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Towards M-Matrix

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Year 2017

Could McKay correspondence generalize in TGD framework?

McKay correspondence states that the McKay graphs for the irreducible representations (irreps) of finite subgroups of G⊂ SU(2) characterizing their fusion algebra is given by extended Dynkin diagram of ADE type Lie group. Minimal conformal models with SU(2) Kac-Moody algebra (KMA) allow a classification by the same diagrams as fusion algebras of primary fields. The resolution of the singularities of complex algebraic surfaces in C3 by blowing implies the emergence of complex lines CP1. The intersection matrix for the CP1s is Dynkin diagram of ADE type Lie group. These results are highly inspiring concerning adelic TGD.

  1. The appearance of Dynkin diagrams in the classification of minimal conformal field theories (CFTs) inspires the conjecture that in adelic physics Galois groups Gal or semidirect products of Gal with a discrete subgroup G of automorphism group SO(3) (having SU(2) as double covering!) classifies TGD generalizations of minimal CFTs. Also discrete subgroups of octonionic automorphism group can be considered. The fusion algebra of irreps of Gal would define also the fusion algebra for KMA for the counterparts of minimal fields. This would provide deep insights to the general structure of adelic physics.
  2. One cannot avoid the question whether the extended ADE diagram could code for a dynamical symmetry of a minimal CFT or its modification? If the Gal singlets formed from the primary fields of minimal model define primary fields in Cartan algebra of ADE type KMA, then standard free field construction would give the charged KMA generators. In TGD framework this conjecture generalizes.
  3. A further conjecture is that the singularities of space-time surface imbedded as 4-surface in its 6-D twistor bundle with twistor sphere as fiber could be classified by McKay graph of Gal. The singular intersection of the Euclidian and Minkowskian regions of space-time surface is especially interesting: the twistor spheres at the common points defining light-like partonic orbits need not be same but have intersections with intersection matrix given by McKay graph for Gal. The basic information about adelic CFT would be coded by the general character of singularities for the twistor bundle.
  4. In TGD also singularities in which the group Gal is reduced to its subgroup Gal/H, where H is normal group are possible and would correspond to phase transition reducing the value of Planck constant. What happens in these phase transitions to single particle states would be dictated by the decomposition of representations of Gal to those of Gal/H and transition matrix elements could be evaluated.

See the new chapter Are higher structures needed in the categorification of TGD? or the article Could McKay correspondence generalize in TGD framework?.

About McKay and Langlands correspondences in TGD framework

In adelic TGD Galois groups for extensions of rationals become discrete symmetry groups acting on dark matter, identified as heff/h=n phases of ordinary matter. n gives the number of sheet of covering assignable to space-time surface. Since Galois group acts on the cognitive representation defined by a discrete set of points of space-time surface with coordinates having values in extension of rationals, the action of Galois group defines n-sheeted covering, where n is the order of Galois group thus identifiable in terms of Planck constant.

Adelic TGD inspires the question whether the representations of Galois groups could correspond to representations of Lie groups defining the ground states of Kac-Moody representations emerging in TGD in two manners: as representations of Kac-Moody algebra assignable the Poincare-, color- and electroweak symmetries on one hand and with dynamical generated from supersymplectic symmetry assignable with the boundaries of causal diamond (CD) and extended Kac-Moody symmetres assignable to the light-like orbits of partonic 2-surfaces defining boundaries between space-time regions with Minkowskian and Euclidian signatures of the induced metric.

McKay correspondence states that the finite discrete subgroups of SU(2) can be characterized by McKay graphs characterizing the fusion rules for the tensor products for the representations of these groups. These graphs correspond to the Dynkin diagrams for Kac-Moody algebras of ADE type group (all roots have same unit length in Dynkin diagram). This inspires the conjecture that finite subgroups of SU(2) indeed correspond to Kac-Moody algebras. Could the representations of discrete subgroups appearing in the McKay graph define also representations for the ground states of corresponding ADE type Kac-Moodyt algebra? More generally, could the Mc-Kay graps of the Galois groups?

Number theoretic Langlands correspondence in turn states roughly that the representations of Galois group for extensions of rationals correspond to the so called automorphic representations of algebraic variants of reductive Lie groups. This is not totally surprising since the matrices defining algebraic matrix group has matrix elements in the extension of rationals. This raises the question how closely the number theoretic Langlands correspondence corresponds to the basic physical picture of TGD.

1. Could normal sub-groups of symplectic group and of Galois groups correspond to each other?

Measurement resolution realized in terms of various inclusion is the key principle of quantum TGD. There is an analogy between the hierarchies of Galois groups, of fractal sub-algebras of supersymplectic algebra (SSA), and of inclusions of hyperfinite factors of type II1 (HFFs). The inclusion hierarchies of isomorphic sub-algebras of SSA and of Galois groups for sequences of extensions of extensions should define hierarchies for measurement resolution. Also the inclusion hierarchies of HFFs are proposed to define hierarcies of measurement resolutions. How closely are these hierarchies related and could the notion of measurement resolution allow to gain new insights about these hierarchies and even about the mathematics needed to realize them?

  1. As noticed, SSA and its isomorphic sub-algebras are in a relation analogous to the between normal sub-group H of group Gal (analog of isomorphic sub-algebra) and the group G/H. One can assign to given Galois extension a hierarchy of intermediate extensions such that one proceeds from given number field (say rationals) to its extension step by step. The Galois groups H for given extension is normal sub-group of the Galois group of its extension. Hence Gal/H is a group. The physical interpretation is following. Finite measurement resolution defined by the condition that H acts trivially on the representations of Gal implies that they are representations of Gal/H. Thus Gal/H is completely analogous to the Kac-Moody type algebra conjecture to result from the analogous pair for SSA.
  2. How does this relate to McKay correspondence stating that inclusions of HFFs correspond to finite discrete sub-groups of SU(2) acting as isometries of regular n-polygons and Platonic solids correspond to Dynkin diagrams of ADE type Super Kac-Moody algebras (SKMAs) determined by ADE Lie group G. Could one identify the discrete groups as Galois groups represented geometrically as sub-groups of SU(2) and perhaps also those of corresponding Lie group? Could the representations of Galois group correspond to a sub-set of representations of G defining ground states of Kac-Moody representations. This might be possible. The sub-groups of SU(2) can however correspond only to a very small fraction of Galois groups.
Can one imagine a generalization of ADE correspondence? What would be required that the representations of Galois groups relate in some natural manner to the representations as Kac-Moody groups.

1.1 Some basic facts about Galois groups and finite groups

Some basic facts about Galois groups mus be listed before continuing. Any finite group can appear as a Galois group for an extension of some number field. It is known whether this is true for rationals (see this).

Simple groups appear as building bricks of finite groups and are rather well understood. One can even speak about periodic table for simple finite groups (see this). Finite groups can be regarded as a sub-group of permutation group Sn for some n. They can be classified to cyclic, alternating , and Lie type groups. Note that alternating group An is the subgroup of permutation group Sn that consists of even permutations. There are also 26 sporadic groups and Tits group.

Most simple finite groups are groups of Lie type that is rational sub-groups of Lie groups. Rational means ordinary rational numbers or their extension. The groups of Lie type (see this) can be characterized by the analogs of Dynkin diagrams characterizing Lie algebras. For finite groups of Lie type the McKay correspondence could generalize.

1.2 Representations of Lie groups defining Kac-Moody ground states as irreps of Galois group?

The goal is to generalize the McKay correspondence. Consider extension of rationals with Galois group Gal. The ground staes of KMA representations are irreps of the Lie group G defining KMA. Could the allow ground states for given Gal be irreps of also Gal?

This constraint would determine which group representations are possible as ground states of SKMA representations for a given Gal. The better the resolution the larger the dimensions of the allowed representations would be for given G. This would apply both to the representations of the SKMA associated with dynamical symmetries and maybe also those associated with the standard model symmetries. The idea would be quantum classical correspondence (QCC) space-time sheets as coverings would realize the ground states of SKMA representations assignable to the various SKMAs.

This option could also generalize the McKay correspondence since one can assign to finite groups of Lie type an analog of Dynkin diagram (see this). For Galois groups, which are discrete finite groups of SU(2) the hypothesis would state that the Kac-Moody algebra has same Dynkin diagram as the finite group in question.

To get some perspective one can ask what kind of algebraic extensions one can assign to ADE groups appearing in the McKay correspondence? One can get some idea about this by studying the geometry of Platonic solids (see this). Also the geometry of Dynkin diagrams telling about the geometry of root system gives some idea about the extension involved.

  1. Platonic solids have p vertices and q faces. One has [p,q]∈ { [3, 3], [4, 3], [3, 4], [5, 3], [3, 5]}. Tetrahedron is self-dual (see this) object whereas cube and octahedron and also dodecahedron and icosahedron are duals of each other. From the table of Wikipedia article one finds that the cosines and sines for the angles between the vectors for the vertices of tetrahedron, cube, and octahedron are rational numbers. For icosahedron and dodecahedron the coordinates of vertices and the angle between these vectors involve Golden Mean φ=(1+51/2)/2 so that algebraic extension must involve 51/2 at least.

    The dihedral angle θ between the faces of Platonic solid [p,q] is given by sin(θ/2)= cos(π/q)/sin(π/p). For tetrahedron, cube and octahedron sin(θ) and cos(θ) involve 31/2. For icosahedron dihedral angle is tan(θ/2)= φ. For instance, the geometry of tetrahedron involves both 21/2 and 31/2. For dodecahedron more complex algebraic numbers are involved.

  2. The rotation matrices for for the triangles of tetrahedron and icosahedron involve cos(2π/3) and sin(2π/3) associated with the quantum phase q= exp(i2π/3) associated with it. The rotation matrices performing rotation for a pentagonal face of dodecahedron involves cos(2π/5) and sin(2π/5) and thus q= exp(i2π/5) characterizing the extension. Both q= exp(i2π/3) and q= exp(i2π/5) are thus involved with icosahedral and dodecahedral rotation matrices. The rotation matrices for cube and for octahedron have rational matrix elements.
  3. The Dynkin diagrams characterize both the finite discrete groups of SU(2) and those of ADE groups. The Dynkin diagrams of Lie groups reflecting the structure of corresponding Weyl groups involve only the angles π/2, 2π/3, π-π/6, 2π- π/6 between the roots. They would naturally relate to quadratic extensions.

    For ADE Lie groups the diagram tells that the roots associated with the adjoint representation are either orthogonal or have mutual angle of 2π/3 and have same length so that length ratios are equal to 1. One has sin(2π/3)= 31/2/2. This suggests that 31/2 belongs to the algebraic extension associated with ADE group always. For the non-simply laced Lie groups of type B, C, F, G the ratios of some root lengths can be 21/2 or 31/2.

For ADE groups assignable to n-polygons (n>5) Galois group must involve the cyclic extension defined by exp(i2π/n). The simplest option is that the extension corresponds to the roots of the polynomial xn= 1.

2. A possible connection with number theoretic Langlands correspondence

I have discussed number theoretic version of Langlands correspondence in \citeallb/Langland,Langlandsnew trying to understand it using physical intuition provided by TGD (the only possible approach in my case). Concerning my unashamed intrusion to the territory of real mathematicians I have only one excuse: the number theoretic vision forces me to do this.

Number theoretic Langlands correspondence relates finite-dimensional representations of Galois groups and so called automorphic representations of reductive algebraic groups defined also for adeles, which are analogous to representations of Poincare group by fields. This is kind of relationship can exist follows from the fact that Galois group has natural action in algebraic reductive group defined by the extension in question.

The "Resiprocity conjecture" of Langlands states that so called Artin L-functions assignable to finite-dimensional representations of Galois group Gal are equal to L-functions arising from so called automorphic cuspidal representations of the algebraic reductive group G. One would have correspondence between finite number of representations of Galois group and finite number of cuspidal representations of G.

This is not far from what I am naively conjecturing on physical grounds: finite-D representations of Galois group are reductions of certain representations of G or of its subgroup defining the analog of spin for the automorphic forms in G (analogous to classical fields in Minkowski space). These representations could be seen as induced representations familiar for particle physicists dealing with Poincare invariance. McKay correspondence encourages the conjecture that the allowed spin representations are irreducible also with respect to Gal. For a childishly naive physicist knowing nothing about the complexities of the real mathematics this looks like an attractive starting point hypothesis.

In TGD framework Galois group could provide a geometric representation of "spin" (maybe even spin 1/2 property) as transformations permuting the sheets of the space-time surface identifiable as Galois covering. This geometrization of number theory in terms of cognitive representations analogous to the use of algebraic groups in Galois correspondence might provide a totally new geometric insights to Langlands correpondence. One could also think that Galois group represented in this manner could combine with the dynamical Kac-Moody group emerging from SSA to form its Langlands dual.

Skeptic physicist taking mathematics as high school arithmetics might argue that algebraic counterparts of reductive Lie groups are rather academic entities. In adelic physics the situation however changes completely. Evolution corresponds to a hierarchy of extensions of rationals reflected directly in the physics of dark matter in TGD sense: that is as phases of ordinary matter with heff/h=n identifiable as order of Galois group for extension of rationals. Algebraic groups and their representations get physical meaning and also the huge generalization of their representation to adelic representations makes sense if TGD view about consciousness and cognition is accepted.

In attempts to understand what Langlands conjecture says one should understand first the rough meaning of many concepts. Consider first the Artin L-functions appearing at the number theoretic side. Consider first the Artin L-functions appearing at the number theoretic side.

  1. L-functions (see this) are meromorphic functions on complex plane that can be assigned to number fields and are analogs of Riemann zeta function factorizing into products of contributions labelled by primes of the number field. The definition of L-function involves Direchlet characters: character is very general invariant of group representation defined as trace of the representation matrix invariant under conjugation of argument.
  2. In particular, there are Artin L-functions (see this) assignable to the representations of non-Abelian Galois groups. One considers finite extension L/K of fields with Galois group G. The factors of Artin L-function are labelled by primes p of K. There are two cases: p is un-ramified or ramified depending on whether the number of primes of L to which p decomposes is maximal or not. The number of ramified primes is finite and in TGD framework they are excellent candidates for physical preferred p-adic primes for given extension of rationals.

    These factors labelled by p analogous to the factors of Riemann zeta are identified as characteristic polynomials for a representation matrix associated with any element in a preferred conjugacy class of G. This preferred conjugacy class is known as Frobenius element Frob(p) for a given prime ideal p , whose action on given algebraic integer in OL is represented as its p:th power. For un-ramified p the characteristic polynomial is explicitly given as determinant det[I-tρ(Frob(p))]-1, where one has t= N(p)-s and N(p) is the field norm of p in the extension L (see this).

    In the ramified case one must restrict the representation space to a sub-space invariant under inertia subgroup, which by definition leaves invariant integers of OL/p that is the lowest part of integers in expansion of powers of p.

At the other side of the conjecture appear representations of algebraic counterparts of reductive Lie groups and their L-functions and the two number theoretic and automorphic L-functions would be identical.
  1. Automorphic form F generalizes the notion of plane wave invariant under discrete subgroup of the group of translations and satisfying Laplace equation defining Casimir operator for translation group. Automorphic representations can be seen as analogs for the modes of classical fields with given mass having spin characterized by a representation of subgroup of Lie group G (SO(3) in case of Poincare group).

    Automorphic functions as field modes are eigen modes of some Casimir operators assignable to G. Algebraic groups would in TGD framework relate to adeles defined by the hierarchy of extensions of rationals (also roots of e can be considered in extensions). Galois groups have natural action in algebraic groups.

  2. Automorphic form (see this) is a complex vector valued function F from topological group to some vector space V. F is an eigen function of certain Casimir operators of G. In the simplest situation these function are invariant under a discrete subgroup Γ⊂ G identifiable as the analog of the subgroup defining spin in the case of induced representations.

    In general situation the automorphic form F transforms by a factor j of automorphy under Γ. The factor can also act in a finite-dimensional representation of group Γ, which would suggest that it reduces to a subgroup of Γ obtained by dividing with a normal subgroup. j satisfies 1-cocycle condition j(g1,g2g3)= j(g1g2,g3) in group cohomology guaranteeing associativity (see this). Cuspidality relates to the conditions on the growth of F at infinity.

  3. Elliptic functions in complex plane characterized by two complex periods are meromorphic functions of this kind. A less trivial situation corresponds to non-compact group G=SL(2,R) and Γ ⊂ SL(2,Q).
There are more groups involved: Langlands group LF and Langlands dual group LG. A more technical formulation says that the automorphic representations of a reductive Lie group G correspond to homomorphisms from so called Langlands group LF (see this) at the number theoretic side to L-group LG or Langlands dual of algebraic G at group theory side (see this). It is important to notice that LG is a complex Lie group. Note also that homomorphism is a representation of Langlands group LF in L-group LG. In TGD this would be analogous to a homomorphism of Galois group defining it as subgroup of the group G defining Kac-Moody algebra.
  1. Langlands group LF of number field is a speculative notion conjectured to be a extension of the Weil group of extension, which in turn is a modification of the absolute Galois group. Unfortunately, I was not able to really understand the Wikipedia definition of Weil group (this). If E/F is finite extension as it is now, the Weil group would be WE/F= WF/WcE, WcE refers to the commutator subgroup WE defining a normal subgroup, and the factor group is expected to be finite. This is not Galois group but should be closely related to it.

    Only finite-D representations of Langlands group are allowed, which suggests that the representations are always trivial for some normal subgroup of LF For Archimedean local fields LF is Weil group, non-Archimedean local fields LF is the product of Weil group of L and of SU(2). The first guess is that SU(2) relates to quaternions. For global fields the existence of LF is still conjectural.

  2. I also failed to understand the formal Wikipedia definition of the L-group LG appearing at the group theory side. For a reductive Lie group one can construct its root datum (X*,Δ,X*, Δc), where X* is the lattice of characters of a maximal torus, X* its dual, Δ the roots, and Δc the co-roots. Dual root datum is obtained by switching X* and X* and Δ and Δc. The root datum for G and LG are related by this switch.

    For a reductive G the Dynkin diagram of LG is obtained from that of G by exchanging the components of type Bn with components of type Cn. For simple groups one has Bn↔ Cn. Note that for ADE groups the root data are same for G and its dual and it is the Kac-Moody counterparts of ADE groups, which appear in McKay correspondence. Could this mean that only these are allowed physically?

  3. Consider now a reductive group over some field with a separable closure K (say k for rationals and K for algebraic numbers). Over K G as root datum with an action of Galois group of K/k. The full group LG is the semi-direct product LG0⋊ Gal(K/k) of connected component as Galois group and Galois group. Gal(K/k) is infinite (absolute group for rationals). This looks hopelessly complicated but it turns it that one can use the Galois group of a finite extension over which G is split. This is what gives the action of Galois group of extension (l/k) in LG having now finitely many components. The Galois group permutes the components. The action is easy to understand as automorphism on Gal elements of G.
Could TGD picture provide additional insights to Langlands duality or vice versa?
  1. In TGD framework the action of Gal on algebraic group G is analogous to the action of Gal on cognitive representation at space-time level permuting the sheets of the Galois covering, whose number in the general case is the order of Gal identifiable as heff/h=n. The connected component LG0 would correspond to one sheet of the covering.
  2. What I do not understand is whether LG =G condition is actually forced by physical contraints for the dynamical Kac-Moody algebra and whether it relates to the notion of measurement resolution and inclusions of HFFs.
  3. The electric-magnetic duality in gauge theories suggests that gauge group action of G on electric charges corresponds in the dual phase to the action of LG on magnetic charges. In self-dual situation one would have G=LG. Intriguingly, CP2 geometry is self-dual (Kähler form is self-dual so that electric and magnetic fluxes are identical) but induced Kähler form is self-dual only at the orbits of partonic 2-surfaces if weak form of electric-magnetic duality holds true. Does this condition leads to LG=G for dynamical gauge groups? Or is it possible to distinguish between the two dynamical descriptions so that Langlands duality would correspond to electric-magnetic duality. Could this duality correspond to the proposed duality of two variants of SH: namely, the electric description provided by string world sheets and magnetic description provided by partonic 2-surfaces carrying monopole fluxes?
See the new chapter Are higher structures needed in the categorification of TGD? or the article with the same title.

Are higher structures needed in the categorification of TGD?

The notion of higher structures promoted by John Baez looks very promising notion in the attempts to understand various structures like quantum algebras and Yangians in TGD framework. The stimulus for this article came from the nice explanations of the notion of higher structure by Urs Screiber. The basic idea is simple: replace "=" as a blackbox with an operational definition with a proof for $A=B$. This proof is called homotopy generalizing homotopy in topological sense. n-structure emerges when one realizes that also the homotopy is defined only up to homotopy in turn defined only up...

In TGD framework the notion of measurement resolution defines in a natural manner various kinds of "="s and this gives rise to resolution hierarchies. Hierarchical structures are characteristic for TGD: hierarchy of space-time sheet, hierarchy of p-adic length scales, hierarchy of Planck constants and dark matters, hierarchy of inclusions of hyperfinite factors, hierarchy of extensions of rationals defining adeles in adelic TGD and corresponding hierarchy of Galois groups represented geometrically, hierarchy of infinite primes, self hierarchy, etc...

In this article the idea of n-structure is studied in more detail. A rather radical idea is a formulation of quantum TGD using only cognitive representations consisting of points of space-time surface with imbedding space coordinates in extension of rationals defining the level of adelic hierarchy. One would use only these discrete points sets and Galois groups. Everything would reduce to number theoretic discretization at space-time level perhaps reducing to that at partonic 2-surfaces with points of cognitive representation carrying fermion quantum numbers.

Even the"{world of classical worlds" (WCW) would discretize: cognitive representation would define the coordinates of WCW point. One would obtain cognitive representations of scattering amplitudes using a fusion category assignable to the representations of Galois groups: something diametrically opposite to the immense complexity of the WCW but perhaps consistent with it. Also a generalization of McKay's correspondence suggests itself: only those irreps of the Lie group associated with Kac-Moody algebra that remain irreps when reduced to a subgroup defined by a Galois group of Lie type are allowed as ground states.

See the new chapter Are higher structures needed in the categorification of TGD? or the article with the same title.

Could categories, tensor networks, and Yangians provide the tools for handling the complexity of TGD?

TGD Universe is extremely simple but the presence of various hierarchies make it to look extremely complex globally. Category theory and quantum groups, in particular Yangian or its TGD generalization are most promising tools to handle this complexity. The arguments developed in the sequel suggest the following overall view.

  1. Positive and negative energy parts of zero energy states can be regarded as tensor networks identifiable as categories. The new element is that one does not have only particles (objects) replaced with partonic 2-surfaces but also strings connecting them (morphisms). Morphisms and functors provide a completely new element not present in standard model. For instance, S-matrix would be a functor between categories. Various hierarchies of of TGD would in turn translate to hierarchies of categories.
  2. TGD view about generalized Feynman diagrams relies on two general ideas. First, the twistor lift of TGD replaces space-time surfaces with their twistor-spaces getting their twistor structure as induced twistor structure from the product of twistor spaces of M4 and CP2. Secondly, topological scattering diagrams are analogous to computations and can be reduced to tree diagrams with braiding. This picture fits very nicely with the picture suggested by fusion categories. At fermionic level the basic interaction is 2+2 scattering of fermions occurring at the vertices identifiable as partonic 2-surface and re-distributes the fermion lines between partonic 2-surfaces. This interaction is highly analogous to what happens in braiding interaction but vertices expressed in terms of twistors depend on momenta of fermions.
  3. Braiding transformations take place inside the light-like orbits of partonic 2-surfaces defining boundaries of space-time regions with Minkowskian and Euclidian signature of induced metric respectively permuting two braid strands. R-matrix satisfying Yang-Baxter equation characterizes this operation algebraically.
  4. Reconnections of fermionic strings connecting partonic 2-surfaces are possible and suggest interpretation in terms of 2-braiding generalizing ordinary braiding: string world sheets get knotted in 4-D space-time forming 2-knots and strings form 1-knots in 3-D space. Reconnection induces an exchange of braid strands defined by the boundaries of the string world sheet and therefore exchange of fermion lines defining boundaries of string world sheets. A generalization of quantum algebras to include also algebraic representation for reconnection is needed. Also reconnection might reduce to a braiding type operation.
Yangians look especially natural quantum algebras from TGD point of view. They are bi-algebras with co-product Δ. This makes the algebra multi-local raising hopes about the understanding of bound states. Δ-iterates of single particle system would give many-particle systems with non-trivial interactions reducing to kinematics.

One should assign Yangian to various Kac-Moody algebras (SKMAs) involved and even with super-conformal algebra (SSA), which however reduces effectively to SKMA for finite-dimensional Lie group if the proposed gauge conditions meaning vanishing of Noether charges for some sub-algebra H of SSA isomorphic to it and for its commutator [SSA,H] with the entire SSA. Strong form of holography (SH) implying almost 2-dimensionality motivates these gauge conditions. Each SKMA would define a direct summand with its own parameter defining coupling constant for the interaction in question.

See the new chapter Could categories, tensor networks, and Yangians provide the tools for handling the complexity of TGD? of "Towards M-matrix" or the article with the same title.

Getting even more quantitative about CP violation

The twistor lift of TGD forces to introduce the analog of Kähler form for M4, call it J. J is covariantly constant self-dual 2-form, whose square is the negative of the metric. There is a moduli space for these Kähler forms parametrized by the direction of the constant and parallel magnetic and electric fields defined by J. J partially characterizes the causal diamond (CD): hence the notation J(CD) and can be interpreted as a geometric correlate for fixing quantization axis of energy (rest system) and spin.

Kähler form defines classical U(1) gauge field and there are excellent reasons to expect that it gives rise to U(1) quanta coupling to the difference of B-L of baryon and lepton numbers. There is coupling strength α1 associated with this interaction. The first guess that it could be just Kähler coupling strength leads to unphysical predictions: α1 must be much smaller. Here I do not yet completely understand the situation. One can however check whether the simplest guess is consistent with the empirical inputs from CP breaking of mesons and antimatter asymmetry. This turns out to be the case.

One must specify the value of α1 and the scaling factor transforming J(CD) having dimension length squared as tensor square root of metric to dimensionless U(1) gauge field F= J(CD)/S. This leads to a series of questions.

How to fix the scaling parameter S?

  1. The scaling parameter relating J(CD) and F is fixed by flux quantization implying that the flux of J(CD) is the area of sphere S2 for the twistor space M4× S2. The gauge field is obtained as F=J/S, where S= 4π R2(S2) is the area of S2.
  2. Note that in Minkowski coordinates the length dimension is by convention shifted from the metric to linear Minkowski coordinates so that the magnetic field B1 has dimension of inverse length squared and corresponds to J(CD)/SL2, where L is naturally be taken to the size scale of CD defining the unit length in Minkowski coordinates. The U(1) magnetic flux would the signed area using L2 as a unit.
How R(S2) relates to Planck length lP? lP is either the radius lP=R of the twistor sphere S2 of the twistor space T=M4× S2 or the circumference lP= 2π R(S2) of the geodesic of S2. Circumference is a more natural identification since it can be measured in Riemann geometry whereas the operational definition of the radius requires imbedding to Euclidian 3-space.

How can one fix the value of U(1) coupling strength α1? As a guideline one can use CP breaking in K and B meson systems and the parameter characterizing matter-antimatter symmetry.

  1. The recent experimental estimate for so called Jarlskog parameter characterizing the CP breaking in kaon system is J≈ 3.0× 10-5. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does not distinguish between different meson so that it is better to talk about orders of magnitude only.
  2. Matter-antimatter asymmetry is characterized by the number r=nB/nγ ∼ 10-10 telling the ratio of the baryon density after annihilation to the original density. There is about one baryon 10 billion photons of CMB left in the recent Universe.
Consider now the identification of α1.
  1. Since the action is obtained by dimensional reduction from the 6-D Kähler action, one could argue α1= αK. This proposal leads to unphysical predictions in atomic physics since neutron-electron U(1) interaction scales up binding energies dramatically.

    U(1) part of action can be however regarded a small perturbation characterized by the parameter ε= R2(S2)/R2(CP2), the ratio of the areas of twistor spheres of T(M4) and T(CP2). One can however argue that since the relative magnitude of U(1) term and ordinary Kähler action is given by ε, one has α1=ε× αK so that the coupling constant evolution for α1 and αK would be identical.

  2. ε indeed serves in the role of coupling constant strength at classical level. αK disappears from classical field equations at the space-time level and appears only in the conditions for the super-symplectic algebra but ε appears in field equations since the Kähler forms of J resp. CP2 Kähler form is proportional to R2(S2) resp. R2(CP2) times the corresponding U(1) gauge field. R(S2) appears in the definition of 2-bein for R2(S2) and therefore in the modified gamma matrices and modified Dirac equation. Therefore ε1/2=R(S2)/R(CP2) appears in modified Dirac equation as required by CP breaking manifesting itself in CKM matrix.

    NTU for the field equations in the regions, where the volume term and Kähler action couple to each other demands that ε and ε1/2 are rational numbers, hopefully as simple as possible. Otherwise there is no hope about extremals with parameters of the polynomials appearing in the solution in an arbitrary extension of rationals and NTU is lost. Transcendental values of ε are definitely excluded. The most stringent condition ε=1 is also unphysical. ε= 22r is favoured number theoretically.

Concerning the estimate for ε it is best to use the constraints coming from p-adic mass calculations.
  1. p-Adic mass calculations predict electron mass as

    me= hbar/R(CP2)(5+Y)1/2 .

    Expressing me in terms of Planck mass mP and assuming Y=0 (Y∈ (0,1)) gives an estimate for lP/R(CP2) as

    lPR(CP2) ≈ 2.0× 10-4 .

  2. From lP= 2π R(S2) one obtains estimate for ε, α1, g1=(4πα1)1/2 assuming αK≈ α≈ 1/137 in electron length scale.

    ε = 2-30 ≈ 1.0× 10-9 ,

    α1=εαK ≈ 6.8× 10-12 ,

    g1= (4πα11/2 ≈ 9.24 × 10-6 .

There are two options corresponding to lP= R(S2) and lP =2π R(S2). Only the length of the geodesic of S2 has meaning in the Riemann geometry of S2 whereas the radius of S2 has operational meaning only if S2 is imbedded to E3. Hence lP= 2π R(S2) is more plausible option.

For ε=2-30 the value of lP2/R2(CP2) is lP2/R2(CP2)=(2π)2 × R2(S2)/R2(CP2) ≈ 3.7× 10-8. lP/R(S2) would be a transcendental number but since it would not be a fundamental constant but appear only at the QFT-GRT limit of TGD, this would not be a problem.

One can make order of magnitude estimates for the Jarlskog parameter J and the fraction r= n(B)/n(γ). Here it is not however clear whether one should use ε or α1 as the basis of the estimate

  1. The estimate based on ε gives J∼ ε1/2 ≈ 3.2× 10-5 ,

    r∼ ε ≈ 1.0× 10-9 .

    The estimate for J happens to be very near to the recent experimental value J≈ 3.0× 10-5. The estimate for r is by order of magnitude smaller than the empirical value.

  2. The estimate based on α1 gives J∼ g1 ≈ 0.92× 10-5 ,

    r∼ α1 ≈ .68× 10-11 .

    The estimate for J is excellent but the estimate for r by more than order of magnitude smaller than the empirical value. One explanation is that αK has discrete coupling constant evolution and increases in short scales and could have been considerably larger in the scale characterizing the situation in which matter-antimatter asymmetry was generated.

Atomic nuclei have baryon number equal the sum B= Z+N of proton and neutron numbers and neutral atoms have B= N. Only hydrogen atom would be also U(1) neutral. The dramatic prediction of U(1) force is that neutrinos might not be so weakly interacting particles as has been thought. If the quanta of U(1) force are not massive, a new long range force is in question. U(1) quanta could become massive via U(1) super-conductivity causing Meissner effect. As found, U(1) part of action can be however regarded a small perturbation characterized by the parameter ε= R2(S2)/R2(CP2). One can however argue that since the relative magnitude of U(1) term and ordinary Kähler action is given by ε, one has α1=ε× αK.

Quantal U(1) force must be also consistent with atomic physics. The value of the parameter α1 consistent with the size of CP breaking of K mesons and with matter antimatter asymmetry is α1= εαK = 2-30αK.

  1. Electrons and baryons would have attractive interaction, which effectively transforms the em charge Z of atom Zeff= rZ, r=1+(N/Z)ε1, ε11/α=ε × αK/α≈ ε for αK≈ α predicted to hold true in electron length scale. The parameter

    s=(1 + (N/Z)ε)2 -1= 2(N/Z)ε +(N/Z)2ε2

    would characterize the isotope dependent relative shift of the binding energy scale.

    The comparison of the binding energies of hydrogen isotopes could provide a stringent bounds of the value of α1. For lP= 2π R(S2) option one would have α1=2-30αK ≈ .68× 10-11 and s≈ 1.4× 10-10. s is by order of magnitude smaller than α4≈ 2.9× 10-9 corrections from QED (see this). The predicted differences between the binding energy scales of isotopes of hydrogen might allow to test the proposal.

  2. B=N would be neutralized by the neutrinos of the cosmic background. Could this occur even at the level of single atom or does one have a plasma like state? The ground state binding energy of neutrino atoms would be α12mν/2 ∼ 10-24 eV for mν =.1 eV! This is many many orders of magnitude below the thermal energy of cosmic neutrino background estimated to be about 1.95× 10-4 eV (see this). The Bohr radius would be hbar/(α1mν) ∼ 106 meters and same order of magnitude as Earth radius. Matter should be U(1) plasma. U(1) superconductor would be second option.
See the new chapter Questions about twistor lift of TGD of "Towards M-matrix" or the article with the same title.

About some unclear issues of TGD

TGD has been in the middle of palace revolution during last two years and it is almost impossible to keep the chapters of the books updated. Adelic vision and twistor lift of TGD are the newest developments and there are still many details to be understood and errors to be corrected. The description of fermions in TGD framework has contained some unclear issues. Hence the motivation for the following brief comments.

Adelic vision and symmetries

In the adelic TGD SH is weakened: also the points of the space-time surface having imbedding space coordinates in an extension of rationals (cognitive representation) are needed so that data are not precisely 2-D. I have believed hitherto that one must use preferred coordinates for the imbedding space H - a subset of these coordinates would define space-time coordinates. These coordinates are determined apart from isometries. Does the number theoretic discretization imply loss of general coordinate invariance and also other symmetries?

The reduction of symmetry groups to their subgroups (not only algebraic since powers of e define finite-dimensional extension of p-adic numbers since ep is ordinary p-adic number) is genuine loss of symmetry and reflects finite cognitive resolution. The physics itself has the symmetries of real physics.

The assumption about preferred imbedding space coordinates is actually not necessary. Different choices of H-coordinates means only different and non-equivalent cognitive representations. Spherical and linear coordinates in finite accuracy do not provide equivalent representations.

Quantum-classical correspondence for fermions

Quantum-classical correspondence (QCC) for fermions is rather well-understood but deserves to be mentioned also here.

QCC for fermions means that the space-time surface as preferred extremal should depend on fermionic quantum numbers. This is indeed the case if one requires QCC in the sense that the fermionic representations of Noether charges in the Cartan algebras of symmetry algebras are equal to those to the classical Noether charges for preferred extremals.

Second aspect of QCC becomes visible in the representation of fermionic states as point like particles moving along the light-like curves at the light-like orbits of the partonic 2-surfaces (curve at the orbit can be locally only light-like or space-like). The number of fermions and antifermions dictates the number of string world sheets carrying the data needed to fix the preferred extremal by SH. The complexity of the space-time surface increases as the number of fermions increases.

Strong form of holography for fermions

It seems that scattering amplitudes can be formulated by assigning fermions with the boundaries of strings defining the lines of twistor diagrams. This information theoretic dimensional reduction from D=4 to D=2 for the scattering amplitudes can be partially understood in terms of strong form of holography (SH): one can construct the theory by using the data at string worlds sheets and/or partonic 2-surfaces at the ends of the space-time surface at the opposite boundaries of causal diamond (CD).

4-D modified Dirac action would appear at fundamental level as supersymmetry demands but would be reduced for preferred extremals to its 2-D stringy variant serving as effective action. Also the value of the 4-D action determining the space-time dynamics would reduce to effective stringy action containing area term, 2-D Kähler action, and topological Kähler magnetic flux term. This reduction would be due to the huge gauge symmetries of preferred extremals. Sub-algebra of super-symplectic algebra with conformal weigths coming as n-multiples of those for the entire algebra and the commutators of this algebra with the entire algebra would annihilate the physical states, and thecorresponding classical Noether charges would vanish.

One still has the question why not the data at the entire string world sheets is not needed to construct scattering amplitudes. Scattering amplitudes of course need not code for the entire physics. QCC is indeed motivated by the fact that quantum experiments are always interpreted in terms of classical physics, which in TGD framework reduces to that for space-time surface.

The relationship between spinors in space-time interior and at boundaries between Euclidian and Minkoskian regions

Space-time surface decomposes to interiors of Minkowskian and Euclidian regions. At light-like 3-surfaces at which the four-metric changes, the 4-metric is degenerate. These metrically singular 3-surfaces - partonic orbits- carry the boundaries of string world sheets identified as carriers of fermionic quantum numbers. The boundaries define fermion lines in the twistor lift of TGD. The relationship between fermions at the partonic orbits and interior of the space-time surface has however remained somewhat enigmatic.

So: What is the precise relationship between induced spinors ΨB at light-like partonic 3-surfaces and ΨI in the interior of Minkowskian and Euclidian regions? Same question can be made for the spinors ΨB at the boundaries of string world sheets and ΨI in interior of the string world sheets. There are two options to consider:

  • Option I: ΨB is the limiting value of ΨI .
  • Option II: ΨB serves as a source of ΨI .
For Opion I it is difficult to understand the preferred role of ΨB.

I have considered Option II already years ago but have not been able to decide.

  1. That scattering amplitudes could be formulated only in terms of sources only, would fit nicely with SH, twistorial amplitude construction, and also with the idea that scattering amplitudes in gauge theories can be formulated in terms of sources of boson fields assignable to vertices and propagators. Now the sources would become fermionic.
  2. One can take gauge theory as a guideline. One adds to free Dirac equation source term γkkAkΨ. Therefore the natural boundary term in the action would be of the form (forgetting overall scale factor)

    SB=ΨIΓα(C-S)AαΨB+ c.c .

    Here the modified gamma matrix is Γα(C-S) (contravariant form is natural for light-like 3-surfaces) is most naturally defined by the boundary part of the action - naturally Chern-Simons term for Kähler action. A denotes the Kähler gauge potential.

  3. The variation with respect to ΨB gives


    at the boundary so that the C-S term and interaction term vanish. This does not however imply vanishing of the source term! This condition can be seen as a boundary condition.

The same argument applies also to string world sheets.

About second quantization of the induced spinor fields

The anti-commutation relations for the induced spinors have been a long-standing issue and during years I have considered several options. The solution of the problem looks however stupifuingly simple. The conserved fermion currents are accompanied by super-currents obtained by replacing Ψ with a mode of the induced spinor field to get unΓαΨ or ΨΓαun with the conjugate of the mode. One obtains infinite number of conserved super currents. One can also replace both Ψ and Ψ in this manner to get purely bosonic conserved currents ΨmΓαun to which one can assign a conserved bosonic charges Qmn.

I noticed this years ago but did not realize that these bosonic charges define naturally anti-commutators of fermionic creation and annihilation operators! The ordinary anti-commutators of quantum field theory follow as a special case! By a suitable unitary transformation of the spinor basis one can diagonalize the hermitian matrix defined by Qmn and by performing suitable scalings one can transform anti-commutation relations to the standard form. An interesting question is whether the diagonalization is needed, and whether the deviation of the diagonal elements from unity could have some meaning and possibly relate to the hierarchy heff=n× h of Planck constants - probably not.

Is statistical entanglement "real" entanglement?

The question about the "reality" of statistical entanglement has bothered me for years. This entanglement is maximal and it cannot be reduced by measurement so that one can argue that it is not "real". Quite recently I learned that there has been a longstanding debate about the statistical entanglement and that the issue still remains unresolved.

The idea that all electrons of the Universe are maximally entangled looks crazy. TGD provides several variants for solutions of this problem. It could be that only the fermionic oscillator operators at partonic 2-surfaces associated with the space-time surface (or its connected component) inside given CD anti-commute and the fermions are thus indistinguishable. The extremist option is that the fermionic oscillator operators belonging to a network of partonic 2-surfaces connected by string world sheets anti-commute: only the oscillator operators assignable to the same scattering diagram would anti-commute.

What about QCC in the case of entanglement. ER-EPR correspondence introduced by Maldacena and Susskind for 4 years ago proposes that blackholes (maybe even elementary particles) are connected by wormholes. In TGD the analogous statement emerged for more than decade ago - magnetic flux tubes take the role of wormholes in TGD. Magnetic flux tubes were assumed to be accompanied by string world sheets. I did not consider the question whether string world sheets are always accompanied by flux tubes.

What could be the criterion for entanglement to be "real"? "Reality" of entanglement demands some space-time correlate. Could the presence of the flux tubes make the entanglement "real"? If statistical entanglement is accompanied by string connections without magnetic flux tubes, it would not be "real": only the presence of flux tubes would make it "real". Or is the presence of strings enough to make the statistical entanglement "real". In both cases the fermions associated with disjoint space-time surfaces or with disjoint CDs would not be indistinguishable. This looks rather sensible.

The space-time correlate for the reduction of entanglement would be the splitting of a flux tube and fermionic strings inside it. The fermionic strings associated with flux tubes carrying monopole flux are closed and the return flux comes back along parallel space-time sheet. Also fermionic string has similar structure. Reconnection of this flux tube with shape of very long flattened square splitting it to two pieces would be the correlate for the state function reduction reducing the entanglement with other fermions and would indeed decouple the fermion from the network.

See the chapter Number Theoretical Vision.

Getting quantitative about breaking of CP, P, and T

The twistor lift of TGD led to the introduction of Kähler form also in M4 factor of imbedding space M4×CP2. The moduli space of causal diamonds (CDs) introduced already early allow to save Poincare invariance at the level of WCW. One of the very nice things is that the self-duality of J(M4) leads to a new mechanism of breaking for P,CP, and T in long scales, where these breakings indeed take place. P corresponds to chirality selection in living matter, CP to matter antimatter asymmetry and T could correspond to preferred arrow of clock time. TGD allows both arrows but T breaking could make other arrow dominant. Also the hierarchy of Planck constant is expected to be important.

Can one say anything quantitative about these various breakings?

  1. J(M4) is proportional to Newton's constant G in the natural scale of Minkowski coordinates defined by twistor sphere of T(M4). Therefore CP breaking is expected to be proportional to lP2/R2 or to its square root lP/R. The estimate for lP/R is X== lP/R≈ 2-12≈ 2.5× 10-4.

    The determinant of CKM matrix is equal to phase factor by unitarity (UU=1) and its imaginary part characterizes CP breaking. The imaginary part of the determinant should be proportional to the Jarlskog invariant J= +/- Im(VusVcbV*ub V*cs) characterizing CP breaking of CKM matrix (see this).

    The recent experimental estimate is J≈ 3.0× 10-5. J/X≈ 0 .1 so that there is and order of magnitude deviation. Earlier experimental estimate used in p-adic mass calculations was by almost order of magnitude larger consistent with the value of X. For B mesons CP breading is about 50 times larger than for kaons and it is clear that Jarlskog invariant does nto distinguish between different meson so that it is better to talk about orders of magnitude only.

    The parameter used to characterize matter antimatter asymmetry (see this) is the ratio R=[n(B-n(B*)]/n(γ)) ≈ 9× 10-11 of the difference of baryon and antibaryon densities to photon density in cosmological scales. One has X3 ≈ 1.4 × 10-11, which is order of magnitude smaller than R.

  2. What is interesting that P is badly broken in long length scales as also CP. The same could be true for T. Could this relate to the thermodynamical arrow of time? In ZEO state function reductions to the opposite boundary change the direction of clock time. Most physicist believe that the arrow of thermodynamical time and thus also clock time is always the same. There is evidence that in living matter both arrows are possible. For instance, Fantappie has introduced the notion of syntropy as time reversed entropy. This suggests that thermodynamical arrow of time could correspond to the dominance of the second arrow of time and be due to self-duality of J(M4) leading to breaking of T. For instance, the clock time spend in time reversed phase could be considerably shorter than in the dominant phase. A quantitative estimate for the ratio of these times might be given some power of the the ratio X=lP/R.
For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.

A new kind of duality of old duality from a new perspective?

M8-H duality maps the preferred extremals in H to those M4× CP2 and vice versa. The tangent spaces of an associative space-time surface in M8 would be quaternionic (Minkowski) spaces.

In M8 one can consider also co-associative space-time surfaces having associative normal space. Could the co-associative normal spaces of associative space-time surfaces in the case of preferred extremals form an integrable distribution therefore defining a space-time surface in M8 mappable to H by M8-H duality? This might be possible but the associative tangent space and the normal space correspond to the same CP2 point so that associative space-time surface in M8 and its possibly existing co-associative companion would be mapped to the same surface of H.

This dead idea however inspires an idea about a duality mapping Minkowskian space-time regions to Euclidian ones. This duality would be analogous to inversion with respect to the surface of sphere, which is conformal symmetry. Maybe this inversion could be seen as the TGD counterpart of finite-D conformal inversion at the level of space-time surfaces. There is also an analogy with the method of images used in some 2-D electrostatic problems used to reflect the charge distribution outside conducting surface to its virtual image inside the surface. The 2-D conformal invariance would generalize to its 4-D quaterionic counterpart. Euclidian/Minkowskian regions would be kind of Leibniz monads, mirror images of each other.

  1. If strong form of holography (SH) holds true, it would be enough to have this duality at the informational level relating only 2-D surfaces carrying the holographic information. For instance, Minkowskian string world sheets would have duals at the level of space-time surfaces in the sense that their 2-D normal spaces in X4 form an integrable distribution defining tangent spaces of a 2-D surface. This 2-D surface would have induced metric with Euclidian signature.

    The duality could relate either a) Minkowskian and Euclidian string world sheets or b) Minkowskian/Euclidian string world sheets and partonic 2-surfaces common to Minkowskian and Euclidian space-time regions. a) and b) is apparently the most powerful option information theoretically but is actually implied by b) due to the reflexivity of the equivalence relation. Minkowskian string world sheets are dual with partonic 2-surfaces which in turn are dual with Euclidian string world sheets.

    1. Option a): The dual of Minkowskian string world sheet would be Euclidian string world sheet in an Euclidian region of space-time surface, most naturally in the Euclidian "wall neighbour" of the Minkowskian region. At parton orbits defining the light-like boundaries between the Minkowskian and Euclidian regions the signature of 4-metric is (0,-1,-1,-1) and the induced 3-metric has signature (0,-1,-1) allowing light-like curves. Minkowskian and Euclidian string world sheets would naturally share these light-like curves aas common parts of boundary.
    2. Option b): Minkowskian/Euclidian string world sheets would have partonic 2-surfaces as duals. The normal space of the partonic 2-surface at the intersection of string world sheet and partonic 2-surface would be the tangent space of string world sheets so that this duality could make sense locally. The different topologies for string world sheets and partonic 2-surfaces force to challenge this option as global option but it might hold in some finite region near the partonic 2-surface. The weak form of electric-magnetic duality could closely relate to this duality.
    In the case of elementary particles regarded as pairs of wormhole contacts connected by flux tubes and associated strings this would give a rather concrete space-time view about stringy structure of elementary particle. One would have a pair of relatively long (Compton length) Minkowskian string sheets at parallel space-time sheets completed to a parallelepiped by adding Euclidian string world sheets connecting the two space-time sheets at two extremely short (CP2 size scale) Euclidian wormhole contacts. These parallelepipeds would define lines of scattering diagrams analogous to the lines of Feynman diagrams.
This duality looks like new but as already noticed is actually just the old electric-magnetic duality seen from number-theoretic perspective.

For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.

About the generalization of dual conformal symmetry and Yangian in TGD

The discovery of dual of the conformal symmetry of gauge theories was crucial for the development of twistor Grassmannian approach. The D=4 conformal generators acting on twistors have a dual representation in which they act on momentum twistors: one has dual conformal symmetry, which becomes manifest in this representation. These two separate symmetries extend to Yangian symmetry providing a powerful constraint on the scattering amplitudes in twistor Grassmannian approach fo N=4 SUSY.

In TGD the conformal Yangian extends to super-symplectic Yangian - actually, all symmetry algebras have a Yangian generalization with multi-locality generalized to multi-locality with respect to partonic 2-surfaces. The generalization of the dual conformal symmetry has however remained obscure. In the following I describe what the generalization of the two conformal symmetries and Yangian symmetry would mean in TGD framework.

One also ends up with a proposal of an information theoretic duality between Euclidian and Minkowskian regions of the space-time surface inspired by number theory: one might say that the dynamics of Euclidian regions is mirror image of the dynamics of Minkowskian regions. A generalization of the conformal reflection on sphere and of the method of image charges in 2-D electrostatics to the level of space-time surfaces allowing a concrete construction reciple for both Euclidian and Minkowskian regions of preferred extremals is in question. One might say that Minkowskian and Euclidian regions are analogous to Leibnizian monads reflecting each other in their internal dynamics.

See the chapter Some Questions Related to the Twistor Lift of TGD or the article with the same title.

About unitarity for scattering amplitudes

The first question is what one means with S-matrix in ZEO. I have considered several proposals for the counterparts of S-matrix. In the original U-matrix, M-matrix and S-matrix were introduced but it seems that U-matrix is not needed.

  1. The first question is whether the unitary matrix is between zero energy states or whether it characterizes zero energy states themselves as time-like entanglement coefficients between positive and negative energy parts of zero energy states associated with the ends of CD. One can argue that the first option is not sensible since positive and negative energy parts of zero energy states are strongly correlated rather than forming a tensor product: the S-matrix would in fact characterize this correlation partially.

    The latter option is simpler and is natural in the proposed identification of conscious entity - self - as a generalized Zeno effect, that is as a sequence of repeated state function reductions at either boundary of CD shifting also the boundary of CD farther away from the second boundary so that the temporal distance between the tips of CD increases. Each shift of this kind is a step in which superposition of states with different distances of upper boundary from lower boundary results followed by a localization fixing the active boundary and inducing unitary transformation for the states at the original boundary.

  2. The proposal is that the the proper object of study for given CD is M-matrix. M-matrix is a product for a hermitian square root of diagonalized density matrix ρ with positive elements and unitary S-matrix S : M= ρ1/2S. Density matrix ρ could be interpreted in this approach as a non-trivial Hilbert space metric. Unitarity conditions are replaced with the conditions MM= ρ and MM=ρ. For the single step in the sequence of reductions at active boundary of CD one has M→ MS (Δ T) so that one has S→ SS(Δ T). S(Δ T) depends on the time interval Δ T measured as the increase in the proper time distance between the tips of CD assignable to the step.
What does unitarity mean in the twistorial approach?
  1. In accordance with the idea that scattering diagrams is a representation for a computation, suppose that the deformations of space-time surfaces defining a given topological diagram as a maximum of the exponent of Kähler function, are the basic objects. They would define different quantum phases of a larger quantum theory regarded as a square root of thermodynamics in ZEO and analogous to those appearing also in QFTs. Unitarity would hold true for each phase separately.

    The topological diagrams would not play the role of Feynman diagrams in unitarity conditions although their vertices would be analogous to those appearing in Feynman diagrams. This would reduce the unitarity conditions to those for fermionic states at partonic 2-surfaces at the ends of CDs, actually at the ends of fermionic lines assigned to the boundaries of string world sheets.

  2. The unitarity conditions be interpreted stating the orthonormality of the basis of zero energy states assignable with given topological diagram. Since 3-surfaces as points of WCW appearing as argument of WCW spinor field are pairs consisting of 3-surfaces at the opposite boundaries of CD, unitarity condition would state the orthonormality of modes of WCW spinor field. If might be even that no mathematically well-defined inner product assignable to either boundary of CD exists since it does not conform with the view provided by WCW geometry. Perhaps this approach might help in identifying the correct form of S-matrix.
  3. If only tree diagrams constructed using 4-fermion twistorial vertex are allowed, the unitarity relations would be analogous to those obtained using only tree diagrams. They should express the discontinuity for T in S=1+iT along unitary cut as Disc(T)= TT. T and T would be T-matrix and its time reversal.
  4. The correlation between the structure of the fermionic scattering diagram and topological scattering diagrams poses very strong restrictions on allowed scattering reactions for given topological scattering diagram. One can of course have many-fermion states at partonic 2-surfaces and this would allow arbitrarily high fermion numbers but physical intuition suggests that for given partonic 2-surface (throat of wormhole contact) the fermion number is only 0, 1, or perhaps 2 in the case of supersymmetry possibly generated by right-handed neutrino.

    The number of fundamental fermions both in initial and final states would be finite for this option. In quantum field theory with only masive particles the total energy in the final state poses upper bound on the number of particles in the final state. When massless particles are allowed there is no upper bound. Now the complexity of partonic 2-surface poses an upper bound on fermions.

    This would dramatically simplify the unitarity conditions but might also make impossible to satisfy them. The finite number of conditions would be in spirit with the general philosophy behind the notion of hyper-finite factor. The larger the number of fundamental fermions associated with the state, the higher the complexity of the topological diagram. This would conform with the idea about QCC. One can make non-trivial conclusions about the total energy at which the phase transitions changing the topology of space-time surface defined by a topological diagram must take place.

For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.

Kerr effect, breaking of T symmetry, and Kähler form of M4

I encountered in Facebook (thanks to Ulla) a link to a very interesting article Here is the abstract.

We prove an instance of the Reciprocity Theorem that demonstrates that Kerr rotation, also known as the magneto-optical Kerr effect, may only arise in materials that break microscopic time reversal symmetry. This argument applies in the linear response regime, and only fails for nonlinear effects. Recent measurements with a modified Sagnac Interferometer have found finite Kerr rotation in a variety of superconductors. The Sagnac Interferometer is a probe for nonreciprocity, so it must be that time reversal symmetry is broken in these materials.

I had to learn some basic condensed matter physics. Magneto-optic Kerr effect occurs when a circularly polarized plane wave - often with normal incidence - reflects from a sample with planar boundary. In magneto-optic Kerr effect there are many options depending on the relative directions of the reflection plane (incidence is not normal in the general case so that one can talk about reflection plane) and magnetization. Also the incoming polarization can be linear or circular. Reflected circular polarized beams suffers a phase change in the reflection: as if they would spend some time at the surface before reflecting. Linearly polarized light reflects as elliptically polarized light.

Kerr angle θK is defined as 1/2 of the difference of the phase angle increments caused by reflection for oppositely circularly polarized plane wave beams. As the name tells, magneto-optic Kerr effect is often associated with magnetic materials.

Kerr effect has been however observed also for high Tc superconductors and this has raised controversy. As a layman in these issues I can naively wonder whether the controversy is created by the expectation that there are no magnetic fields inside the super-conductor. Anti-ferromagnetism is however important for high Tc superconductivity. In TGD based model for high Tc superconductors the supracurrents would flow along pairs of flux tubes with the members of S=0 (S=1) Cooper pairs at parallel flux tubes carrying magnetic fields with opposite (parallel) magnetic fluxes. Therefore magneto-optic Kerr effect could be in question after all.

The author claims to have proven that Kerr effect in general requires breaking of microscopic time reversal symmetry. Time reversal symmetry breaking (TRSB) caused by the presence of magnetic field and in the case of unconventional superconductors is explained nicely here. Magnetic field is required. Magnetic field is generated by a rotating current and by right-hand rule time reversal changes the direction of the current and also of magnetic field. For spin 1 Cooper pairs the analog of magnetization is generated, and this leads to T breaking.

This result is very interesting from the point of TGD. The reason is that twistorial lift of TGD requires that imbedding space M4× CP2 has Kähler structure in generalized sense. M4 has the analog of Kähler form, call it J(M4). J(M4) is assumed to be self-dual and covariantly constant as also CP2 Kähler form, and contributes to the Abelian electroweak U(1) gauge field (electroweak hypercharge) and therefore also to electromagnetic field.

J(M4) implies breaking of Lorentz invariance since it defines decomposition M4= M2× E2 Implying preferred rest frame and preferred spatial direction identifiable as direction of spin quantization axis. In zero energy ontology (ZEO) one has moduli space of causal diamonds (CDs) and therefore also moduli space of Kähler forms and the breaking of Lorentz invariance cancels. Note that a similar Kähler form is conjectured in quantum group inspired non-commutative quantum field theories and the problem is the breaking of Lorentz invariance.

What is interesting that the action of P,CP, and T on Kähler form transforms it from self-dual to anti-self-dual form and vice versa. If J(M4) is self-dual as also J(CP2), all these 3 discrete symmetries are broken in arbitrarily long length scales. On basis of tensor property of J(M4) one expects P: (J(M2),J(E2)→ (J(M2),-J(E2) and T: (J(M2),J(E2)→ (-J(M2),J(E2). Under C one has (J(M2),J(E2)→ (-J(M2),-J(E2). This gives CPT: (J(M2),J(E2)→ (J(M2),J(E2) as expected.

One can imagine several consequences at the level of fundamental physics.

  1. One implication is a first principle explanation for the mysterious CP violation and matter antimatter asymmetry not predicted by standard model (see below).
  2. A new kind of parity breaking is predicted. This breaking is separate from electroweak parity breaking and perhaps closely related to the chiral selection in living matter.
  3. The breaking of T might in turn relate to Kerr effect if the argument of authors is correct. It could occur in high Tc superconductors in macroscopic scales. Also large heff/h=n scaling up quantum scales in high Tc superconductors could be involved as with the breaking of chiral symmetry in living matter. Strontium ruthenate for which Cooper pairs are in S=1 state is is indeed found to exhibit TRSB (for references and explanation see this).

    In TGD based model of high Tc superconductivity the members of the Cooper pair are at parallel magnetic flux tubes with the same spin direction of magnetic field. The magnetic fields and thus the direction of spin component in this direction changes under T causing TRSB. The breaking of T for S=1 Cooper pairs is not spontaneous but would occur at the level of physics laws: the time reversed system finds itself experiences in the original self-dual J(M4)) rather than in (-J(M2),J(E2)) demanded by T symmetry.

For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.

Key ideas related to the twistor lift of TGD

The generalization of twistor approach from M4 to H=M4× CP2 involves the replacement of twistor space of M4 with that of H. M8-H duality allows also an alternative approach in which one constructs twistor space of octonionic M8. Note that M4,E4, S4, and CP2 are the unique 4-D spaces allowing twistor space with Kähler structure. This makes TGD essentially unique.

Ordinary twistor approach has two problems.

  1. It applies only if the particles are massless. In TGD particles are massless in 8-D sense but the projection of 8-momentum to given M4 is in general massive in 4-D sense. This solves the problem. Note that the 4-D M4 momenta can be light-like for a suitable choice of M4⊂ H. There exist even a choice of M2 for which this is the case. For given M2 the choices of quaternionic M4 are parametrized by CP2.
  2. The twistor approach has second problem: it works nicely in signature (2,2) rather (1,3) for Minkowski space. For instance, twistor Fourier transform cannot be defined as an ordinary integral. The very nice results by Nima Arkani-Hamed et al about positive Grassmannian follow only in the signature (2,2).

    One can always find M2⊂ M8 in which the 8-momentum lies and is therefore light-like in 2-D sense. Furthermore, the light-like 8-momenta and thus 2-momenta are prediced already at classical level to be complex. M2 as subspace of momentum space M8 effectively extends to its complex version with signature (2,2)!

    At classical space-time level the presence of preferred M2 reflects itself in the properties of massless extremals with M4= M2× E2 decomposition such that light-like momentum is in M2 and polarization in E2.

    4-D conformal invariance is restricted to its 2-D variant in M2. Twistor space of M4 reduces to that of M2. This is SO(2,2)/SO(2,1)=RP3. This is 3-D RP3, the real variant of twistor space CP3. Complexification of light-like momenta replaces RP3 with CP3.

Light-like M8-momenta are in question but they are not arbitrary.
  1. They must lie in some quaternionic plane containing fixed M2, which corresponds to the plane spanned by real octonion unit and some imaginary unit. . This condition is analogous to the condition that the space-time surfaces as preferred extremals in M8 have quaternionic tangent planes.
  2. In particular, the wave functions can be expressed as products of plane waves in M2, wave functions in the plane of transverse momenta in E2⊂ M4, where M4 is quaternionic plane containing M2 and wave function in the space for the choices of M4, which is CP2. One obtains exactly the same result in M4× CP2 if delocalization in transversal E2 momenta taking place of quarks inside hadrons takes place. Transversal wave function can also concentrate on single momentum value.

    It should be noticed that quaternionicity forces number theoretical spontaneous compactification. It would be very clumsy to realize the condition that allowed 8-momenta are qiuaternionic. Instead going to M4× CP2, "spontaneously compactifying", description everyting becomes easy.

  3. What is amusing that the geometric twistor space M4× S2 of M4 having bundle projections to M4 and ordinary twistor spaces is nothing but the space of choices of causal diamonds with preferred M2 and fixed rest frame (time axis connecting the tips). M4 point fixes the tip of causal diamond (CD) and S2 the spatial direction fixing M2 plane. In case of CP2 the point of twistor space fixes point of CP2 as analog for tip of CD: the complex CP2 coordinates have origin at this point. The point of twistor sphere of SU(3)/U(1)× U(1) codes for the selection of quantization axis for hypercharge Y and isospin I3. The corresponding subgroup U(1)× U(1) affects only the phases of the preferred complex coordinates transforming linearly under SU(2)× U(1).

    At the level of momentum space M4 twistor codes for the momentum and helicity of particle. For CP2 it codes for the selection of M4⊂ M8 and for em charge as analog of helicity. Now one has actually wave function for the selections of CP2 point labelled by the color numbers of the particle.

Number theoretical vision inspires the idea that scattering ampitudes define representations for algebraic computations leading from initial set of algebraic objects to to final set of objects. If so, the amplitudes should not depend on how the computation is done and there should exist a minimal computation possibly represented by a tree diagram. There would be no summation over the equivalent diagrams: one can choose any-one of them and the best choice is the simplest one.

To develop this idea one must understand what scattering diagrams are. The scattering diagrams involve two kinds of lines.

  1. There are topological "lines" corresponding to light-like orbits of partonic 2-surfaces playing the role of lines of Feynman diagrams. The topological diagram formed by these lines gives boundary conditions for 4-surface: at these light-like partonic orbits Euclidian space-time region changes to Minkowskian one. Vertices correspond to 2-surfaces at which these 3-D lines meet just like line in the case of Feynman diagrams.
  2. There are also fermion lines assignable to fundamental fermions serving as building bricks of elementary particles. They correspond to the boundaries of string world sheets at the orbits of partonic 2-surfaces. Fundamental fermion-fermion scattering takes place via classical interactions at partonic 2-surfaces: there is no 4-vertex in the usual sense (this would lead to non-renormalizable theory).

    The conjecture is that he 4-vertex is described by twistor amplitude fixed apart from over all scaling factor. Fermion lines are along parton orbits. Boson lines correspond to pairs of fermion and antifermion at the same parton orbit.

    As a matter fact, the situation is more complex for elementary particles since they correspond to pairs of wormhole contacts connected by monopole magnetic tubes and wormhole contacts has two wormhole throats - partonic 2-surfaces.

For the idea about diagrams as representations of computations to make sense, there should exist moves which allow to glide the 4-fermion vertex and associated flux tubes along the topological line of scattering diagrams in the vicinity of the second end of the loop. Second move should allow to snip away the loop. Is this possible? The possibility to find M2 for which momentum is light-like is central in the argument claiming that this is indeed possible.

The basic problem is that the kinematics for 4-fermion vertices need not be consistent with the gliding of vertex past another one so that this move is not possible.

  1. Clearly, one must assume something. If all momenta along at vertices along fermion line are in same M2 then they parallel as light-like M2-momenta. Kinematical conditions allow the gliding of two vertices of this kind past each other as is easy to show. The scattering would mean only redistribution of parallel light-like momenta in this particular M2.

    This kind of scattering would be more general than the scattering in integrable quantum field theories in M2: in this case the scattering would not affect the momenta but would induce phase shifts: particles would spend some time in the vertex before continuing. What is crucial for having non-trivial scatterings, is that in the general frame M2⊂ M4 ⊂ M8 the momenta would be massive and also different.

  2. The condition would be that all four-fermion vertices along given fermion line correspond to the same preferred M2. M2:s can differ only for fermionic sub-diagrams which do not have common vertices.

    Note however that tree diagrams for which lines can have different M2s can give rise to non-trivial scattering. One can take tree diagram and assign to the internal lines networks with same M2s as the internal line has. It is quite possible that for general graphs allowing different M2s in internal lines and loops, the reduction to tree graph is not possible.

    At least this idea could define precisely what the equivalence of diagrams, if vertices in which M2:s can be different are allowed. One can of course argue, that there is not deep reason for not allowing more general loopy graphs in which the incoming lines can have arbitrary M2:s.

One implication is that the BCFW recursion formula allowing to generate loop diagrams from those with lower number of loops must be trivial in TGD - this of course only if one accepts that BCFW formula makes sense in TGD. This requires that the entangled removal appearing as second term in the right hand side of BCFW formula and adding loop gives zero. One can develop and argument for why this must be the case in TGD framework. Also the second term corresponding to removal of BCFW bridge should give zero so that allowed diagrams cannot have BCFW bridges.

In TGD Universe allowed diagrams would represent closed objects in what one might call BCFW homology. The operation appearing at the right hand side of BCFW recursion formula is indeed boundary operation, whose square by definition gives zero.

For background see chapter Some questions related to the twistor lift of TGD or the article with the same title.

Issues related to the precise formulation of twistor lift of TGD

During last two weeks I have worked hardly to deduce the implications of some observations relating to the twistor lift of Kähler action. Some of these observations were very encouraging but some observations were a cold shower forcing a thorough criticism of the first view about the details of the twistor lift of TGD.

New formulation of Kähler action

The first observation was that the correct formulation of 6-D Kähler action in the framework of adelic physics implies that the classical physics of TGD does not depend on the overall scaling of Kähler action.

  1. Kähler form has dimension length squared. Kähler form projected to the space-time surface defines Mawell field, which should be however dimensionless. I had assumed that one can just divide Kähler form by CP2 radius squared to achieve this. The skeptic realizes immediately that this parameter is free coupling parameter albeit CP2 radius is good guess for it. The correct formulation of the action principle must keep Kähler form dimensional and divides Kähler action by a dimensional parameter with dimension 4: this is new coupling contant type parameter besides αK. The classical field equations do not depend at all on this scaling parameter. The exponent of action defining vacuum functional however depends on it.
  2. What is so nice that all couplings disappear from classical field equations in the new formulation, and number theoretical universality (NTU) is automatically achieved. In particular, the preferred extremals need not be minimal surface extremals of Kähler action to achieve this as in the original proposal for the twstor lift. It is enough that they are so asymptotically - near the boundaries of CDs, where they behave like free particles. In the interior they couple to Kähler force. This also nicely conforms with the physical idea that they are 4-D generalizations for orbits of particles in induced Kähler field.
  3. I also realized that the exchange of conserved quantities between Euclidian and Minkowskian space-time regions is not possible for the original version of twistor lift. This does not sound physical: quantal interactions should have classical correlates. The reason for the catastrophe is simple. Metric determinant appearing in action integral is identified as g41/2. In Minkowskian regions it is purely imaginary but real in Euclidian regions. Boundary conditions lead to decoupling of Minkowskian and Euclidian regions.

    This forced to return to an old nagging question whether one should use a) g41/2 (imaginary in Minkowskian regions) or b)|g41/2| in the action. For real αK the option a) is unavoidable and the need to have exponent of imaginary action in Minkowskian regions indeed motivated option a).

    For complex αK forced by other considerations the situation however changes - something that I had not noticed. Complex αK allows |g41/2|. The study of so called CP2 extremals assuming that 1/αK= s, s=1/2+iy zero of Riemann zeta shows that NTU is realized in the sense that the exponent of action exists in some extension of rationals, provided that the imaginary part of zero of zeta satisfies y= qπ, q rational, implying that the exponent of y is root of unity. This possibility has been considered already earlier. This is highly non-trivial hypothesis about zeros of zeta.

  4. Option b) allows transfer of conserved quantities between Minkowskian and Euclidian regions as required. Option a) also predicts separate conservation of Noether charges for Kähler action and volume term. This can make sense only asymptotically. Therefore only Option b) remains under serious consideration. In the new picture the interaction region in particle physics experiences corresponds to the region, where there is coupling between volume and Kähler terms: extrenal particles correpond to minimal surface extremals of Kähler action and all known extremals indeed are such.

Realizing NTU

The independence of the classical physics on the scale of the action in the new formulation inspires a detailed discussion of the number theoretic vision.

  1. Quantum Classical Correspondence (QCC) breaks the invariance with respect to the scalings via fermionic anti-commutation relations and NTU can fix the spectrum of values of the over-all scaling parameter of the action. Fermionic anticommutation relations introduce the constraint removing the projective invariance.
  2. One ends up to a condition guaranteeing NTU of the action exponentiale xp(S). One must have S= q1+iq2π , qi rational. This guarantees that exp(S) is in some extension of rationals and therefore number theoretically universal. S itself is however not number theoretically universal. The overall scaling parameter for action contrained by fermionic anticommutations must have a value allowing to satisfy the condition.

  3. The vision about scattering amplitude as a representation of computation however suggests the action exponential disappears from twistorial scattering amplitudes altogether as it does in quantum field theories. This would require that one defines scattering amplitude - actually zero energy state - by allowing functional integral only around single maximum of action. Whether this makes sense is not obvious but ZEO might allow it. I have not yet discussed seriously the constraints from unitary - or its generalization to ZEO, and these constraints might force sum over several maxima.

    This looks at first a catastrophe but the scattering amplitudes depend on the preferred extremal in implicit manner. For instance, the heff/h= n depends on extremal. Also quantum classical correspondence (QCC) realized as boundary conditions stating that the classical Noether charges are equal to the eigenvalues of fermionic charges in Cartan algebra bring in the dependence of scattering amplitudes on preferred extremal. Furhermore, the maxima of Kähler function could correspond to the points of WCW for which WCW coordinates are in the extension of rationals: if the exponent of action is such a coordinate this could be the case.

    One could see the situation in two manners. The standard view in which preferred extremals are maxima of Kähler function, whose exponentials however disappear from the scattering amplitudes, and the number theoretic view in which maxima correspond to WCW points in the intersection of real and various p-adic WCWs defining cognitive representation at the level of WCW similar to that provided by the discretization at the level of space-time surface. Maybe there is a maximization of cognitive information (classical correlate for NMP): say in the sense that the number of points in the intersection of real and p-adic space-time surfaces is maximal for the preferred extremals.

    This kind of connection would mean deep connection between cognition and sensory perception, p-adic physics and real physics, and geometric and number theoretic views about physics.

Trouble with cosmological constant

Also an unpleasant observation about cosmological constant forces to challenge the original view about twistor lift.

  1. The original vision for the p-adic evolution of cosmological constant assumed that αK(M4) and αK(CP2) are different for the twistor lift. This is definitely somewhat ad hoc choice but in principle possible. If one assumes that the Kähler form has also M4 part J(M4) this option becomes very artificial. In fact, the assumption that the twistor space M4× S2 associated with M4 allows Kähler structure, J(M4) must be non-vanishing and is completely fixed. It is now clear that J(M4) allows to understand both CP breaking and parity breaking (in particular chiral selection in living matter). The introduction of moduli space for CDs means also introduction of moduli space for the choices of J(M4), which is nothing but the twistor space T(M4)!
  2. One indeed finds in the more geometric formulation of 6-D Kähler action that single value of αK is the only natural choice. The nice outcome guaranteeing NTU is that the preferred extremals do not depend on the coupling parameters at all. In the original version one had to assume that extremals of Kähler action are also minimal surfaces to guarantee this.
  3. One however loses the original proposal for the p-adic length scale evolution of cosmological constant explaining why it is so small in cosmological scale. The solution to the problem would be that the entire 6-D action decomposing to 4-D Kähler action and volume term is identified in terms of cosmological constant. The cancellation of Kähler electric contribution and remaining contributions would explain why the cosmological constant is so small in cosmological scales and also allows to understand p-adic coupling constant evolution of cosmological constant.

    One important implication is that there are two kind string like objects. Those for which string tension is very large and which are analogous to the strings of super-string theories and those for which string tension is small due to the cancellation of Kähler action and volume term. These strings appear in all scales and they also mediate gravitational interaction. Also hadronic strings are this kind of strings as also elementary particles as string like objects. In this framework one additional reason for the superstring tragedy becomes manifest: they predict only the strings giving rise to a gigantic cosmological constant.

To sum up, it is fair to say that the twistor lift of TGD has now achieved rather stable form. There are also a lot of details to be polished but this requires only hard work and a lot of counter argumentation. What is so fascinating is that the formalism produces now rather precise predictions and new detailed fresh insights to the basic problems of standard model. The problems of cosmological constant and CP breaking represent only two examples in this respect. There is also an explicit proposal for twistor four-fermion amplitudes and one can understand how the QFT picture with central role played by the loops emerges although there are no loops at the fundamental level: when particles are approximated by point like objects, some tree diagrams are contracted to loop diagrams. Consider only exchange between two particle lines replaced with single line in pointlike this approximation.

See the articles About twistor lift of TGD and and See the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" or the article with the same title. See also the article About twistor lift of TGD.

What causes CP violation?

CP violation and matter antimatter asymmetry involving it represent white regions in the map provided by recent day physics. Standard model does not predict CP violation necessarily accompanied by the violation of time reflection symmetry T by CPT symmetry assumed to be exact. The violation of T must be distinguished from the emergence of time arrow implies by the randomness associated with state function reduction.

CP violation was originally observed for mesons via the mixing of neutral kaon and antikaon having quark content nsbar and nbars. The lifetimes of kaon and antikaon are different and they transform to each other. CP violation has been also observed for neutral mesons of type nbbar. Now it has been observed also for baryons Λb with quark composition u-d-b and its antiparticle (see this). Standard model gives the Feynman graphs describing the mixing in standard model in terms of CKM matrix (see this).

The CKM mixing matrix associated with weak interactions codes for the CP violation. More precisely, the small imaginary part for the determinant of CKM matrix defines the invariant coding for the CP violation. The standard model description of CP violation involves box diagrams in which the coupling to heavy quarks takes place. b quark gives rise to anomalously large CP violation effect also for mesons and this is not quite understood. Possible new heavy fermions in the loops could explain the anomaly.

Quite generally, the origin of CP violation has remained a mystery as also CKM mixing. In TGD framework CKM mixing has topological explanation in terms of genus of partonic 2-surface assignable to quark (sphere, torus or sphere with two handles). Topological mixings of U and D type quarks are different and the difference is not same for quarks and antiquarks. But this explains only CKM mixing, not CP violation.

Classical electric field - not necessary electromagnetic - prevailing inside hadrons could cause CP violation. So called instantons are basic prediction of gauge field theories and could cause strong CP violation since self-dual gauge field is involved with electric and magnetic fields having same strength and direction. That this strong CP violation is not observed is a problem of QCD. There are however proposals that instantons in vacuum could explain the CP violation of hadron physics (see this).

What says TGD? I have considered this here and in the earlier blog posting (see this).

  1. M4 and CP2 are unique in allowing twistor space with Kähler structure (in generalized sense for M4). If the twistor space T(M4)= M4× S2 having bundle projections to both M4 and to the conventional twistor space CP3, or rather its non-compact version) allows Kähler structure then also M4 allow the generalized Kähler structure and the analog symplectic structure.

    This boils down to the existence of self-dual and covariantly constant U(1) gauge field J(M4) for which electric and magnetic fields E and B are equal and constant and have the same direction. This field is not dynamical like gauge fields but would characterize the geometry of M4. J(M4) implies violation Lorentz invariance. TGD however leads to a moduli space for causal diamonds (CDs) effectively labelled by different choices of direction for these self-dual Maxwell fields. The common direction of E and B could correspond to that for spin quantization axis. J(M4) has nothing to do with instanton field. It should be noticed that also the quantum group inspired attempts to build quantum field theories for which space-time geometry is non-commutative introduce the analog of Kähler form in M4, and are indeed plagued by the breaking of Lorentz invariance. Here there is no moduli space saving the situation (see this) .

  2. The choice of quantization axis would therefore have a correlate at the level of "world of classical worlds" (WCW). Different choices would correspond to different sectors of WCW. The moduli space for the choices of preferred point of CP2 and color quantization axis corresponds to the twistor space T(CP2)= SU(3)/U(1)× U(1) of WCW. One could interpret also the twistor space T(M4)= M4× S2 as the space with given point representing the position of the tip of CD and the direction of the quantization axis of angular momentum. This choice requires a characterization of a unique rest system and the directions of quantization axis and time axes defines plane M2 playing a key role in TGD approach to twstorialization(see this) .
  3. The prediction would be CP violation for a given choice of J(M4). Usually this violation would be averaged out in the average over the moduli space for the choices of M2 but in some situation this would not happen. Why the CP violation does not average out when there is CKM mixing of quarks? Why the parity violation due to the preferred direction is not compensated by C violation meaning that the directions of E and B fields would be exactly opposite for quarks and antiquarks. Could the fact that quarks are not free but inside hadron induce CP violation? Could a more abstract formulation say that the wave function in the moduli space for J(M4) (wave function for the choices of spin quantization axis!) is not CP symmetric and this is reflected in the CKM matrix.
  4. An important delicacy is that J(M4) can be both self-dual and anti-self-dual depending on whether the magnetic and electric field have same or opposite directions. It will be found that reflection P and CP transform self-dual J(M4) to anti-self-dual one. If only self-dual J(M4) is allowed, one has both parity breaking and CP violations at the level of WCW.
Can one understand the emergence of CP violation in TGD framework?
  1. Zero energy state is pair of two positive and negative energy parts. Let us assume that positive energy part is fixed - one can call corresponding boundary of CD passive. This state corresponds to the outcome of state function reduction fixing the direction of quantization axes and producing eigenstates of measured observables, for instance spin. Single system at passive boundary is by definition unentangled with the other systems. It can consists of entangled subsystems hadrons are basic example of systems having entanglement in spin degrees of freedom of quarks: only the total spin of hadron is precisely defined.

    The states at the active boundary of CD evolve by repeated unitary steps by the action of the analog of S-matrix and are not anymore eigenstates of single particle observables but entangled. There is a sequence of trivial state function reductions at passive boundary inducing sequence of unitary time evolutions to the state at the active boundary of CD and shifting it. This gives rise to self as a generalized Zeno effect.

    Classically the time evolution of hadron corresponds to a superposition of space-time surfaces inside CD. The passive ends of the space-time surface or rather, the quantum superposition of them - is fixed. At the active end one has a superposition of 3-surfaces defining classical correlates for quantum states at the active end: this superposition changes in each unitary step during repeated measurements not affecting the passive end. Also time flows, which means that the distance between the tips of CD defining clock-time increases as the active boundary of CD shifts farther away.

  2. The classical field equations for space-time surface follow from an action, which at space-time level is sum of Kähler action and volume term. If Kähler form at space-time surface is induced (projected to space-time surface) from J=J(M4)+J(CP2), the classical time evolution is CP violating. CKM mixing is induced by different topological mixings for U and D type quarks (recall that 3 particle generations correspond to different genera for partonic 2-surfaces: sphere, torus, and sphere with two handles). J(M4)+J(CP2) defines the electroweak U(1) component of electric field so that J(M4) contributes to U(1) part of em field and is thus physically observable.
  3. Topological mixing of quarks corresponds to a superposition of time evolutions for the partonic 2-surfaces, which can also change the genus of partonic 2-surface defined as the number of handles attached to 2-sphere. For instance, sphere can transform to torus or torus to a sphere with two handles. This induces mixing of quantum states. For instance, one can say that a spherical partonic 2-surface containing quark would develop to quantum superposition of sphere, torus, and sphere with two handles. The sequence of state function reductions leaving the passive boundary of CD unaffected (generalized Zeno effect) by shifting the active boundary from its position after the first state function reduction to the passive boundary could but need not give rise to a further evolution of CKM matrix.
If the topological mixings are different for U and D type quarks, one obtains CKM mixing. How could the classical time evolution for quarks and for antiquarks as their CP transforms differ? To answer the question one must look how J(M4) transforms under C, P, T and CP.
  1. J(M4)=(J0z, Jxy= ε J0z), ε=+/- 1, characterizes hadronic space-time sheet (all space-time sheets in fact). Since J(M4) is tensor, P changes only the sign of J0z giving J(M4)→ (-J0z, Jxy). Since C changes the signs of charges and therefore the signs of fields created by them, one expects J(M4)→ -J(M4) under C. CP would give J(M4)→ (J0z, -Jxy) transforming selfdual J(M4) to anti-selfdual J(M4). If WCW has no anti-self-dual sector, CP is violated at the level of WCW.
  2. If CPT leaves J(M4) invariant, one must have J(M4) → (J0z, -Jxy) under T rather than J(M4)→ (-J0z, Jxy). The anti-unitary character of T could correspond for additional change of sign under T. Otherwise CPT should act as J(M4)→ -J(M4) and only (CPT)2 would correspond to unity.
  3. Same considerations apply to J(CP2) but the difference would be that induced J(M4) for space-time surfaces, which are small deformations of M4 covariantly constant in good approximation. Also for string world sheets corresponding to small cosmological constant J(M4)× J(M4)-2≈ 0 holds true in good approximation and induced J(M4) at string world sheet is in good approximation covariantly constant. If the string world sheet is just M2 characterizing J(M4) the condition is exact and was has Kähler electric field induced by J(M4) but no corresponding magnetic field. This would make the CP breaking effect large.
If CP is not violated, particles and their CP transforms correspond to different sectors of WCW with self dual and anti-self dual J(M4). If only self-dual sector of WCW is present then CP is violated. Also P is violated at the level of WCW and this parity breaking is different from that associated with weak interactions and could relate to the geometric parity breaking manifesting itself via chiral selection in living matter. Classical time evolutions induce different CKM mixings for quarks and antiquarks reflecting itself in the small imaginary part of the determinant of CKM matrix. CP breaking at the level of WCW could explain also matter-antimatter asymmetry. For instance, antimatter could be dark with different value of heff/h=n.

See the new chapter Some Questions Related to the Twistor Lift of TGD or the article with the same title. See also the article About twistor lift of TGD.

Criticizing the TGD based twistorial construction of scattering amplitudes

I have developed a rather detailed vision about twistorial construction of scattering amplitudes of fundamental fermions in TGD framework. These amplitudes serve as building bricks of scattering amplitudes of elementary particles. The construction allows to solve the basic problems of ordinary twistor approach.

Some of the key notions are 8-D light-likeness allowing to get rid of the problems produced by the mass of particles in 4-D sense, M8-M4× CP2 duality having nice interpretation in twistor space of $H$, quantum criticality demanding the vanishing of loops associated with functional integral and together with Kähler property implying that functional integral reduces to mere action exponential around given maximum of K\"ahler function, and number theoretical universality (NTU) suggesting that scattering diagrams could be seen as representations of computations reducible to minimal computation represented by tree diagram. One ends up with an explicit representations for the fundamentl 4-fermion scattering amplitude.

The vision is discussed in Questions related to twistor lift TGD. For the necessary background see About twistor lift of TGD. One can however criticize the proposed vision.

What about loops of QFT?

The idea about cancellation of loop corrections in functional integral and moves allowing to transform scattering diagrams represented as networks of partonic orbits meeting at partonic 2-surfaces defining topological vertices is nice.

Loops are however unavoidable in QFT description and their importance is undeniable. Photon-photon cattering is described by a loop diagram in which fermions appear in box like loop. Magnetic moment of muon) involves a triangle loop. A further interesting case is CP violation for mesons involving box-like loop diagrams.

Apart from divergence problems and problems with bound states, QFT works magically well and loops are important. How can one understand QFT loops if there are no fundamental loops? How could QFT emerge from TGD as an approximate description assuming lengths scale cutoff?

The key observation is that QFT basically replaces extended particles by point like particles. Maybe loop diagrams can be "unlooped" by introducing a better resolution revealing the non-point like character of the particles. What looks like loop for a particle line becomes in an improved resolution a tree diagram describing exchange of particle between sub-lines of line of the original diagram. In the optimal resolution one would have the scattering diagrams for fundamental fermions serving as building bricks of elementary particles.

To see the concrete meaning of the "unlooping" in TGD framework, it is necessary to recall the qualitative view about what elementary particles are in TGD framework.

  1. The fundamental fermions are assigned to the boundaries of string world sheets at the light-like orbits of partonic 2-surfaces: both fermions and bosons are built from them. The classical scatterings of fundamental fermions at the 2-D partonic 2-surface defining the vertices of topological scattering diagrams give rise to scattering amplitudes at the level of fundamental fermions and twistor lift with 8-D light-likeness suggests essentially unique expressions for the 4-fermion vertex.
  2. Elementary particle is modelled as a pair of wormhole contacts (Euclidian signature of metric) connecting two space-time sheets with throats at the two sheets connected by monopole flux tubes. All elementary particles are hadronlike systems but at recent energies the substructure is not visible. The fundamental fermions at the wormhole throats at given space-time sheet are connected by strings. There are altogether 4 wormhole throats per elementary particle in the simplest model.

    Elementary boson corresponds to fundamental fermion and antifermion at opposite wormhole throats with very small size (CP2 size). Elementary fermion has only single fundamental fermion at either throat. There is νLνbarR pair or its CP conjugate at the other end of the flux tube to neutralize the weak isospin. The flux tube has length of order Compton length (or elementary particle or of weak boson) gigantic as compared to the size of the wormhole contact.

  3. The vertices of topological diagram involve joining of the stringy diagrams associated with elementary particles at their ends defined by wormhole contacts. Wormhole contacts defining the ends of partonic orbits of say 3 interacting particles meet at the vertex - like lines in Feynman diagram - and fundamental fermion scattering redistributes fundamental fermions between the outgoing partonic orbits.
  4. The important point is that there are 2× 2=4 manners for the wormhole contacts at the ends of two elementary particle flux tubes to join together. This makes a possible a diagrams in which particle described by a string like object is emitted at either end and glued back at the other end of string like object. This is basically tree diagram at the level of wormhole contacts but if one looks it at a resolution reducing string to a point, it becomes a loop diagram.
  5. Improvement of the resolution reveals particles inside particles, which can scatter by tree diagrams. This allows to "unloop" the QFT loops. By increasing resolution new space-time sheets with smaller size emerge and one obtains "unlooped" loops in shorter scales. The space-time sheets are characterized by p-adic length scale and primes near powers of 2 are favored. p-Adic coupling constant evolution corresponds to the gradual "unlooping" by going to shorter and shorter p-adic length scales revealing smaller and smaller space-time sheets.
The loop diagrams of QFTs could thus be seen as a direct evidence of the fractal many-sheeted space-time and quantum criticality and number theoretical universality (NTU) of TGD Universe. Quantum critical dynamics makes the dynamics universal and this explains the unreasonable success of QFT models as far as length scale dependence of couplings constants is considered. The weak point of QFT models is that they are not able to describe bound states: this indeed requires that the extended structure of particles as 3-surfaces is taken into account.

Can action exponentials really disappear?

The disappearance of the action exponentials from the scattering amplitudes can be criticized. In standard approach the action exponentials associated with extremals determine which configurations are important. In the recent case they should be the 3-surfaces for which Kähler action is maximum and has stationary phase. But what would select them if the action exponentials disappear in scattering amplitudes?

The first thing to notice is that one has functional integral around a maximum of vacuum functional and the disappearance of loops is assumed to follow from quantum criticality. This would produce exponential since Gaussian and metric determinants cancel, and exponentials would cancel for the proposal inspired by the interpretation of diagrams as computations. One could in fact define the functional integral in this manner so that a discretization making possible NTU would result.

Fermionic scattering amplitudes should depend on space-time surface somehow to reveal that space-time dynamics matters. In fact, QCC stating that classical Noether charges for bosonic action are equal to the eigenvalues of quantal charges for fermionic action in Cartan algebra would bring in the dependence of scattering amplitudes on space-time surface via the values of Noether charges. For four-momentum this dependence is obvious. The identification of heff/h=n as order of Galois group would mean that the basic unit for discrete charges depends on the extension characterizing the space-time surface.

Also the cognitive representations defined by the set of points for which preferred imbedding space coordinates are in this extension. Could the cognitive representations carry maximum amount of information for maxima? For instance, the number of the points in extension be maximal. Could the maximum configurations correspond to just those points of WCW, which have preferred coordinates in the extension of rationals defining the adele? These 3-surfaces would be in the intersection of reality and p-adicities and would define cognitive representation.

These ideas suggest that the usual quantitative criterion for the importance of configurations could be equivalent with a purely number theoretical criterion. p-Adic physics describing cognition and real physics describing matter would lead to the same result. Maximization for action would correspond to maximization for information.

Irrespective of these arguments, the intuitive feeling is that the exponent of the bosonic action must have physical meaning. It is number theoretically universal if action satisfies S= q1+iq2π. This condition could actually be used to fix the dependence of the coupling parameters on the extension of rationals (see this). By allowing sum over several maxima of vacuum functional these exponentials become important. Therefore the above ideas are interesting speculations but should be taken with a big grain of salt.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix".

Two observations about twistor lift of Kähler action

During last couple years a kind of palace revolution has taken place in the formulation and interpretation of TGD. The notion of twistor lift and 8-D generalization of twistorialization have dramatically simplified and also modified the view about what classical TGD and quantum TGD are.

The notion of adelic physics suggests the interpretation of scattering diagrams as representations of algebraic computations with diagrams producing the same output from given input are equivalent. The simplest possible manner to perform the computation corresponds to a tree diagram. As will be found, it is now possible to even propose explicit twistorial formulas for scattering formulas since the horrible problems related to the integration over WCW might be circumvented altogether.

From the interpretation of p-adic physics as physics of cognition, heff/h=n could be interpreted as the order of Galois group. Discrete coupling constant evolution would correspond to phase transitions changing the extension of rationals and its Galois group. TGD inspired theory of consciousness is an essential part of TGD and the crucial Negentropy Maximization Principle in statistical sense follows from number theoretic evolution as increase of the order of Galois group for extension of rationals defining adeles.

During the re-processing of the details related to twistor lift, it became clear that the earlier variant for the twistor lift can be criticized and allows an alternative. This option led to a simpler view about twistor lift, to the conclusion that minimal surface extremals of Kähler action represent only asymptotic situation near boundaries of CD (external particles in scattering), and also to a re-interpretation for the p-adic evolution of the cosmological constant: cosmological term would correspond to the entire 4-D action and the cancellation of Kähler action and cosmological term would lead to the small value of the effective cosmological constant. The pleasant observation was that the correct formulation of 6-D Kähler action in the framework of adelic physics implies that the classical physics of TGD does not depend on the overall scaling of Kähler action but that quantum classical correspondence implies this dependence. It is however too early to select between the two options.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix".

Questions related to the quantum aspects of twistorialization

The progress in the understanding of the classical aspects of twistor lift of TGD makes possible to consider in detail the quantum aspects of twistorialization of TGD and for the first time an explicit proposal for the part of scattering diagrams assignable to fundamental fermions emerges.

  1. There are several notions of twistor. Twistor space for M4 is T(M4) =M4× S2 (see this) having projections to both M4 and to the standard twistor space T1(M4) often identified as CP3. T(M4)=M4× S2 is necessary for the twistor lift of space-time dynamics. CP2 gives the factor T(CP2)= SU(3)/U(1)× U(1) to the classical twistor space T(H). The quantal twistor space T(M8)= T1(M4)× T(CP2) assignable to momenta. The possible way out is M8-H duality relating the momentum space M8 (isomorphic to the tangent space H) and H by mapping space-time associative and co-associative surfaces in M8 to the surfaces which correspond to the base spaces of in H: they construction would reduce to holomorphy in complete analogy with the original idea of Penrose in the case of massless fields.
  2. The standard twistor approach has problems. Twistor Fourier transform reduces to ordinary Fourier transform only in signature (2,2) for Minkowski space: in this case twistor space is real RP3 but can be complexified to CP3. Otherwise the transform requires residue integral to define the transform (in fact, p-adically multiple residue calculus could provide a nice manner to define integrals and could make sense even at space-time level making possible to define action).

    Also the positive Grassmannian requires (2,2) signature. In M8-H relies on the existence of the decomposition M2⊂ M2= M2× E2⊂ M8. M2 could even depend on position but M2(x) should define an integrable distribution. There always exists a preferred M2, call it M20, where 8-momentum reduces to light-like M2 momentum. Hence one can apply 2-D variant of twistor approach. Now the signature is (1,1) and spinor basis can be chosen to be real! Twistor space is RP3 allowing complexification to CP3 if light-like complex momenta are allowed as classical TGD suggests!

  3. A further problem of the standard twistor approach is that in M4 twistor approach does not work for massive particles. In TGD all particles are massless in 8-D sense. In M8 M4-mass squared corresponds to transversal momentum squared coming from E4⊂ M4× E4 (from CP2 in H). In particular, Dirac action cannot contain anyo mass term since it would break chiral invariance.

    Furthermore, the ordinary twistor amplitudes are holomorphic functions of the helicity spinors λi and have no dependence on &lambda tile;i: no information about particle masses! Only the momentum conserving delta function gives the dependence on masses. These amplitudes would define as such the M4 parts of twistor amplitudes for particles massive in TGD sense. The simplest 4-fermion amplitude is unique.

Twistor approach gives excellent hopes about the construction of the scattering amplitudes in ZEO. The construction would split into two pieces corresponding to the orbital degrees of freedom in "world of classical worlds" (WCW) and to spin degrees of freedom in WCW: that is spinors, which correspond to second quantized induced spinor fields at space-time surface (actually string world sheets- either at fundamental level or for effective action implied by strong form of holography (SH)).
  1. At WCW level there is a perturbative functional integral over small deformations of the 3-surface to which space-time surface is associated. The strongest assumption is that this 3-surface corresponds to maximum for the real part of action and to a stationary phase for its imaginary part: minimal surface extremal of Kähler action would be in question. A more general but number theoretically problematic option is that an extremal for the sum of Kähler action and volume term is in question.

    By Kähler geometry of WCW the functional integral reduces to a sum over contributions from preferred extremals with the fermionic scattering amplitude multiplied by the ration Xi/X, where X=∑i Xi is the sum of the action exponentials for the maxima. The ratios of exponents are however number theoretically problematic.

    Number theoretical universality is satisfied if one assigns to each maximum independent zero energy states: with this assumption ∑ Xi reduces to single Xi and the dependence on action exponentials becomes trivial! ZEO allow this. The dependence on coupling parameters of the action essential for the discretized coupling constant evolution is only via boundary conditions at the ends of the space-time surface at the boundaries of CD.

    Quantum criticality of TGD demands that the sum over loops associated with the functional integral over WCW vanishes and strong form of holography (SH) suggests that the integral over 4-surfaces reduces to that over string world sheets and partonic 2-surfaces corresponding to preferred extremals for which the WCW coordinates parametrizing them belong to the extension of rationals defining the adele. Also the intersections of the real and various p-adic space-time surfaces belong to this extension.

  2. Second piece corresponds to the construction of twistor amplitude from fundamental 4-fermion amplitudes. The diagrams consists of networks of light-like orbits of partonic two surfaces, whose union with the 3-surfaces at the ends of CD is connected and defines a boundary condition for preferred extremals and at the same time the topological scattering diagram.

    Fermionic lines correspond to boundaries of string world sheets. Fermion scattering at partonic 2-surfaces at which 3 partonic orbits meet are analogs of 3-vertices in the sense of Feynman and fermions scatter classically. There is no local 4-vertex. This scattering is assumed to be described by simplest 4-fermion twistor diagram. These can be fused to form more complex diagrams. Fermionic lines runs along the partonic orbits defining the topological diagram.

  3. Number theoretic universality suggests that scattering amplitudes have interpretation as representations for computations. All space-time surfaces giving rise to the same computation wold be equivalent and tree diagrams corresponds to the simplest computation. If the action exponentials do not appear in the amplitudes as weights this could make sense but would require huge symmetry based on two moves. One could glide the 4-vertex at the end of internal fermion line along the fermion line so that one would eventually get the analog of self energy loop, which should allow snipping away. An argument is developed stating that this symmetry is possible if the preferred M20 for which 8-D momentum reduces to light-like M2-momentum having unique direction is same along entire fermion line, which can wander along the topological graph.

    The vanishing of topological loops would correspond to the closedness of the diagrams in what might be called BCFW homology. Boundary operation involves removal of BCFW bridge and entangled removal of fermion pair. The latter operation forces loops. There would be no BCFW bridges and entangled removal should give zero. Indeed, applied to the proposed four fermion vertex entangled removal forces it to correspond to forward scattering for which the proposed twistor amplitude vanishes.

To sum up, the twistorial approach leads to a proposal for an explicit construction of scattering amplitudes for the fundamental fermions. Bosons and fermions as elementary particles are bound states of fundamental fermions assignable to pairs of wormhole contacts carrying fundamental fermions at the throats. Clearly, this description is analogous to a quark level description of hadron. Yangian symmetry with multilocal generators is expected to crucial for the construction of the many-fermion states giving rise to elementary particles. The problems of the standard twistor approach find a nice solution in terms of M8-H duality, 8-D masslessness, and holomorphy of twistor amplitudes in λi and their indepence on &lambda tilde;i.

See the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix".

A new view about color, color confinement, and twistors

To my humble opinion twistor approach to the scattering amplitudes is plagued by some mathematical problems. Whether this is only my personal problem is not clear.

  1. As Witten shows, the twistor transform is problematic in signature (1,3) for Minkowski space since the the bi-spinor μ playing the role of momentum is complex. Instead of defining the twistor transform as ordinary Fourier integral, one must define it as a residue integral. In signature (2,2) for space-time the problem disappears since the spinors μ can be taken to be real.
  2. The twistor Grassmannian approach works also nicely for (2,2) signature, and one ends up with the notion of positive Grassmannians, which are real Grassmannian manifolds. Could it be that something is wrong with the ordinary view about twistorialization rather than only my understanding of it?
  3. For M4 the twistor space should be non-compact SU(2,2)/SU(2,1)× U(1) rather than CP3= SU(4)/SU(3)× U(1), which is taken to be. I do not know whether this is only about short-hand notation or a signal about a deeper problem.
  4. Twistorilizations does not force SUSY but strongly suggests it. The super-space formalism allows to treat all helicities at the same time and this is very elegant. This however forces Majorana spinors in M4 and breaks fermion number conservation in D=4. LHC does not support N=1 SUSY. Could the interpretation of SUSY be somehow wrong? TGD seems to allow broken SUSY but with separate conservation of baryon and lepton numbers.
In number theoretic vision something rather unexpected emerges and I will propose that this unexpected might allow to solve the above problems and even more, to understand color and even color confinement number theoretically. First of all, a new view about color degrees of freedom emerges at the level of M8.
  1. One can always find a decomposition M8=M20× E6 so that the complex light-like quaternionic 8-momentum restricts to M20. The preferred octonionic imaginary unit represent the direction of imaginary part of quaternionic 8-momentum. The action of G2 to this momentum is trivial. Number theoretic color disappears with this choice. For instance, this could take place for hadron but not for partons which have transversal momenta.
  2. One can consider also the situation in which one has localized the 8-momenta only to M4 =M20× E2. The distribution for the choices of E2 ⊂ M20× E2=M4 is a wave function in CP2. Octonionic SU(3) partial waves in the space CP2 for the choices for M20× E2 would correspond ot color partial waves in H. The same interpretation is also behind M8-H correspondence.
  3. The transversal quaternionic light-like momenta in E2⊂ M20× E2 give rise to a wave function in transversal momenta. Intriguingly, the partons in the quark model of hadrons have only precisely defined longitudinal momenta and only the size scale of transversal momenta can be specified.

    The introduction of twistor sphere of T(CP2) allows to describe electroweak charges and brings in CP2 helicity identifiable as em charge giving to the mass squared a contribution proportional to Qem2 so that one could understand electromagnetic mass splitting geometrically.

    The physically motivated assumption is that string world sheets at which the data determining the modes of induced spinor fields carry vanishing W fields and also vanishing generalized Kähler form J(M4) +J(CP2). Em charge is the only remaining electroweak degree of freedom. The identification as the helicity assignable to T(CP2) twistor sphere is natural.

  4. In general case the M2 component of momentum would be massive and mass would be equal to the mass assignable to the E6 degrees of freedom. One can however always find M20× E6 decomposition in which M2 momentum is light-like. The naive expectation is that the twistorialization in terms of M2 works only if M2 momentum is light-like, possibly in complex sense. This however allows only forward scattering: this is true for complex M2 momenta and even in M4 case.

    The twistorial 4-fermion scattering amplitude is however holomorphic in the helicity spinors λi and has no dependence on λtilde;i. Therefore carries no information about M2 mass! Could M2 momenta be allowed to be massive? If so, twistorialization might make sense for massive fermions!

M20 momentum deserves a separate discussion.
  1. A sharp localization of 8-momentum to M20 means vanishing E2 momentum so that the action of U(2) would becomes trivial: electroweak degree of freedom would simply disappear, which is not the same thing as having vanishing em charge (wave function in T(CP2) twistorial sphere S2 would be constant). Neither M20 localization nor localization to single M4 (localization in CP2) looks plausible physically - consider only the size scale of CP2. For the generic CP2 spinors this is impossible but covariantly constant right-handed neutrino spinor mode has no electro-weak quantum numbers: this would most naturally mean constant wave function in CP2 twistorial sphere.

    For the preferred extremals of twistor lift of TGD either M4 or CP2 twistor sphere can effectively collapse to a point. This would mean disappearence of the degrees of freedom associated with M4 helicity or electroweak quantum numbers.

  2. The localization to M4⊃ M20 is possible for the tangent space of quaternionic space-time surface in M8. This could correlate with the fact that neither leptonic nor quark-like induced spinors carry color as a spin like quantum number. Color would emerge only at the level of H and M8 as color partial waves in WCW and would require de-localization in the CP2 cm coordinate for partonic 2-surface. Note that also the integrable local decompositions M4= M2(x)× E2(x) suggested by the general solution ansätze for field equations are possible.
  3. Could it be possible to perform a measurement localization the state precisely in fixed M20 always so that the complex momentum is light-like but color degrees of freedom disappear? This does not mean that the state corresponds to color singlet wave function! Can one say that the measurement eliminating color degrees of freedom corresponds to color confinement. Note that the subsystems of the system need not be color singlets since their momenta need not be complex massless momenta in M20. Classically this makes sense in many-sheeted space-time. Colored states would be always partons in color singlet state.
  4. At the level of H also leptons carry color partial waves neutralized by Kac-Moody generators, and I have proposed that the pion like bound states of color octet excitations of leptons explain so called lepto-hadrons. Only right-handed covariantly constant neutrino is an exception as the only color singlet fermionic state carrying vanishing 4-momentum and living in all possible M20:s, and might have a special role as a generator of supersymmetry acting on states in all quaternionic subs-spaces M4.
  5. Actually, already p-adic mass calculations performed for more than two decades ago forced to seriously consider the possibility that particle momenta correspond to their projections o M20⊂ M4. This choice does not break Poincare invariance if one introduces moduli space for the choices of M20⊂ M4 and the selection of M20 could define quantization axis of energy and spin. If the tips of CD are fixed, they define a preferred time direction assignable to preferred octonionic real unit and the moduli space is just S2. The analog of twistor space at space-time level could be understood as T(M4)=M4× S2 and this one must assume since otherwise the induction of metric does not make sense.
What happens to the twistorialization at the level of M8 if one accepts that only M20 momentum is sharply defined?
  1. What happens to the conformal group SO(4,2) and its covering SU(2,2) when M4 is replaced with M20⊂ M8? Translations and special conformational transformation span both 2 dimensions, boosts and scalings define 1-D groups SO(1,1) and R respectively. Clearly, the group is 6-D group SO(2,2) as one might have guessed. Is this the conformal group acting at the level of M8 so that conformal symmetry would be broken? One can of course ask whether the 2-D conformal symmetry extends to conformal symmetries characterized by hyper-complex Virasoro algebra.
  2. Sigma matrices are by 2-dimensionality real (σ0 and σ3 - essentially representations of real and imaginary octonionic units) so that spinors can be chosen to be real. Reality is also crucial in signature (2,2), where standard twistor approach works nicely and leads to 3-D real twistor space.

    Now the twistor space is replaced with the real variant of SU(2,2)/SU(2,1)× U(1) equal to SO(2,2)/SO(2,1), which is 3-D projective space RP3 - the real variant of twistor space CP3, which leads to the notion of positive Grassmannian: whether the complex Grassmannian really allows the analog of positivity is not clear to me. For complex momenta predicted by TGD one can consider the complexification of this space to CP3 rather than SU(2,2)/SU(2,1)× U(1). For some reason the possible problems associated with the signature of SU(2,2)/SU(2,1)× U(1) are not discussed in literature and people talk always about CP3. Is there a real problem or is this indeed something totally trivial?

  3. SUSY is strongly suggested by the twistorial approach. The problem is that this requires Majorana spinors leading to a loss of fermion number conservation. If one has D=2 only effectively, the situation changes. Since spinors in M2 can be chosen to be real, one can have SUSY in this sense without loss of fermion number conservation! As proposed earlier, covariantly constant right-handed neutrino modes could generate the SUSY but it could be also possible to have SUSY generated by all fermionic helicity states. This SUSY would be however broken.
  4. The selection of M20 could correspond at space-time level to a localization of spinor modes to string world sheets. Could the condition that the modes of induced spinors at string world sheets are expressible using real spinor basis imply the localization? Whether this localization takes place at fundamental level or only for effective action being due to SH, is a question to be settled. The latter options looks more plausible.
To sum up, these observation suggest a profound re-evalution of the beliefs related to color degrees of freedom, to color confinement, and to what twistors really are.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD.

How does the twistorialization at imbedding space level emerge?

One objection against twistorialization at imbedding space level is that M4-twistorialization requires 4-D conformal invariance and massless fields. In TGD one has towers of particle with massless particles as the lightest states. The intuitive expectation is that the resolution of the problem is that particles are massless in 8-D sense as also the modes of the imbedding space spinor fields are. M8-H duality indeed provides a solution of the problem. Massless quaternionic momentum in M8 can be for a suitable choice of decomposition M8= M4× E4 be reduce to massless M4 momentum and one can describe the information about 8-momentum using M4 twistor and CP2 twistor.

Second objection is that twistor Grassmann approach uses as twistor space the space T1(M4) =SU(2,2)/SU(2,1)× U(1) whereas the twistor lift of classical TGD uses T(M4)=M4× S2. The formulation of the twistor amplitudes in terms of strong form of holography (SH) using the data assignable to the 2-D surfaces - string world sheets and partonic 2-surfaces perhaps - identified as surfaces in T(M4)× T(CP2) requires the mapping of these twistor spaces to each other - the incidence relations of Penrose indeed realize this map.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD.

Twistor lift and the reduction of field equations and SH to holomorphy

It has become clear that twistorialization has very nice physical consequences. But what is the deep mathematical reason for twistorialization? Understanding this might allow to gain new insights about construction of scattering amplitudes with space-time surface serving as analogs of twistor diatrams.

Penrose's original motivation for twistorilization was to reduce field equations for massless fields to holomorphy conditions for their lifts to the twistor bundle. Very roughly, one can say that the value of massless field in space-time is determined by the values of the twistor lift of the field over the twistor sphere and helicity of the massless modes reduces to cohomology and the values of conformal weights of the field mode so that the description applies to all spins.

I want to find the general solution of field equations associated with the Kähler action lifted to 6-D Kähler action. Also one would like to understand strong form of holography (SH). In TGD fields in space-time are are replaced with the imbedding of space-time as 4-surface to H. Twistor lift imbeds the twistor space of the space-time surface as 6-surface into the product of twistor spaces of M4 and CP2. Following Penrose, these imbeddings should be holomorphic in some sense.

Twistor lift T(H) means that M4 and CP2 are replaced with their 6-D twistor spaces.

  1. If S2 for M4 has 2 time-like dimensions one has 3+3 dimensions, and one can speak about hyper-complex variants of holomorphic functions with time-like and space-like coordinate paired for all three hypercomplex coordinates. For the Minkowskian regions of the space-time surface X4 the situation is the same.
  2. For T(CP2) Euclidian signature of twistor sphere guarantees this and one has 3 complex coordinates corresponding to those of S2 and CP2. One can also now also pair two real coordinates of S2 with two coordinates of CP2 to get two complex coordinates. For the Euclidian regions of the space-time surface the situation is the same.
Consider now what the general solution could look like. Let us continue to use the shorthand notations S21= S2(X4); S22= S2(CP2);S23= S2(M4).
  1. Consider first solution of type (1,0) so that coordinates of S22 are constant. One has holomorphy in hypercomplex sense (light-like coordinate t-z and t+z correspond to hypercomplex coordinates).
    1. The general map T(X4) to T(M4) should be holomorphic in hyper-complex sense. S21 is in turn identified with S23 by isometry realized in real coordinates. This could be also seen as holomorphy but with different imaginary unit. One has analytical continuation of the map S21→ S23 to a holomorphic map. Holomorphy might allows to achieve this rather uniquely. The continued coordinates of S21 correspond to the coordinates assignable with the integrable surface defined by E2(x) for local M2(x)× E2(x) decomposition of the local tangent space of X4. Similar condition holds true for T(M4). This leaves only M2(x) as dynamical degrees of freedom. Therefore one has only one holomorphic function defined by 1-D data at the surface determined by the integrable distribution of M2(x) remains. The 1-D data could correspond to the boundary of the string world sheet.
    2. The general map T(X4) to T(CP2) cannot satisfy holomorphy in hyper-complex sense. One can however provide the integrable distribution of E2(x) with complex structure and map it holomorphically to CP2. The map is defined by 1-D data.
    3. Altogether, 2-D data determine the map determining space-time surface. These two 1-D data correspond to 2-D data given at string world sheet: one would have SH.
  2. What about solutions of type (0,1) making sense in Euclidian region of space-time? One has ordinary holomorphy in CP2 sector.
    1. The simplest picture is a direct translation of that for Minkowskian regions. The map S21→ S22 is an isometry regarded as an identification of real coordinates but could be also regarded as holomorphy with different imaginary unit. The real coordinates can be analytically continued to complex coordinates on both sides, and their imaginary parts define coordinates for a distribution of transversal Euclidian spaces E22(x) on X4 side and E2(x) on M4 side. This leaves 1-D data.
    2. What about the map to T(M4)? It is possible to map the integrable distribution E22(x) to the corresponding distribution for T(M4) holomorphically in the ordinary sense of the word. One has 1-D data. Altogether one has 2-D data and SH and partonic 2-surfaces could carry these data. One has SH again.
  3. The above construction works also for the solutions of type (1,1), which might make sense in Euclidian regions of space-time. It is however essential that the spheres S22 and S23 have real coordinates.
SH thus would thus emerge automatically from the twistor lift and holomorphy in the proposed sense.
  1. Two possible complex units appear in the process. This suggests a connection with quaternion analytic functions suggested as an alternative manner to solve the field equations. Space-time surface as associative (quaterionic) or co-associate (co-quaternionic) surface is a further solution ansatz.

    Also the integrable decompositions M2(x)× E2(x) resp. E21(x)× E22(x) for Minkowskian resp. Euclidian space-time regions are highly suggestive and would correspond to a foliation by string wold sheets and partonic 2-surfaces. This expectation conforms with the number theoretically motivated conjectures.

  2. The foliation gives good hopes that the action indeed reduces to an effective action consisting of an area term plus topological magnetic flux term for a suitably chosen stringy 2-surfaces and partonic 2-surfaces. One should understand whether one must choose the string world sheets to be Lagrangian surfaces for the Kähler form including also M4 term. Minimal surface condition could select the Lagrangian string world sheet, which should also carry vanishing classical W fields in order that spinors modes can be eigenstates of em charge.

    The points representing intersections of string world sheets with partonic 2-surfaces defining punctures would represent positions of fermions at partonic 2-surfaces at the boundaries of CD and these positions should be able to vary. Should one allow also non-Lagrangian string world sheets or does the space-time surface depend on the choice of the punctures carrying fermion number (quantum classical correspondence)?

  3. The alternative option is that any choice produces of the preferred 2-surfaces produces the same scattering amplitudes. Does this mean that the string world sheet area is a constant for the foliation - perhaps too strong a condition - or could the topological flux term compensate for the change of the area?

    The selection of string world sheets and partonic 2-surfaces could indeed be also only a gauge choice. I have considered this option earlier and proposed that it reduces to a symmetry identifiable as U(1) gauge symmetry for Kähler function of WCW allowing addition to it of a real part of complex function of WCW complex coordinates to Kähler action. The additional term in the Kähler action would compensate for the change if string world sheet action in SH. For complex Kähler action it could mean the addition of the entire complex function.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD.

More details about the induction of twistor structure

The notion of twistor lift of TGD (see this and this) has turned out to have powerful implications concerning the understanding of the relationship of TGD to general relativity. The meaning of the twistor lift really has remained somewhat obscure. There are several questions to be answered. What does one mean with twistor space? What does the induction of twistor structure of H=M4× CP2 to that of space-time surface realized as its twistor space mean?

In TGD one replaces imbedding space H=M4× CP2 with the product T= T(M4)× T(CP2) of their 6-D twistor spaces, and calls T(H) the twistor space of H. For CP2 the twistor space is the flag manifold T(CP2)=SU(3)/U(1)× U(1) consisting of all possible choices of quantization axis of color isospin and hypercharge.

  1. The basic idea is to generalize Penrose's twistor program by lifting the dynamics of space-time surfaces as preferred extremals of Kähler action to those of 6-D Kähler action in twistor space T(H). The conjecture is that field equations reduce to the condition that the twistor structure of space-time surface as 4-manifold is the twistor structure induced from T(H).

    Induction requires that dimensional reduction occurs effectively eliminating twistor fiber S2 (X4) from the dynamics. Space-time surfaces would be preferred extremals of 4-D Kähler action plus volume term having interpretation in terms of cosmological constant. Twistor lift would be more than an mere alternative formulation of TGD.

  2. The reduction would take place as follows. The 6-D twistor space T(X4) has S2 as fiber and can be expressed locally as a Cartesian product of 4-D region of space-time and of S2. The signature of the induced metric of S2 should be space-like or time-like depending on whether the space-time region is Euclidian or Minkowskian. This suggests that the twistor sphere of M4 is time-like as also standard picture suggests.
  3. Twistor structure of space-time surface is induced to the allowed 6-D surfaces of T(H), which as twistor spaces T(X4) must have fiber space structure with S2 as fiber and space-time surface X4 as base. The Kähler form of T(H) expressible as a direct sum

    J(T(H)= J(T(M4))⊕ J(T(CP2)

    induces as its projection the analog of Kähler form in the region of T(X4) considered.

    There are physical motivations (CP breaking, matter antimatter symmetry, the well-definedness of em charge) to consider the possibility that also M4 has a non-trivial symplectic/Kähler form of M4 obtained as a generalization of ordinary symplectic/Kähler form (see this). This requires the decomposition M4=M2× E2 such that M2 has hypercomplex structure and E2 complex structures.

    This decomposition might be even local with the tangent spaces M2(x) and E2(x) integrating to locally orthogonal 2-surfaces. These decomposition would define what I have called Hamilton-Jacobi structure (see this). This would give rise to a moduli space of M4 Kähler forms allowing besides covariantly constant self-dual Kähler forms with decomposition (m0,m3) and (m1, m2) also more general self-dual closed Kähler forms assignable to integrable local decompositions. One example is spherically symmetric stationary self-dual Kähler form corresponding to the decomposition (m0,rM) and (θ,φ) suggested by the need to get spherically symmetric minimal surface solutions of field equations. Also the decomposition of Robertson-Walker coordinates to (a,r) and (θ,π) assignable to light-cone M4+ can be considered.

    The moduli space giving rise to the decomposition of WCW to sectors would be finite-dimensional if the integrable 2-surfaces defined by the decompositions correspond to orbits of subgroups of the isometry group of M4 or CD. This would allow planes of M4, and radial half-planes and spheres of M4 in spherical Minkowski coordinates and of M4+ in Robertson-Walker coordinates. These decomposition could relate to the choices of measured quantum numbers inducing symmetry breaking to the subgroups in question. These choices would chose a sector of WCW (see this) and would define quantum counterpart for a choice of quantization axes as distinct from ordinary state function reduction with chosen quantization axes.

  4. The induced Kähler form of S2 fiber of T(X4) is assumed to reduce to the sum of the induced Kähler forms from S2 fibers of T(M4) and T(CP2). This requires that the projections of the Kähler forms of M4 and CP2 to S2(X4) are trivial. Also the induced metric is assumed to be direct sum and similar conditions holds true.These conditions are analogous to those occurring in dimensional reduction.

    Denote the radii of the spheres associated with M4 and CP2 as RP=klP and R and the ratio RP/R by ε. Both the Kähler form and metric are proportional to Rp2 resp. R2 and satisfy the defining condition JkrgrsJsl= -gkl. This condition is assumed to be true also for the induced Kähler form of J(S2(X4).

This is the general description. How many solutions to these conditions are obtained? It seems that there are essentiablly 3 solutions. The projection of the twistor space of space-time surface to the twistor sphere of either M4 or CP2 is trivial and the solution in which it is trivial to both and twistor spheres correspond to each other by a one-to-one isometry (see this).

For details see the new chapter Some Questions Related to the Twistor Lift of TGD or the article Some questions related to the twistor lift of TGD.

Symplectic structure for M4, CP breaking, matter-antimatter asymmetry, and electroweak symmetry breaking

The preparation of an article about number theoretic aspects of TGD forced to go through various related ideas and led to a considerable integration of the ideas. In this note idea about the symplectic structure of M4 is discussed although it is not directly related to number theoretic aspects of TGD.

  1. Twistor lift of TGD suggests strongly a symmetry between M4 and CP2. In particular, M4 should have the analog of symplectic structure.
  2. It has been already noticed that this structure could allow to understand both CP breaking and matter-antimatter asymmetry from first principles. A further study showed that it can also allow to understand electroweak symmetry breaking.
Consider now the delicacies of this picture.
  1. Should assign also to M4 the analog of symplectic structure giving an additional contribution to the induced Kähler form? The symmetry between M4 and CP2 suggests this, and this term could be highly relevant for the understanding of the observed CP breaking and matter antimatter asymmetry. Poincare invariance is not lost since the needed moduli space for M4 Kähler forms would be the moduli space of CDs forced by ZEO in any case, and M4 Kähler form would serve as the correlate for fixing rest system and spin quantization axis in quantum measurement.
  2. Also induced spinor fields are present. The well-definedness of electro-magnetic charge for the spinor modes forces in the generic case the localization of the modes of induced spinor fields at string world sheets (and possibly to partonic 2-surfaces) at which the induced charged weak gauge fields and possibly also neutral Z0 gauge field vanish. The analogy with branes and super-symmetry force to consider two options.

    Option I: The fundamental action principle for space-time surfaces contains besides 4-D action also 2-D action assignable to string world sheets, whose topological part (magnetic flux) gives rise to a coupling term to Kähler gauge potentials assignable to the 1-D boundaries of string world sheets containing also geodesic length part. Super-symplectic symmetry demands that modified Dirac action has 1-, 2-, and 4-D parts: spinor modes would exist at both string boundaries, string world sheets, and space-time interior. A possible interpretation for the interior modes would be as generators of space-time super-symmetries.

    This option is not quite in the spirit of SH and string tension appears as an additional parameter. Also the conservation of em charge forces 2-D string world sheets carrying vanishing induced W fields and this is in conflict with the existence of 4-D spinor modes unless they satisfy the same condition. This looks strange.

    Option II: Stringy action and its fermionic counterpart are effective actions only and justified by SH. In this case there are no problems of interpretation. SH requires only that the induced spinor fields at string world sheets determine them in the interior much like the values of analytic function at curve determine it in an open set of complex plane. At the level of quantum theory the scattering amplitudes should be determined by the data at string world sheets. If induced W fields at string world sheets are vanishing, the mixing of different charge states in the interior of X4 would not make itself visible at the level of scattering amplitudes! In this case 4-D spinor modes do not define space-time super-symmetries.

    This option seems to be the only logical one. It is also simplest and means that quantum TGD would reduce to string model apart from number theoretical discretization of space-time surface bringing in dark matter as heff/h=n phases with n identifiable as factor of the order of the Galois group of extension of rationals. This would also lead to adelic physics, predict preferred extensions and identify corresponding ramified primes as preferred p-adic primes.

  3. Why the string world sheets coding for effective action should carry vanishing weak gauge fields? If M4 has the analog of Kähler structure, one can speak about Lagrangian sub-manifolds in the sense that the sum of the symplectic forms of M4 and CP2 projected to Lagrangian sub-manifold vanishes. Could the induced spinor fields for effective action be localized to generalized Lagrangian sub-manifolds? This would allow both string world sheets and 4-D space-time surfaces but SH would select 2-D Lagrangian manifolds. At the level of effective action the theory would be incredibly simple.

    Induced spinor fields at string world sheets could obey the "dynamics of avoidance" in the sense that both the induced weak gauge fields W,Z0 and induced Kähler form (to achieve this U(1) gauge potential must be sum of M4 and CP2 parts) would vanish for the regions carrying induced spinor fields. They would coupleonly to the induced em field (!) given by the vectorial R12 part of CP2 spinor curvature for D=2,4. For D=1 at boundaries of string world sheets the coupling to gauge potentials would be non-trivial since gauge potentials need not vanish there. Spinorial dynamics would be extremely simple and would conform with the vision about symmetry breaking of weak group to electromagnetic gauge group.

    The projections of canonical currents of Kähler action to string world sheets would vanish, and the projections of the 4-D modified gamma matrices would define just the induced 2-D metric. If the induced metric of space-time surface reduces to an orthogonal direct sum of string world sheet metric and metric acting in normal space, the flow defined by 4-D canonical momentum currents is parallel to string world sheet. These conditions could define the "boundary" conditions at string world sheets for SH.

To sum up, the notion M4 symplectic structure is now on rather firm basis both physically and mathematically.

See the chapter About twistor lift of TGD or the article with the same title.

Questions about TGD

In FB I was made a question about general aspects of TGD. It was impossible to answer the question with few lines and I decided to write a blog posting. I am sorry for typos in the hastily written text. A more detailed article Can one apply Occam’s razor as a general purpose debunking argument to TGD? tries to emphasize the simplicity of the basic principles of TGD and of the resulting theory.

A. In what aspects TGD extends other theory/theories of physics?

I will replace "extends" with "modifies" since TGD also simplifies in many respects. I shall restrict the considerations to the ontological level which to my view is the really important level.

  1. Space-time level is where TGD started from. Space-time as an abstract 4-geometry is replaced as space-time as 4-surface in M4× CP2. In GRT space-time is small deformation of Minkowski space.

    In TGD both Relativity Principle (RP) of Special Relativity (SRT) and General Coordinate Invariance (GCI) and Equivalence Principle (EP) of General Relativity hold true. In GRT RP is given up and leads to the loss of conservation laws since Noether theorem cannot be applied anymore: this is what led to the idea about space-time as surface in H. Strong form of holography (SH) is a further principle reducing to strong form of GCI (SGCI).

  2. TGD as a physical theory extends to a theory of consciousness and cognition. Observer as something external to the Universe becomes part of physical system - the notion of self - and quantum measurement theory which is the black sheet of quantum theory extends to a theory of consciousness and also of cognition relying of p-adic physics as correlate for cognition. Also quantum biology becomes part of fundamental physics and consciousness and life are seen as basic elements of physical existence rather than something limited to brain.

    One important aspect is a new view about time: experienced time and geometric time are not one and same thing anymore although closely related. ZEO explains how the experienced flow and its direction emerges. The prediction is that both arrows of time are possible and that this plays central role in living matter.

  3. p-Adic physics is a new element and an excellent candidate for a correlate of cognition. For instance, imagination could be understood in terms of non-determinism of p-adic partial differential equations for p-adic variants of space-time surfaces. p-Adic physics and fusion of real and various p-adic physics to adelic physics provides fusion of physics of matter with that of cognition in TGD inspired theory of cognition. This means a dramatic extension of ordinary physics. Number Theoretical Universality states that in certain sense various p-adic physics and real physics can be seen as extensions of physics based on algebraic extensions of rationals (and also those generated by roots of e inducing finite-D extensions of p-adics).
  4. Zero energy ontology (ZEO) in which so called causal diamonds (CDs, analogs Penrose diagrams) can be seen as being forced by very simple condition: the volume action forced by twistorial lift of TGD must be finite. CD would represent the perceptive field defined by finite volume of imbedding space H=M4× CP2.

    ZEO implies that conservation laws formulated only in the scale of given CD do not anymore fix select just single solution of field equations as in classical theory. Theories are strictly speaking impossible to test in the old classical ontology. In ZEO testing is possible be sequence of state function reductions giving information about zero energy states.

    In principle transition between any two zero energy states - analogous to events specified by the initial and final states of event - is in principle possible but Negentropy Maximization Principle (NMP) as basic variational principle of state function reduction and of consciousness restricts the possibilities by forcing generation of negentropy: the notion of negentropy requires p-adic physics.

    Zero energy states are quantum superpositions of classical time evolutions for 3-surfaces and classical physics becomes exact part of quantum physics: in QFTs this is only the outcome of stationary phase approximation. Path integral is replaced with well-defined functional integral- not over all possible space-time surface but pairs of 3-surfaces at the ends of space-time at opposite boundaries of CD.

    ZEO leads to a theory of consciousness as quantum measurement theory in which observer ceases to be outsider to the physical world. One also gets rid of the basic problem caused by the conflict of the non-determinism of state function reduction with the determinism of the unitary evolution. This is obviously an extension of ordinary physics.

  5. Hierarchy of Planck constants represents also an extension of quantum mechanics at QFT limi. At fundamental level one actually has the standard value of h but at QFT limit one has effective Planck constant heff =n× h, n=1,2,... this generalizes quantum theory. This scaling of h has a simple topological interpretation: space-time surface becomes n-fold covering of itself and the action becomes n-multiple of the original which can be interpreted as heff=n×h.

    The most important applications are to biology, where quantum coherence could be understood in terms of a large value of heff/h. The large n phases resembles the large N limit of gauge theories with gauge couplings behaving as α ∝ 1/N used as a kind of mathematical trick. Also gravitation is involved: heff is associated with the flux tubes mediating various interactions (being analogs to wormholes in ER-EPR correspondence). In particular, one can speak about hgr, which Nottale introduced originally and heff= hgr plays key role in quantum biology according to TGD.

B. In what sense TGD is simplification/extension of existing theory?

  1. Classical level: Space-time as 4-surface of H means a huge reduction in degrees of freedom. There are only 4 field like variables - suitably chosen 4 coordinates of H=M4× CP2. All classical gauge fields and gravitational field are fixed by the surface dynamics. There are no primary gauge fields or gravitational fields nor any other fields in TGD Universe and they appear only at the QFT limit.

    GRT limit would mean that many-sheeted space-time is replaced by single slightly curved region of M4. The test particle - small particle like 3-surface - touching the sheets simultaneously experience sum of gravitational forces and gauge forces. It is natural to assume that this superposition corresponds at QFT limit to the sum for the deviations of induced metrics of space-time sheets from flat metric and sum of induce gauge potentials. These would define the fields in standard model + GRT. At fundamental level effects rather than fields would superpose. This is absolutely essential for the possibility of reducing huge number field like degrees of freedom. One can obviously speak of emergence of various fields.

    A further simplification is that only preferred extremals for which data coding for them are reduced by SH to 2-D string like world sheets and partonic 2-surfaces are allowed. TGD is almost like string model but space-time surfaces are necessary for understanding the fact that experiments must be analyzed using classical 4-D physics. Things are extremely simple at the level of single space-time sheet.

    Complexity emerges from many-sheetedness. From these simple basic building bricks - minimal surface extremals of Kähler action (not the extremal property with respect to Kähler action and volume term strongly suggested by the number theoretical vision plus analogs of Super Virasoro conditions in initial data) - one can engineer space-time surfaces with arbitrarily complex topology - in all length scales. An extension of existing space-time concept emerges. Extremely simple locally, extremely complex globally with topological information added to the Maxwellian notion of fields (topological field quantization allowing to talk about field identify of system/field body/magnetic body.

    Another new element is the possibility of space-time regions with Euclidian signature of the induced metric. These regions correspond to 4-D "lines" of general scattering diagrams. Scattering diagrams has interpretation in terms of space-time geometry and topology.

  2. The construction of quantum TGD using canonical quantization or path integral formalism failed completely for Kähler action by its huge vacuum degeneracy. The presence of volume term still suffers from complete failure of perturbation theory and extreme non-linearity. This led to the notion of world of classical worlds (WCW) - roughly the space of 3-surfaces. Essentially pairs of 3-surfaces at the boundaries of given CD connected by preferred extremals of action realizing SH and SGCI.

    The key principle is geometrization of the entire quantum theory, not only of classical fields geometrized by space-time as surface vision. This requires geometrization of hermitian conjugation and representation of imaginary unit geometrically. Kähler geometry for WCW makes this possible and is fixed once Kähler function defining Kähler metric is known. Kähler action for a preferred extremal of Kähler action defining space-time surface as an analog of Bohr orbit was the first guess but twistor lift forced to add volume term having interpretation in terms of cosmological constant.

    Already the geometrization of loop spaces demonstrated that the geometry - if it exists - must have maximal symmetries (isometries). There are excellent reasons to expect that this is true also in D=3. Physics would be unique from its mathematical existence!

  3. WCW has also spinor structure. Spinors correspond to fermionic Fock states using oscillator operators assignable to the induced spinor fields - free spinor fiels. WCW gamma matrices are linear combinations of these oscillator operators and Fermi statistics reduces to spinor geometry.

  4. There is no quantization in TGD framework at the level of WCW. The construction of quantum states and S-matrix reduces to group theory by the huge symmetries of WCW. Therefore zero energy states of Universe (or CD) correspond formally to classical WCW spinor fields satisfying WCW Dirac equation analogous to Super Virasoro conditions and defining representations for the Yangian generalization of the isometries of WCW (so called super-symplectic group). In ZEO stated are analogous to pairs of initial and final states and the entanglement coefficients between positive and negative energy parts of zero energy states expected to be fixed by Yangian symmetry define scattering matrix and have purely group theoretic interpretation. If this is true, entire dynamics would reduce to group theory in ZEO.

C. What is the hypothetical applicability of the extension - in energies, sizes, masses etc?

TGD is a unified theory and is meant to apply in all scales. Usually the unifications rely on reductionistic philosophy and try to reduce physics to Planck scale. Also super string models tried this and failed: what happens at long length scales was completely unpredictable (landscape catastrophe).

Many-sheeted space-time however forces to adopt fractal view. Universe would be analogous to Mandelbrot fractal down to CP2 scale. This predicts scaled variants of say hadron physics and electroweak physics. p-Adic length scale hypothesis and hierarchy of phases of matter with heff=n×h interpreted as dark matter gives a quantitative realization of this view.

  1. p-Adic physics shows itself also at the level of real physics. One ends up to the vision that particle mass squared has thermal origin: the p-adic variant of particle mass square is given as thermal mass squared given by p-adic thermodynamics mappable to real mass squared by what I call canonical identification. p-Adic length scale hypothesis states that preferred p-adic primes characterizing elementary particles correspond to primes near to power of 2: p=about 2k. p-Adic length scale is proportional to p1/2.

    This hypothesis is testable and it turns out that one can predict particle mass rather accurately. This is highly non-trivial since the sensitivity to the integer k is exponential. So called Mersenne primes turn out to be especially favoured. This part of theory was originally inspired by the regularities of particle mass spectrum. I have developed arguments for why the crucial p-adic length scale hypothesis - actually its generalization - should hold true. A possible interpretation is that particles provide cognitive representations of themselves by p-adic thermodynamics.

  2. p-Adic length scale hypothesis leads also to consider the idea that particles could appear as different p-adically scaled up variants. For instance, ordinary hadrons to which one can assign Mersenne prime M107=2107-1 could have fractally scaled variants. M89 and MG,107 (Gaussian prime) would be two examples and there are indications at LHC for these scaled up variants of hadron physics. These fractal copies of hadron physics and also of electroweak physics would correspond to extension of standard model.
  3. Dark matter hierarchy predicts zoomed up copies of various particles. The simplest assumption is that masses are not changed in the zooming up. One can however consider that binding energy scale scales non-trivially. The dark phases would emerge are quantum criticality and give rise to the associated long range correlations (quantum lengths are typically scaled up by heff/h=n).

D. What is the leading correction/contribution to physical effects due to TGD onto particles, interactions, gravitation, cosmology?

  1. Concerning particles I already mentioned the key predictions.
    1. The existence of scaled variants of various particles and entire branches of physics. The fundamental quantum numbers are just standard model quantum numbers code by CP2 geometry.

    2. Particle families have topological description meaning that space-time topology would be an essential element of particle physics. The genus of partonic 2-surfaces (number of handles attached to sphere) is g=0,1,2,... and would give rise to family replication. g<2 partonic 2-surfaces have always global conformal symmetry Z2 and this suggests that they give rise to elementary particles identifiable as bound states of g handles. For g>2 this symmetry is absent in the generic case which suggests that they can be regarded as many-handle states with mass continuum rather than elementary particles. 2-D anyonic systems could represent an example of this.
    3. A hierarchy of dynamical symmetries as remnants of super-symplectic symmetry however suggests itself. The super-symplectic algebra possess infinite hierarchy of isomorphic sub-algebras with conformal weights being n-multiples of for those for the full algebra (fractal structure again possess also by ordinary conformal algebras). The hypothesis is that sub-algebra specified by n and its commutator with full algebra annihilate physical states and that corresponding classical Noether charges vanish. This would imply that super-symplectic algebra reduces to finite-D Kac-Moody algebra acting as dynamical symmetries. The connection with ADE hierarchy of Kac-Moody algebras suggests itself. This would predict new physics. Condensed matter physics comes in mind.
    4. Number theoretic vision suggests that Galois groups for the algebraic extensions of rationals act as dynamical symmetry groups. They would act on algebraic discretizations of 3-surfaces and space-time surfaces necessary to realize number theoretical universality. This would be completely new physics.
  2. Interactions would be mediated at QFT limit by standard model gauge fields and gravitons. QFT limit however loses all information about many-sheetedness and there would be anomalies reflecting this information loss. In many-sheeted space-time light can propagate along several paths and the time taken to travel along light-like geodesic from A to B depends on space-time sheet since the sheet is curved and warped. Neutrinos and gamma rays from SN1987A arriving at different times would represent a possible example of this. It is quite possible that the outer boundaries of even macroscopic objects correspond to boundaries between Euclidian and Minkowskian regions at the space-time sheet of the object.

    The failure of QFTs to describe bound states of say hydrogen atom could be second example: many-sheetedness and identification of bound states as single connected surface formed by proton and electron would be essential and taken into account in wave mechanical description but not in QFT description.

  3. Concerning gravitation the basic outcome is that by number theoretical vision all preferred extremals are extremals of both Kähler action and volume term. This is true for all known extremals what happens if one introduces the analog of Kähler form in M4 is an open question).

    Minimal surfaces carrying no K&aum;lher field would be the basic model for gravitating system. Minimal surface equation are non-linear generalization of d'Alembert equation with gravitational self-coupling to induce gravitational metric. In static case one has analog for the Laplace equation of Newtonian gravity. One obtains analog of gravitational radiation as "massless extremals" and also the analog of spherically symmetric stationary metric.

    Blackholes would be modified. Besides Schwartschild horizon which would differ from its GRT version there would be horizon where signature changes. This would give rise to a layer structure at the surface of blackhole.

  4. Concerning cosmology the hypothesis has been that RW cosmologies at QFT limit can be modelled as vacuum extremals of Kä hler action. This is admittedly ad hoc assumption inspired by the idea that one has infinitely long p-adic length scale so that cosmological constant behaving like 1/p as function of p-adic length scale assignable with volume term in action vanishes and leaves only Kähler action. This would predict that cosmology with critical is specified by a single parameter - its duration as also over-critical cosmology. Only sub-critical cosmologies have infinite duration.

    One can look at the situation also at the fundamental level. The addition of volume term implies that the only RW cosmology realizable as minimal surface is future light-cone of M4. Empty cosmology which predicts non-trivial slightly too small redshift just due to the fact that linear Minkowski time is replaced with lightcone proper time constant for the hyperboloids of M4+. Locally these space-time surfaces are however deformed by the addition of topologically condensed 3-surfaces representing matter. This gives rise to additional gravitational redshift and the net cosmological redshift. This also explains why astrophysical objects do not participate in cosmic expansion but only comove. They would have finite size and almost Minkowski metric.

    The gravitational redshift would be basically a kinematical effect. The energy and momentum of photons arriving from source would be conserved but the tangent space of observer would be Lorentz-boosted with respect to source and this would course redshift.

    The very early cosmology could be seen as gas of arbitrarily long cosmic strings in H (or M4) with 2-D M4 projection. Horizon would be infinite and TGD suggests strongly that large values of heff makes possible long range quantum correlations. The phase transition leading to generation of space-time sheets with 4-D M4 projection would generate many-sheeted space-time giving rise to GRT space-time at QFT limit. This phase transition would be the counterpart of the inflationary period and radiation would be generated in the decay of cosmic string energy to particles.

See the new chapter Can one apply Occam's razor as a general purpose debunking argument to TGD? or article with the same title.

Generalized Kähler structure for M4 and CP breaking and matter antimatter asymmetry

In the following I will consider some questions related to the twistor lift of TGD and end up to a possible vision about general mechanism of CP breaking and generation of matter antimatter asymmetry.

  1. Can the analog of Kähler form J(M4) assignable to M4 suggested by the symmetry between M4 and CP2 and by number theoretical vision appear in the theory? What would be the physical implications? The basic objection is the loss of Poincare invariance is lost. This can be however avoided by introducing the moduli space for Kähler forms. This moduli space is actually the moduli space of causal diamonds (CDs) forced in any case by zero energy ontology (ZEO) and playing central role in the generalization of quantum measurement theory to a theory of consciousness and in the explanation of the relationship between geometric and subjective time.

    Why J(M4) would be needed? J(M4) corresponds to parallel constant electric and magnetic fields in given direction. Constant E and B=E fix directions of quantization axes for energy (rest system) and spin. One implication is transversal localization of imbedding space spinor modes: imbedding space spinor modes are products of harmonic oscillator Gaussians in transversal degrees of freedom very much like quarks inside hadrons.

    Also CP breaking is implied by the electric field and the question is whether this could explain the observed CP breaking as appearing already at the level of imbedding space M4× CP2. The estimate for the CP mass splitting of neutral kaon and anti-kaon is of correct order of magnitude. Whether stationary spherically symmetric metric as minimal surface allows a sensible physical generalization is a killer test for the hypothesis.

  2. How does gravitational coupling emerge at fundamental level? The answer is obvious: string area action is scaled by 1/G as in string models. The objection is that p-adic mass calculations suggest that string tension is determined by CP2 size R: the analog of string tension appearing in mass formula given by p-adic mass calculations would be by a factor about 10-8 smaller than that estimated from string tension. The discrepancy evaporates by noticing that p-adic mass calculations rely on p-adic thermodynamics at imbedding space level whereas string world sheets appear at space-time level.
  3. Could one regard the localization of spinor modes to string world sheets as a localization to Lagrangian sub-manifolds of space-time surface having by definition vanishing induced Kähler form: J(M4)+J(CP2)=0. Lagrangian sub-manifolds would be commutative in the sense of Poisson bracket. Could string world sheets be minimal surfaces satisfying J(M4)+J(CP2)=0. The Lagrangian condition allows also more general solutions - even 4-D space-time surfaces and one obtains analog of brane hierarchy. Could one allow spinor modes also at these analogs of branes. Is Lagrangian condition equivalent with the original condition that induced W boson fields making the em charge of induced spinor modes ill-defined vanish and allowing also solution with other dimensions. How Lagrangian property relates to the idea that string world sheets correspond to complex (commutative) surfaces of quaternionic space-time surface in octonionic imbedding space.
1. Can the Kähler form of M4 appear in Kähler action?

I have already earlier considered the question whether the analog of Kähler form assignable to M4 could appear in Kähler action. Could one replace the induced Kähler form J(CP2) with the sum J=J(M4)+J(CP2) such that the latter term would give rise to a new component of Kähler form both in space-time interior at the boundaries of string world sheets regarded as point-like particles? This could be done both in the Kähler action for the interior of X4 and also in the topological magnetic flux term ∈t J associated with string world sheet and reducing to a boundary term giving couplings to U(1) gauge potentials Aμ(CP2) and Aμ(M4) associated with J(CP2) and J(M4). The interpretation of this coupling is an interesting challenge.

Consider first the objections against introducing J(M4) to the Kähler action at imbedding space level.

  1. J(M4) would would break translational and Lorentz symmetries at the level of imbedding space since J(M4) cannot be Lorentz invariant. For imbedding space spinor modes this term would bring in coupling to the self-dual Kähler form in M4. The simplest choice is A=(At=z, Az=0,Ax=y,Ay=0) defining decomposition M4 =M2× E2. For Dirac equation in M4 one would have free motion in preferred time-like (t,z)-plane plane M2 in whereas in x- and y-directions (E2 plane) would one have harmonic oscillator potentials due to the gauge potentials of electric and magnetic fields. One would have something very similar to quark model of hadron: quark momenta would have conserved longitudinal part and non-conserved transversal part. The solution spectrum has scaling invariance Ψ(mk)→ Ψ(λ mk) so that there is no preferred scale and the transversal scales scale as 1/E and 1/kx.
  2. Since J(M4) is not Lorentz invariant Lorentz boosts would produce new M2× E2 decomposition. If one assumes above kind of linear gauge as gauge invariance suggests, the choices with fixed second tip of causal diamond (CD) define finite-dimensional moduli space SO(3,1)/SO(1,1)× SO(2) having in number theoretic vision an interpretation as a choice of preferred hypercomplex plane and its orthogonal complement. This is the moduli space for hypercomplex structures in M4 with the choices of origins parameterized by M4. The introduction of the moduli space would allow to preserve Poincare invariance.
  3. If one generalizes the condition for Kähler metric to J2(M4)=-g(M4) fixing the scaling of J, the coupling to A(M4) is also large and suggests problems with the large breaking of Poincare symmetry for the spinor modes of the imbedding space for given moduli. The transversal localization by the self-dual magnetic and electric fields for J(M4) would produce wave packets in transversal degrees of freedom: is this physical?

    This moduli space is actually the moduli space introduced for causal diamonds (CDs) in zero energy ontology (ZEO) forced by the finite value of volume action: fixing of the line connecting the tips of CD the Lorentz boost fixing the position for the second tip of CD parametrizes this moduli space apart from division with the group of transformations leaving the planes M2 and E2 having interpretation a plane defined by light-like momentum and polarization plane associated with a given CD invariant.

  4. Why this kind of symmetry breaking for Poincare invariance? A possible explanation proposed already earlier is that quantum measurement involves a selection of quantization axis. This choice necessarily breaks the symmetries and J(M4) would be an imbedding space correlate for the selection of rest frame and quantization axis of spin. This conforms with the fact that CD is interpreted as the perceptive field of conscious entity at imbedding space level: the contents of consciousness would be determined by the superposition of space-time surfaces inside CD. The choice of J(M4) for CD would select preferred rest system (quantization axis for energy as a line connecting tips of CD) via electric part of J(M4) and quantization axis of spin (via magnetic part of J(M4). The moduli space for CDs would be the space for choices of these particular quantization axis and in each state function reduction would mean a localization in this moduli space. Clearly, this reduction would be higher level reduction and correspond to a decision of experimenter.
To summarize, for J(M4)=0 Poincare symmetries are realized at the level of imbedding space but obviously broken slightly by the geometry of CD. The allowance of J(M4)≠ 0 implies that both translational and rotational symmetries are reduced for a given CD: the interpretation would be in terms of a choice of quantization axis in state function reduction. They are however lifted to the level of moduli space of CDs and exact in this more abstract sense. This is nothing new: already the introduction of ZEO and CDs force by volume term in action forced by twistor lift of TGD implies the same. Also the view about state function reduction requires wave functions in the moduli space of CDs. This is also essential for understanding how the arrow of geometric time is inherited from that of subjective time in TGD inspired theory of consciousness.

What about the situation at space-time level?

  1. The introduction of J(M4) part to Kähler action has nice number theoretic aspects. In particular, J selects the preferred complex and quaternionic sub-space of octonionic space of imbedding space. The simplest possibility is that the Kähler action is defined by the Kähler form J(M4)+J(CP2).

    Since M4 and CP2 Kähler geometries decouple it should be possible to take the counterpart of Kähler coupling strength in M4 to be much larger than in CP2 degrees of freedom so that M4 Kähler action is a small perturbation and slowly varying as a functional of preferred extremal. This option is however not in accordance with the idea that entire Kähler form is induced.

  2. Whether the proposed ansätze for general solutions make still sense is not clear. In particular, can one still assume that preferred extremals are minimal surfaces? Number theoretical vision strongly suggests - one could even say demands - the effective decoupling of Kähler action and volume term. This would imply the universality of quantum critical dynamics. The solutions would not depend at all on the coupling parameters except through the dependence on boundary conditions. The coupling between the dynamics of Kähler action and volume term would come also from the conservation conditions at light-like 3-surfaces at which the signature of the induced metric changes.
  3. At space-time level the field equations get more complex if the M4 projection has dimension D(M4)>2 and also for D(M4)=2 if it carries non-vanishing induced J(M4). One would obtain cosmic strings of form X2× Y2 as minimal surface extremals of ordinary Kähler action or X2 Lagrangian manifold of M4 as also CP2 type vacuum extremals and their deformations with M4 projection Lagrangian manifold. Thus the differences would not be seen for elementary particle and string like objects. Simplest string worlds sheet for which J(M4) vanishes would correspond to a piece of plane M2.

    M4 is the simplest minimal surface extremal of Kähler action necessarily involving also J(M4). The action in this case vanishes identically by self-duality (in Euclidian signature self-duality does not imply this). For perturbations of M4 such as spherically symmetric stationary metric the contribution of M4 Kähler term to the action is expected to be small and the come mainly from cross term mostly and be proportional to the deviation from flat metric. The interpretation in terms of gravitational contribution from M4 degrees of freedom could make sense.

  4. What about massless extremals (MEs)? How the induced metric affects the situation and what properties second fundamental form has? Is it possible to obtain a situation in which the energy momentum tensor Tαβ and second fundamental form Hkαβ have in common components which are proportional to light-like vector so that the contraction TαβHkαβ vanishes?

    Minimal surface property would help to satisfy the conditions. By conformal invariance one would expect that the total Kähler action vanishes and that one has JαγJγβ = a× gαβ+b × kαkβ. These conditions together with light-likeness of Kähler current guarantee that field equations are satisfied.

    In fact, one ends up to consider a generalization of MEs by starting from a generalization of holomorphy. Complex CP2 coordinates ξi would be functions of light-like M2 coordinate u+=k• m, k light-like vector, and of complex coordinate w for E2 orthogonal to M2. Therefore the CP2 projection would 3-D rather than 2-D now.

    The second fundamental form has only components of form Hku+w, Hku+w* and Hkww, Hkw*w*. The CP2 contribution to the induced metric has only components of form Δ gu+w, Δ g+w*, and gw*w. There is also contribution gu+u-=1, where v is the light-like dual of u in plane M2. Contravariant metric can be expanded as a power series for in the deviation (Δ gu+w, Δ gu+w*) of the metric from (gu+u-, gww*). Only components of form gu+,ui and gww* are obtained and their contractions with the second fundamental form vanish identically since there are no common index pairs with simultaneously non-vanishing components. Hence it seems that MEs generalize!

    I have asked earlier whether this construction might generalize for ordinary MEs. One can introduce what I have called Hamilton-Jacobi structure for M4 consisting of locally orthogonal slicings by integrable 2-surfaces having tangent space having local decomposition M2x× E2x with light-like direction depending on point x. An objection is that the direction of light-like momentum depends on position: this need not be inconsistent with momentum conservation but would imply that the total four-momentum is not light-like anymore. Topological condensation for MEs and at MEs could imply this kind modification.

  5. There is also a topological magnetic flux type term for string world sheet. Topological term can be transformed to a boundary term coupling classical particles at the boundary of string world sheet to CP2 Kähler gauge potential (added to the equation for a light-like geodesic line). Now also the coupling to M4 gauge potential would be obtained. The condition J(M4)+ J(CP2)=0 at string world sheets is very attractive manner to identify string world sheets as analogs of Lagrangian manifolds but does not imply the vanishing of the net U(1) couplings at boundary since the induce gauge potentials are in general different.

    Also topological term including also M4 Kähler magnetic flux for string world sheet contributes also to the modified Dirac equation since the gamma matrices are modified gamma matrices required by super-conformal symmetries and defined as contractions of canonical momentum densities with imbedding space gamma matrices. This is true both in space-time interior, at string world sheets and at their boundaries. CP2 (M4) term gives a contribution proportional to CP2 (M4) gamma matrices.

    At imbedding space level transversal localization would be the outcome and a good guess is that the same happens also now. This is indeed the case for M4 defining the simplest extremal. The general interpretation of M4 Kähler form could be as a quantum tool for transversal dynamical localization of wave packets in Kähler magnetic and electric fields of M4. Analog for decoherence occurring in transversal degrees of freedom would be in question. Hadron physics could be one application.

How to test this idea?
  1. It might be possible to kill the idea by showing that one does not obtain spherically symmetric Schwartschild type metric as a minimal surface extremal of generalized Kähler action: these extremals are possible for ordinary Kähler action. For the canonical imbedding of M4 field equations are satisfied since energy momentum tensor vanishes identically. For the small deformations the presence of J(M4) would reduce rotational symmetry to cylindrical symmetry.
  2. J(M4) could make its presence manifest in the physics of right-handed neutrino having no direct couplings to electroweak gauge fields. Mixing with left handed neutrino is however induced by mixing of M4 and CP2 gamma matrices. The transversal localization of right-handed neutrino in a background, which is a small deformation of M4 could serve as an experimental signature.
  3. CP breaking in hadronic systems is one of the poorly understood aspects of fundamental physics and relates closely to the mysterious matter-antimatter asymmetry. The constant electric part of self dual J(M4) implies CP breaking. I have earlier considered the possibility that Kähler electric fields could cause this breaking but this breaking would be local. Second possibility is that matter and antimatter correspond to different values of heff and are dark relative to each other.

    Could J(M4) explain the observed CP breaking as appearing already at the level of imbedding space M4× CP2 and could this breaking explain hadronic CP breaking and matter anti-matter asymmetry? Could M4 part of Kähler electric field induce different heff/h=n for particles and antiparticles?

To answer these questions one can study Dirac equation at imbedding space level coupled to the gauge potential A(M4) for J(M4).
  1. The coupling of Kähler form to leptons is 3 times larger than to to quarks as in the case of A(CP2). This would give coupling k=1 for quarks an k=3 for leptons. k corresponds to fermion number which is opposite for fermions and antifermions having therefore opposite values of k at the respective space-time sheets.
  2. The potential satisfies ∂μAμ(M4)=0. Let the non-vanishing components of the Kähler gauge potential be (A0,Az)=ε (x,+/- y). The sign fact ε+/- 1 corresponds to self dual and antiself-dual options, let us assume self-duality as in the case of CP2 Kähler form. Scalar d'Alembertian reads as (∂μμ+ AμAμ)Ψ= -m2 Ψ.
  3. Assuming momentum eigenstate in time and z-direction (plane M2), one obtains by separation of variables (H1+H2)Ψ= (E-m2-kz2)Ψ. Hx= -∂x2+k2x2 and Hy= -∂y2+k2y2) are oscillator Hamiltonians. The spectrum is of Hx+Hy is given by kT2= (n1+n2+1)21/2|k| and one obtains E2=m2 +kz2 +kT2. This contribution is CP invariant and same for fermions and anti-fermions. The special feature is the presence of zero point transversal momentum. It is not possible to have a particle, which would be completely at rest. One can also say that m2 is increased 21/2|k| hbar2/L2, L= 1 m if standard convention for metric is used. For other conventions the numerical value of CP2 radius is scale by L/Lnew. L must correspond to some physical scale assignable to particle: secondary p-adic length scale is the natural identification.
  4. Spinor d'Alembertian contains also dipole moment term kX=JmuνΣμν giving a contribution, which depends on the sign of k: E2=m2 +kz2 +kT2+ kX. The term is sum of magnetic and electric dipole moment terms. The coupling k changes sign in CP operation and be of opposite sign for fermions and anti-fermions. One has a breaking of CP for given spin state. The dependence of X on spin state gives a test for the theory and also for the predicted CP breaking.
  5. Scaling covariance allows in principle all values L. To estimate the size of the effect one must fix the length scale L. CP2 size has only different value using L as unit and in flat background it does not matter. L should correspond to the size scale of the CD associated with particle. The secondary p-adic length scale of fermion defining also the size scale of its magnetic body is a natural guess so that Δ E2/E2≈ 2Δ E/E≈ Δ m/m ∼ 2/p1/2, p≈ 2k would hold true. This mass splitting is very small. For weak bosons having k=89 the mass splitting would be of order 3× 10-4 eV. For small values of p at ultrahigh energies the scale of CP breaking is larger, which conforms with the idea that matter-antimatter-asymmetry has emerged in very early cosmology.

    The recent experiment found that the mass difference Δ m/m for proton and antiproton satisfies Δ m <69× 10-12m ≈ 6.9× 10-2 eV (see this) so that this gives no constraints. Kaon-antikaon mass difference is estimated to be about 3.5× 10-6 eV (see this). This would correspond to a p-adic length scale k=96. Top quark is mainly responsible for the mixing of neutral kaon and its antiparticle in the model of based on loops involving decay to virtual quark pairs. The estimate from p-adic mass calculations for top quark mass scale is k=94 so that the order of magnitude estimate has correct of order of magnitude (being by factor 4 too large). This is an encouraging sign.

    How the mass splitting of neutral kaons would result? In quark model kaon and antikaon can be regarded as sdbbar and dsbar pairs. The net spins vanishes but the mass splitting due to electric moment dipole moment term X is non-vanishing due to the different sign of coupling k. The sign of the mass splitting is also opposite for kaon and antikaon.

  6. One can also consider the modified Dirac equation for canonically imbedded M4 which is simplest preferred extremal. The coupling to J(M4) to modified Dirac equation in space-time interior with gamma matrices replaced with modified gamma matrices are obtained as contractions of canonical momentum currents with M4 gamma matrices. Completely analogous phenomenon happens for CP2 type extremals. Tαβ=0 so that the modified gamma comes from Jαβ Jk~lβmlγk. These give just ordinary gamma matrices so that the two Dirac equations are identical.

2. About string like objects

String like objects and partonic 2-surfaces carry the information about quantum states and about space-time surfaces as preferred extremals if strong form of holography (SH) holds true. SH has of course some variants. The weakest variant states that fundamental information carrying objects are metrically 2-D. The light-like 3-surfaces separating space-time regions with Minkowskian and Euclidian signature of the induced metric are indeed metrically 2-D, and could thus carry information about quantum state.

An attractive possibility is that this information is basically topological. For instance, the value of Planck constant heff=n× h would tell the number sheets of the singular covering defining this surface such that the sheets co-incide at partonic 2-surfaces at the ends of space-time surface at boundaries of CD. In the following some questions related to string world sheets are considered. The information could be also number theoretical. Galois group for the algebraic extension of rationals defining particular adelic physics would transform to each other the number theoretic discretizations of light-like 3-surfaces and give rise to covering space structure. The action is trivial at partonic 2-surfaces should be trivial if one wants singular covering: this would mean that discretizations of partonic 2-surfaces consist of rational points. heff/h=n could in this case be a factor of the order of Galois group.

The original observation was that string world sheets should carry vanishing W boson fields in order that the em charge for the modes of the induced spinor field is well-defined. This condition can be satisfied in certain situations also for the entire space-time surface. This raises several questions. What is the fundamental condition forcing the restriction of the spinor modes to string world sheets - or more generally, to surface of given dimension? Is this restriction dynamical. Can one have an analog of brane hierarchy in which also higher-D objects can carry modes of induced spinor field Could the analogs of Lagrangian sub-manifolds of X4 ⊂ M4× CP2 satisfying J(M4)+J(CP2)=0 define string world sheets and their variants with varying dimension? The additional condition would be minimal surface property.

2.1 How does the gravitational coupling emerge?

The appearance of G=lP2 has coupling constant remained for a long time actually somewhat of a mystery in TGD. lP defines the radius of the twistor sphere of M4 replaced with its geometric twistor space M4× S2 in twistor lift. G makes itself visible via the coefficients ρvac= 8π Λ/G volume term but not directly and if preferred extremals are minimal surface extremals of Kähler action ρvac makes itself visible only via boundary conditions. How G appears as coupling constant?

Somehow the M4 Kähler form should appear in field equations. 1/G could naturally appear in the string tension for string world sheets as string models suggest. p-Adic mass calculations identify the analog of string tension as something of order of magnitude of 1/R2. This identification comes from the fact that the ground states of super-conformal representations correspond to imbedding space spinor modes, which are solutions of Dirac equation in M4× CP2. This argument is rather convincing and allows to expect that the p-adic mass scale is not determined by string tension and it can be chosen to be of order 1/G just as in string models.

2.2 Non-commutative imbedding space and strong form of holography

The precise formulation of strong form of holography (SH) is one of the technical problems in TGD. A comment in FB page of Gareth Lee Meredith led to the observation that besides the purely number theoretical formulation based on commutativity also a symplectic formulation in the spirit of non-commutativity of imbedding space coordinates can be considered. One can however use only the notion of Lagrangian manifold and avoids making coordinates operators leading to a loss of General Coordinate Invariance (GCI).

Quantum group theorists have studied the idea that space-time coordinates are non-commutative and tried to construct quantum field theories with non-commutative space-time coordinates (see this). My impression is that this approach has not been very successful. In Minkowski space one introduces antisymmetry tensor Jkl and uncertainty relation in linear M4 coordinates mk would look something like [mk, ml] = lP2Jkl, where lP is Planck length. This would be a direct generalization of non-commutativity for momenta and coordinates expressed in terms of symplectic form Jkl.

1+1-D case serves as a simple example. The non-commutativity of p and q forces to use either p or q. Non-commutativity condition reads as [p,q]= hbar Jpq and is quantum counterpart for classical Poisson bracket. Non-commutativity forces the restriction of the wave function to be a function of p or of q but not both. More geometrically: one selects Lagrangian sub-manifold to which the projection of Jpq vanishes: coordinates become commutative in this sub-manifold. This condition can be formulated purely classically: wave function is defined in Lagrangian sub-manifolds to which the projection of J vanishes. Lagrangian manifolds are however not unique and this leads to problems in this kind of quantization. In TGD framework the notion of "World of Classical Worlds" (WCW) allows to circumvent this kind of problems and one can say that quantum theory is purely classical field theory for WCW spinor fields. "Quantization without quantization would have Wheeler stated it.

GCI poses however a problem if one wants to generalize quantum group approach from M4 to general space-time: linear M4 coordinates assignable to Lie-algebra of translations as isometries do not generalize. In TGD space-time is surface in imbedding space H=M4× CP2: this changes the situation since one can use 4 imbedding space coordinates (preferred by isometries of H) also as space-time coordinates. The analog of symplectic structure J for M4 makes sense and number theoretic vision involving octonions and quaternions leads to its introduction. Note that CP2 has naturally symplectic form.

Could it be that the coordinates for space-time surface are in some sense analogous to symplectic coordinates (p1,p2,q1,q2) so that one must use either (p1,p2) or (q1,q2) providing coordinates for a Lagrangian sub-manifold. This would mean selecting a Lagrangian sub-manifold of space-time surface? Could one require that the sum Jμν(M4)+ Jμν(CP2) for the projections of symplectic forms vanishes and forces in the generic case localization to string world sheets and partonic 2-surfaces. In special case also higher-D surfaces - even 4-D surfaces as products of Lagrangian 2-manifolds for M4 and CP2 are possible: they would correspond to homologically trivial cosmic strings X2× Y2⊂ M4× CP2, which are not anymore vacuum extremals but minimal surfaces if the action contains besides Käction also volume term.

But why this kind of restriction? In TGD one has strong form of holography (SH): 2-D string world sheets and partonic 2-surfaces code for data determining classical and quantum evolution. Could this projection of M4 × CP2 symplectic structure to space-time surface allow an elegant mathematical realization of SH and bring in the Planck length lP defining the radius of twistor sphere associated with the twistor space of M4 in twistor lift of TGD? Note that this can be done without introducing imbedding space coordinates as operators so that one avoids the problems with general coordinate invariance. Note also that the non-uniqueness would not be a problem as in quantization since it would correspond to the dynamics of 2-D surfaces.

The analog of brane hierarchy for the localization of spinors - space-time surfaces; string world sheets and partonic 2-surfaces; boundaries of string world sheets - is suggesetive. Could this hierarchy correspond to a hierarchy of Lagrangian sub-manifolds of space-time in the sense that J(M4)+J(CP2)=0 is true at them? Boundaries of string world sheets would be trivially Lagrangian manifolds. String world sheets allowing spinor modes should have J(M4)+J(CP2)=0 at them. The vanishing of induced W boson fields is needed to guarantee well-defined em charge at string world sheets and that also this condition allow also 4-D solutions besides 2-D generic solutions. This condition is physically obvious but mathematically not well-understood: could the condition J(M4)+J(CP2)=0 force the vanishing of induced W boson fields? Lagrangian cosmic string type minimal surfaces X2× Y2 would allow 4-D spinor modes. If the light-like 3-surface defining boundary between Minkowskian and Euclidian space-time regions is Lagrangian surface, the total induced Kähler form Chern-Simons term would vanish. The 4-D canonical momentum currents would however have non-vanishing normal component at these surfaces. I have considered the possibility that TGD counterparts of space-time super-symmetries could be interpreted as addition of higher-D right-handed neutrino modes to the 1-fermion states assigned with the boundaries of string world sheets.

It is relatively easy to construct an infinite family of Lagrangian string world sheets satisfying J(M4) +J(CP2)=0 using generalized symplectic transformations of M4 and CP2 as Hamiltonian flows to generate new ones from a given Lagrangian string world sheets. One must pose minimal surface property as a separate condition. Consider a piece of M2 with coordinates (t,z) and homologically non-trivial geodesic sphere S2 of CP2 with coordinates (u= cos(Θ),Φ). One has J(M4)tz=1 and J= 1. Identify string world sheet via map (u,Φ)= (kz,ω t) from M2 to S2. The induced CP2 Kahler form is J(CP2)tz= kω. kω=-1 guarantees J(M4) +J(CP2)=0. The strings have necessarily finite length from L=1/k≤ z≤ L. One can perform symplectic transformations of CP2 and symplectic transformations of M4 to obtain new string world sheets. In general these are not minimal surfaces and this condition would select some preferred string world sheets.

An alternative - but of course not necessarily equivalent - attempt to formulate this picture would be in terms of number theoretic vision. Space-time surfaces would be associative or co-associative depending on whether tangent space or normal space in imbedding space is associative - that is quaternionic. These two conditions would reduce space-time dynamics to associativity and commutativity conditions. String world sheets and partonic 2-surfaces would correspond to maximal commutative or co-commutative sub-manifolds of imbedding space. Commutativity (co-commutativity) would mean that tangent space (normal space as a sub-manifold of space-time surface) has complex tangent space at each point and that these tangent spaces integrate to 2-surface. SH would mean that data at these 2-surfaces would be enough to construct quantum states. String world sheet boundaries would in turn correspond to real curves of the complex 2-surfaces intersecting partonic 2-surfaces at points so that the hierarchy of classical number fields would have nice realization at the level of the classical dynamics of quantum TGD.

To sum up, one cannot exclude the possibility that J(M4) is present implying a universal transversal localization of imbedding space spinor harmonics and the modes of spinor fields in the interior of X4: this could perhaps relate to somewhat mysterious de-coherence interaction producing locality and to CP breaking and matter-antimatter asymmetry. The moduli space for M4 Kähler structures proposed by number theoretic considerations would save from the loss of Poincare invariance and the number theoretic vision based on quaternionic and octonionic structure would have rather concrete realization. This moduli space would only extend the notion of "world of classical worlds" (WCW).

For background see the chapter Questions related to the twistor lift of TGD or the article with the same title.

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