrete phase factors Un characterizing the values of φ coordinate. One has a collection of n rays emanating from origin instead of entire plane. One obtains infinite number of variants of E2 labelled by n characterizing the angular resolution.

    The p-adicization of Cartesian representation of real Euclidian 2-space defined using (x,y) coordinates would give only the first quadrant since negative x and y have no p-adic counterparts. In both cases cognitive representations lose a lot of information for purely number theoretical reasons. Cognitive analog of Uncertainty Principle is suggestive.

  4. Obviously General Coordinate Invariance is broken at the level of cognition which is actually not so surprising after all since the worlds in which mathematician has chosen to used Cartesian resp. spherical coordinates must differ in some delicate manner! Of course, the resulting discretized spaces are very different.

How should one p-adicize?

The p-adicization of various spaces and amplitudes is needed. p-Adicization means that one assigns to a real (or complex) number a p-adic variant by some rule. In the case of trigonometric and hyperbolic angles one can use discretization and algebraic extension but what about other kinds of coordinates? There are two guesses.

  1. Consider only rationals or their algebraic extension and maps only them to their p-adic counterparts in the needed extension of p-adic numbers. This correspondence is however extremely discontinuous since real numbers which are arbitrary distant can be arbitrary near p-adically and vice versa. What is nice that this map respects symmetries suggesting that one has symmetries below some rational cutoff defining measurement resolution. This conforms with the general philosophy about measurement resolution realized in terms of inclusions of hyper-finite factors realizing measurement resolution as analog of dynamical gauge symmetry.
  2. Use canonical identification mapping p-adic numbers to p-adic numbers by canonical identification: x= ∑ xnpn → ∑ xnp-n is the first option. It indeed works only for non-negative real numbers: hence positivity! Canonical identification is continuous but does not respect differentiability nor symmetries. Direct identification via common rationals in turn does not respect continuity.
The resolution of the problems is a compromize based on the use of two cutoffs. In some length scale range above p-adic UV cutoff LUV and scale L a direct correspondence between common rationals is assumed. Between L IR cutoff LIR canonical identification is used. Outside the range [LUV, LIR] there is no correspondence and conditions like smoothness dictate the details at both sides.

p-Adic space-time surfaces as cognitive maps of real ones and real space-time surfaces as correlates for realized intentions

One wants to talk about real topological invariants also in p-adic context: p-adic space-time surface should be a kind of cognitive representation of real space-time surface.

  1. Space-time surfaces are extremals of Kähler action (real/p-adic) but have a discrete set of common rational points. The notion of p-adic manifold for which p-adic regions are mapped to real chart leafs (rather than to p-adic ones!) formalizes this concept and allows to avoid the problem that p-adic balls are either disjoint or nested so that the usual construction of manifold does not make sense. The problem that there exists an endless variety of coordinate choices and each would give different notion of p-adic manifold.
  2. The cure comes from space-time as a surface property, and from symmetries allowing to induce manifold structure from the level of imbedding space the level of space-time surfaces. In TGD imbedding space is M4 × CP2. CP2 allows very natural p-adicizations by using complex coordinates transforming linearly under maximal subgroup of isometries and one obtains discrete variants of coset space with points labeled by phases in the algebraic extensions of p-adic numbers. One can also have a generalization in which each discrete point corresponds to a continuum of p-adic units.

    In the case of M4 one has more options but if one requires that the coordinate which is mapped by canonical identification to its real counterpart, the natural choices is the cosmic coordinates assignable to the future or past light-cone of causal diamond. Light-cone proper time would correspond to non-negative coordinate and the remaining coordinates would be ordinary angles hyperbolic angle and mappable to phase factors and real exponential which exists if one introduces finite number of roots of e and discretizes the hyperbolic angle. Note that p-adic causal diamond (CD) is p-adically non-trivial manifold requiring two chart leafs and this might deeply relate to the fact that state function reductions occur to either boundary of CD.

  3. Surface property implies that the correspondence between real and p-adic space-time surfaces is induced from that between the corresponding imbedding spaces and thus dictated to high degree by symmetries. Preferred imbedding space coordinates and their discretizations induce coordinates and discretizations at space-time surfaces so that there is a huge reduction in the number of different but cognitively non-equivalent discretizations.
  4. This could make possible in practice to define the notion of p-adic space-time surface as a cognitive map of real space-time surface and real space-surface as a realization of intention represented by p-adic space-time surface. The real extremal of Kähler action is mapped to p-adic one only in a given resolution. A subset of discrete points of the space-time surface is mapped to p-adic ones and vice versa and this discrete skeleton is continued to a p-adic extremal of Kähler action. The outcome is interpreted as cognitive representation or its inverse (transformation of intention to action) and need not be unique. The mapping taking p-adic skeleton to real one is up to some cutoff pinary digit just identification along common rationals respecting symmetry and above that canonical identification up to the highest allowed pinary digit.

To sum up, number theoretic universality condition for scattering amplitudes could help to understand the success of twistor Grassmann approach. The existence of p-adic variants of the amplitudes in finite algebraic extensions is a powerful constraint and I have argued that they are satisfied for the polylogarithms used. Positive Grassmannians and positivity of the amplitudes might be also seen as a manner to satisfy these constraints.

For background see the chapter Some fresh ideas about twistorialization of TGD or the article Positivity of N=4 scattering amplitudes from number theoretical universality?.

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