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Note: Newest contributions are at the top!
Also I have been forced to consider this possibility.
See the new chapter Is Non-Associative Physics and Language Possible Only in Many-Sheeted Space-Time?.
The understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During years I have made several attempts to understand coupling evolution.
See the new chapter Does Riemann Zeta Code for Generic Coupling Constant Evolution? or the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?.
With the development of the vision about number theoretically universal view about functional integration in WCW , a concrete vision about the exponent of Kähler action in Euclidian and Minkowskian space-time regions. The basic requirement is that exponent of Kähler action belongs to an algebraic extension of rationals and therefore to that of p-adic numbers and does not depend on the ordinary p-adic numbers at all - this at least for sufficiently large primes p. Functional integral would reduce in Euclidian regions to a sum over maxima since the troublesome Gaussian determinants that could spoil number theoretic universality are cancelled by the metric determinant for WCW.
The adelically exceptional properties of Neper number e, Kähler metric of WCW, and strong form of holography posing extremely strong constraints on preferred extremals, could make this possible. In Minkowskian regions the exponent of imaginary Kähler action would be root of unity. In Euclidian space-time regions expressible as power of some root of e which is is unique in sense that ep is ordinary p-adic number so that e is p-adically an algebraic number - p:th root of ep.
These conditions give conditions on Kähler coupling strength αK= gK2/4π (hbar=1)) identifiable as an analog of critical temperature. Quantum criticality of TGD would thus make possible number theoretical universality (or vice versa).
This approach leads to different algebraic structure of αK than the earlier arguments.
See the chapter Coupling Constant Evolution in Quantum TGD and the chapter Unified Number Theoretic Vision of "Physics as Generalized Number Theory" or the article Could one realize number theoretical universality for functional integral?.
Field equations as conservation laws, Frobenius integrability conditions, and a connection with quaternion analyticity
The following represents qualitative picture of field equations of TGD trying to emphasize the physical aspects. What is new is the discussion of the possibility that Frobenius integrability conditions are satisfied and correspond to quaternion analyticity.
For details see the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" or the article Could One Define Dynamical Homotopy Groups in WCW?.
I learned that Agostino Prastaro has done highly interesting work with partial differential equations, also those assignable to geometric variational principles such as Kähler action in TGD. I do not understand the mathematical details but the key idea is a simple and elegant generalization of Thom's cobordism theory, and it is difficult to avoid the idea that the application of Prastaro's idea might provide insights about the preferred extremals, whose identification is now on rather firm basis.
One could also consider a definition of what one might call dynamical homotopy groups as a genuine characteristics of WCW topology. The first prediction is that the values of conserved classical Noether charges correspond to disjoint components of WCW. Could the natural topology in the parameter space of Noether charges zero modes of WCW metric) be p-adic and realize adelic physics at the level of WCW? An analogous conjecture was made on basis of spin glass analogy long time ago. Second surprise is that the only the 6 lowest dynamical homotopy/homology groups of WCW would be non-trivial. The Kähler structure of WCW suggets that only Π0, Π2, and Π4 are non-trivial.
The interpretation of the analog of Π1 as deformations of generalized Feynman diagrams with elementary cobordism snipping away a loop as a move leaving scattering amplitude invariant conforms with the number theoretic vision about scattering amplitude as a representation for a sequence of algebraic operation can be always reduced to a tree diagram. TGD would be indeed topological QFT: only the dynamical topology would matter.
See the chapter Recent View about Kähler Geometry and Spin Structure of WCW or the article Could one define dynamical homotopy groups in WCW?.
U- and M-matrices are key objects in zero energy ontology (ZEO). M-matrix for large causal diamonds (CDs) is the counterpart of thermal S-matrix and proportional to scale dependent S-matrix: the dependence on the scale of CD characterized by integer is S(n)=Sn in accordance with the idea that S corresponds to the counterpart of ordinary unitary time evolution operator. U-matrix characterizes the time evolution as dispersion in the moduli space of CDs tending to increase the size of CD and giving rise to the experience arrow of geometric time and also to the notion of self in TGD insprired theory of consciousness.
The original view about the relationship between U- and M-matrices was a purely formal guess: M-matrices would define the orthonormal rows of U-matrix. This guess is not correct physically and one must consider in detail what U-matrix really means.
What one can say about M-matrices?
What can one say about U-matrix?
It seems that the simple picture is not quite correct yet. One should obtain somehow an integration over angle in order to obtain Kronecker delta.
What about the identification of S?
What about quantum classical correspondence and zero modes?
The one-one correspondence between the basis of quantum states and zero modes realizes quantum classical correspondence.
One of the old TGD inspired really crazy ideas about scattering amplitudes is that Universe is doing some sort of arithmetics so that scattering amplitude are representations for computational sequences of minimum length. The idea is so crazy that I have even given up its original form, which led to an attempt to assimilate the basic ideas about bi-algebras, quantum groups, Yangians and related exotic things. The work with twistor Grassmannian approach inspired a reconsideration of the original idea seriously with the idea that super-symplectic Yangian could define the arithmetics. I try to describe the background, motivation, and the ensuing reckless speculations in the following.
Do scattering amplitudes represent quantal algebraic manipulations?
The question is whether it could be indeed possible to characterize particle reactions as computations involving transformation of tensor products to products in vertices and co-products to tensor products in co-vertices (time reversals of the vertices). A couple of examples gives some idea about what is involved.
Generalized Feynman diagram as shortest possible algebraic manipulation connecting initial and final algebraic objects
There is a strong motivation for the interpretation of generalized Feynman diagrams as shortest possible algebraic operations connecting initial and final states. The reason is that in TGD one does not have path integral over all possible space-time surfaces connecting the 3-surfaces at the ends of CD. Rather, one has in the optimal situation a space-time surface unique apart from conformal gauge degeneracy connecting the 3-surfaces at the ends of CD (they can have disjoint components).
Path integral is replaced with integral over 3-surfaces. There is therefore only single minimal generalized Feynman diagram (or twistor diagram, or whatever is the appropriate term). It would be nice if this diagram had interpretation as the shortest possible computation leading from the initial state to the final state specified by 3-surfaces and basically fermionic states at them. This would of course simplify enormously the theory and the connection to the twistor Grassmann approach is very suggestive. A further motivation comes from the observation that the state basis created by the fermionic Clifford algebra has an interpretation in terms of Boolean quantum logic and that in ZEO the fermionic states would have interpretation as analogs of Boolean statements A→ B.
To see whether and how this idea could be realized in TGD framework, let us try to find counterparts for the basic operations ⊗ and • and identify the algebra involved. Consider first the basic geometric objects.
This was not the whole story yet
The proposed amplitude represents only the value of WCW spinor field for single pair of 3-surfaces at the opposite boundaries of given CD. Hence Yangian construction does not tell the whole story.
I decided to take here from a previous posting an argument allowing to conclude that super string models are unable to describe macroscopic gravitation involving formation of gravitationally bound states. Therefore superstrings models cannot have desired macroscopic limit and are simply wrong. This is of course reflected also by the landscape catastrophe meaning that the theory ceases to be a theory in macroscopic scales. The failure is not only at the level of superstring models: it is at the level of quantum theory itself. Instead of single value of Planck constant one must allow a hierarchy of Planck constants predicted by TGD. My sincere hope is that this message could gradually leak through the iron curtain to the ears of the super string gurus.
Superstring action has bosonic part proportional to string area. The proportionality constant is string tension proportional to 1/hbar G and is gigantic. One expects only strings of length of order Planck length be of significance.
It is now clear that also in TGD the action in Minkowskian regions contains a string area. In Minkowskian regions of space-time strings dominate the dynamics in an excellent approximation and the naive expectation is that string theory should give an excellent description of the situation.
String tension would be proportional to 1/hbar G and this however raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required non-perturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.
In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.
I have however proposed that gravitons - at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has heff= hgr=GMm/v0, where v0/c<1 (v0 has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v0/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v0/G2Mm. For v0= GMm/hbar, which remains below unity for Mm/mPl2 one would have hgr/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water) - probably not an accident - would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. hgr=heff hypothesis is indeed central in TGD based view about living matter.
To conclude, it seems that superstring theory with single value of Planck constant cannot give rise to macroscopic gravitationally bound matter and would be therefore simply wrong much better than to be not-even-wrong.
TGD differs in several respects from quantum field theories and string models. The basic mathematical difference is that the mathematically poorly defined notion of path integral is replaced with the mathematically well-defined notion of functional integral defined by the Kähler function defining Kähler metric for WCW ("world of classical worlds"). Apart from quantum jump, quantum TGD is essentially theory of classical WCW spinor fields with WCW spinors represented as fermionic Fock states. One can say that Einstein's geometrization of physics program is generalized to the level of quantum theory.
It has been clear from the beginning that the gigantic super-conformal symmetries generalizing ordinary super-conformal symmetries are crucial for the existence of WCW Kähler metric. The detailed identification of Kähler function and WCW Kähler metric has however turned out to be a difficult problem. It is now clear that WCW geometry can be understood in terms of the analog of AdS/CFT duality between fermionic and space-time degrees of freedom (or between Minkowskian and Euclidian space-time regions) allowing to express Kähler metric either in terms of Kähler function or in terms of anti-commutators of WCW gamma matrices identifiable as super-conformal Noether super-charges for the symplectic algebra assignable to δ M4+/-× CP2. The string model description of gravitation emerges and also the TGD based view about dark matter becomes more precise.
Kähler function, Kähler action, and connection with string models
The definition of Kähler function in terms of Kähler action is possible because space-time regions can have also Euclidian signature of induced metric. Euclidian regions with 4-D CP2 projection - wormhole contacts - are identified as lines of generalized Feynman diagrams - space-time correlates for basic building bricks of elementary particles. Kähler action from Minkowskian regions is imaginary and gives to the functional integrand a phase factor crucial for quantum field theoretic interpretation. The basic challenges are the precise specification of Kähler function of "world of classical worlds" (WCW) and Kähler metric.
There are two approaches concerning the definition of Kähler metric: the conjecture analogous to AdS/CFT duality is that these approaches are mathematically equivalent.
Realization of super-conformal symmetries
The detailed realization of various super-conformal symmetries has been also a long standing problem but recent progress leads to very beautiful overall view.
Interior dynamics for fermions, the role of vacuum extremals, dark matter, and SUSY
The key role of CP2-type and M4-type vacuum extremals has been rather obvious from the beginning but the detailed understanding has been lacking. Both kinds of extremals are invariant under symplectic transformations of δ M4× CP2, which inspires the idea that they give rise to isometries of WCW. The deformations CP2-type extremals correspond to lines of generalized Feynman diagrams. M4 type vacuum extremals in turn are excellent candidates for the building bricks of many-sheeted space-time giving rise to GRT space-time as approximation. For M4 type vacuum extremals CP2 projection is (at most 2-D) Lagrangian manifold so that the induced Kähler form vanishes and the action is fourth-order in small deformations. This implies the breakdown of the path integral approach and of canonical quantization, which led to the notion of WCW.
If the action in Minkowskian regions contains also string area, the situation changes dramatically since strings dominate the dynamics in excellent approximation and string theory should give an excellent description of the situation: this of course conforms with the dominance of gravitation.
String tension would be proportional to 1/hbar G and this raises a grave classical counter argument. In string model massless particles are regarded as strings, which have contracted to a point in excellent approximation and cannot have length longer than Planck length. How this can be consistent with the formation of gravitationally bound states is however not understood since the required non-perturbative formulation of string model required by the large valued of the coupling parameter GMm is not known.
In TGD framework strings would connect even objects with macroscopic distance and would obviously serve as correlates for the formation of bound states in quantum level description. The classical energy of string connecting say the two wormhole contacts defining elementary particle is gigantic for the ordinary value of hbar so that something goes wrong.
I have however proposed that gravitons - at least those mediating interaction between dark matter have large value of Planck constant. I talk about gravitational Planck constant and one has heff= hgr=GMm/v0, where v0/c<1 (v0 has dimensions of velocity). This makes possible perturbative approach to quantum gravity in the case of bound states having mass larger than Planck mass so that the parameter GMm analogous to coupling constant is very large. The velocity parameter v0/c becomes the dimensionless coupling parameter. This reduces the string tension so that for string world sheets connecting macroscopic objects one would have T ∝ v0/G2Mm. For v0= GMm/hbar, which remains below unity for Mm/mPl2 one would have hgr/h=1. Hence the action remains small and its imaginary exponent does not fluctuate wildly to make the bound state forming part of gravitational interaction short ranged. This is expected to hold true for ordinary matter in elementary particle scales. The objects with size scale of large neutron (100 μm in the density of water) - probably not an accident - would have mass above Planck mass so that dark gravitons and also life would emerge as massive enough gravitational bound states are formed. hgr=heff hypothesis is indeed central in TGD based view about living matter. In this framework superstring theory with single value of Planck constant would not give rise to macroscopic gravitationally bound matter and would be thus simply wrong.
If one assumes that for non-standard values of Planck constant only n-multiples of super-conformal algebra in interior annihilate the physical states, interior conformal gauge degrees of freedom become partly dynamical. The identification of dark matter as macroscopic quantum phases labeled by heff/h=n conforms with this.
The emergence of dark matter corresponds to the emergence of interior dynamics via breaking of super-conformal symmetry. The induced spinor fields in the interior of flux tubes obeying Kähler Dirac action should be highly relevant for the understanding of dark matter. The assumption that dark particles have essentially same masses as ordinary particles suggests that dark fermions correspond to induced spinor fields at both string world sheets and in the space-time interior: the spinor fields in the interior would be responsible for the long range correlations characterizing heff/h=n. Magnetic flux tubes carrying dark matter are key entities in TGD inspired quantum biology. Massless extremals represent second class of M4 type non-vacuum extremals.
This view forces once again to ask whether space-time SUSY is present in TGD and how it is realized. With a motivation coming from the observation that the mass scales of particles and sparticles most naturally have the same p-adic mass scale as particles in TGD Universe I have proposed that sparticles might be dark in TGD sense. The above argument leads to ask whether the dark variants of particles correspond to states in which one has ordinary fermion at string world sheet and 4-D fermion in the space-time interior so that dark matter in TGD sense would almost by definition correspond to sparticles!
Nima Arkani-Hamed et al demonstrate that various twistorially represented on-mass-shell amplitudes (allowing light-like complex momenta) constructible by taking products of the 3-particle on mass-shell amplitude and its conjugate can be assigned with unique permutations of the incoming lines. The article describes the graphical representation of the amplitudes and its generalization. For 3-particle amplitudes, which correspond to ++- and +-- twistor amplitudes, the corresponding permutations are cyclic permutations, which are inverses of each other. One actually introduces double cover for the labels of the particles and speaks of decorated permutations meaning that permutation is always a right shift in the integer and in the range [1,2× n].
Amplitudes as representation of permutations
It is shown that for on mass shell twistor amplitudes the definition using on-mass-shell 3-vertices as building bricks is highly reducible: there are two moves for squares defining 4-particle sub-amplitudes allowing to reduce the graph to a simpler one. The first move is topologically like the s-t duality of the old-fashioned string models and second one corresponds to the transformation black ↔ white for a square sub-diagram with lines of same color at the ends of the two diagonals and built from 3-vertices.
One can define the permutation characterizing the general on mass shell amplitude by a simple rule. Start from an external particle a and go through the graph turning in in white (black) vertex to left (right). Eventually this leads to a vertex containing an external particle and identified as the image P(a) of the a in the permutation. If permutations are taken as right shifts, one ends up with double covering of permutation group with 2× n! elements - decorated permutations. In this manner one can assign to any any line of the diagram two lines. This brings in mind 2-D integrable theories where scattering reduces to braiding and also topological QFTs where braiding defines the unitary S-matrix. In TGD parton lines involve braidings of the fermion lines so that an assignment of permutation also to vertex would be rather nice.
BCFW bridge has an interpretation as a transposition of two neighboring vertices affecting the lines of the permutation defining the diagram. One can construct all permutations as products of transpositions and therefore by building BCFW bridges. BCFW bridge can be constructed also between disjoint diagrams as done in the BCFW recursion formula.
Can one generalize this picture in TGD framework? There are several questions to be answered.
Fermion lines for fermions massless in 8-D sense
What does one mean with particle line at the level of fermions?
One can consider two candidates for fundamental vertices depending on whether one identifies the lines of Feynman diagram as fermion lines or as light-like orbits of partonic 2-surfaces. The latter vertices reduces microscopically to the fermionic 4-vertices.
Partonic surfaces as 3-vertices
At space-time level one could identify vertices as partonic 2-surfaces.
OZI rule implies correspondence between permutations and amplitudes
The realization of the permutation in the same manner as for N=4 amplitudes does not work in TGD. OZI rule following from the absence of 4-fermion vertices however implies much simpler and physically quite a concrete manner to define the permutation for external fermion lines and also generalizes it to include braidings along partonic orbits.
Non-planar Feynman diagrams have remained a challenge for the twistor approach. The problem is simple: there is no canonical ordering of the extrenal particles and the loop integrand involving tricky shifts in integrations to get finite outcome is not unique and well-defined so that twistor Grasmann approach encounters difficulties.
Recently Nima Arkani-Hamed et al have have also considered non-planar MHV diagrams (having minimal number of "wrong" helicities) of N=4 SUSY, and shown that they can be reduced to non-planar diagrams for different permutations of vertices of planar diagrams ordered naturally. There are several integration regions identified as positive Grassmannians corresponding to different orderings of the external lines inducing non-planarity. This does not however hold true generally.
At the QFT limit the crossings of lines emerges purely combinatorially since Feynman diagrams are purely combinatorial objects with the ordering of vertices determining the topological properties of the diagram. Non-planar diagrams correspond to diagrams, which do not allow crossing-free imbedding to plane but require higher genus surface to get rid of crossings.
The twistor approach to TGD require that the expression of light-likeness of M4 momenta in terms of twistors generalizes to 8-D case. The light-likeness condition for twistors states that the 2× 2 matrix representing M4 momentum annihilates a 2-spinor defining the second half of the twistor. The determinant of the matrix reduces to momentum squared and its vanishing implies the light-likeness. This should be generalized to a situation in one has M4 and CP2 twistor, which are not light-like separately but light-likeness in 8-D sense holds true (allowing massive particles in M4 sense and thus generalization of twistor approach for massive particles).
The case of M8=M4× E4
M8-H duality suggests that it might be useful to consider first the twistorialiation of 8-D light-likeness first the simpler case of M8 for which CP2 corresponds to E4. It turns out that octonionic representation of gamma matrices provide the most promising formulation.
In order to obtain quadratic dispersion relation, one must have 2× 2 matrix unless the determinant for the 4× 4 matrix reduces to the square of the generalized light-likeness condition.
The case of M8=M4× CP2
What about twistorialization in the case of M4× CP2? The introduction of wave functions in the twistor space of CP2 seems to be enough to generalize Witten's construction to TGD framework and that algebraic variant of twistors might be needed only to realize quantum classical correspondence. It should correspond to tangent space counterpart of the induced twistor structure of space-time surface, which should reduce effectively to 4-D one by quaternionicity of the space-time surface.
To sum up, the generalization of the notion of twistor to 8-D context allows description of massive particles using twistors but requires that octonionic Dirac equation is introduced. If one requires that octonionic and ordinary description of Dirac equation are equivalent, the description is possible only at surfaces having at most 1-D CP2 projection - geodesic circle for the most stringent option. The boundaries of string world sheets are such surfaces and also string world sheets themselves if they have 1-D CP2 projection, which must be geodesic circle if also induce gauge potentials are required to vanish. In spirit with M8-H duality, string boundaries give rise to classical M8 twistorizalization analogous to the standard M4 twistorialization and generalize 4-momentum to massless 8-momentum whereas imbedding space spinor harmonics give description in terms of four-momentum and color quantum numbers. One has SO(4)-SU(3) duality: a wave function in the space of 8-momenta corresponds to SO(4) description of hadrons at low energies as opposed to that for quarks at high energies in terms of color. The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle (EP) in the most concrete form: gravitational four-momentum equals to inertial four-momentum. EP for internal quantum numbers is clearly more delicate. In twistorialization also helicity is brought and for CP2 degrees of freedom M8 helicity means that electroweak spin is described in terms of helicity.
Biologists have a principle known as "ontogeny recapitulates phylogeny" (ORP) stating that the morphogenesis of the individual reflects evolution of the species. The principle seems to be realized also in theoretical physics - at least in TGD Universe. ORP would now say that the evolution of theoretical physics via the emergence of increasingly complex notion of particle reflects the structure physics itself. Point like particles are really there as points at partonic 2-surfaces carrying fermion number: their 1-D orbits correspond to the boundaries of string world sheets; 2-D hyper-complex string world sheets in flat space (M4× S1) are there and carry induced spinors; also complex (or co-complex) partonic 2-surfaces (Euclidian string world sheets) and carry particle numbers; 3-D space-like surfaces at the ends of causal diamonds (CDs) and the 3-D light-like orbits of partonic 2-surfaces are there; 4-D space-time surfaces are there as quaternionic or co-quaternionic sub-manifolds of 8-D octonionic imbedding space: there the hierarchy ends since there are no higher-dimensional classical number fields. ORP would thus also realize evolution of mathematics at the level of physics.
The M4 projection of the 8-D M8 momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle in the most concrete form: gravitational four-momentum equals to inertial four-momentum.
The twistor Grassmann approach has led to a phenomenal progress in the understanding of the scattering amplitudes of gauge theories, in particular the N=4 SUSY.
As a non-specialist I have been frustrated about the lack of concrete picture, which would help to see how twistorial amplitudes might generalize to TGD framework. A pleasant surprise in this respect was the proposal of a particle interpretation for the twistor amplitudes by Nima Arkani Hamed et al in the article "Unification of Residues and Grassmannian Dualities" .
In this interpretation incoming particles correspond to spheres CP1 so that n-particle state corresponds to (CP1)n/Gl(2) (the modding by Gl(2) might be seen as a kind of formal generalization of particle identity by replacing permutation group S2 with Gl(2) of 2× 2 matrices). If the number of "wrong" helicities in twistor diagram is k, this space is imbedded to CPk-1n/Gl(k) as a surface having degree k-1 using Veronese map to achieve the imbedding. The imbedding space can be identified as Grassmannian G(k,n) . This surface defines the locus of the multiple residue integral defining the twistorial amplitude.
The particle interpretation brings in mind the extension of single particle configuration space E3 to its Cartesian power E3n/Sn for n-particle system in wave mechanics. This description could make sense when point-like particle is replaced with 3-surface or partonic 2-surface: one would have Cartesian product of WCWs divided my Sn. The generalization might be an excellent idea as far calculations are considered but is not in spirit with the very idea of string models and TGD that many-particle states correspond to unions of 3-surfaces in H (or light-like boundaries of causal diamond (CD) in Zero Energy Ontology (ZEO).
Witten's twistor string theory
Witten's twistor string theory is more in spirit with TGD at fundamental level since it is based on the identification of generalization of vertices as 2-surfaces in twistor space.
In the addition to the article Classical part of the twistor story a proposal generalizing Witten's approach to TGD is discussed. The following gives a very concise summary of the article.
Consider first various aspects of twistorialization.
Generalization of Witten's construction in TGD framework
The generalization of the Witten's geometric construction of scattering amplitudes relying on the induction of the twistor structure of the imbedding space to that associated with space-time surface looks surprisingly straight-forward and would provide more precise formulation of the notion of generalized Feynman diagrams forcing to correct some wrong details.
The emergence of fundamental 4-fermion vertex and of boson exchanges
The emergence of the fundamental 4-fermion vertex and of boson exchanges deserves a more detailed discussion.
The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. The obvious ideas coming in mind are appropriately defined quaternionic and octonion analyticity. I have used a considerable amount of time to consider these possibilities but had to give up the idea about octonion analyticity could somehow allow to preferred extemals.
One can argue that quaternion analyticity is the more natural option in the sense that the local octonionic imbedding space coordinate (or at least M8 or E8 coordinate, which is enough if M8-H duality holds true) would for preferred extremals be expressible in the form
o(q)= u(q) + v(q)× I .
Here q is quaternion serving as a coordinate of a quaternionic sub-space of octonions, and I is octonion unit belonging to the complement of the quaternionic sub-space, and multiplies v(q) from right so that quaternions and qiaternionic differential operators acting from left do not notice these coefficients at all. A stronger condition would be that the coefficients are real. u(q) and v(q) would be quaternionic Taylor- of even Laurent series with coefficients multiplying powers of q from right for the same reason.
I ended up to this idea after finding two very interesting articles discussing the generalization of Cauchy-Riemann equations. The first article was about so called triholomorphic maps between 4-D almost quaternionic manifolds. The article gave as a reference an article about quaternionic analogs of Cauchy-Riemann conditions discussed by Fueter long ago (somehow I have managed to miss Fueter's work just like I missed Hitchin's work about twistorial uniqueness of M4 and CP2), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.
The so called Cauhy-Riemann-Fueter conditions generalize Cauchy-Riemann conditions. These conditions are however not unique.
The first form of Cauchy-Rieman-Fueter conditions
Cauhy-Riemann-Fueter (CRF) conditions generalize Cauchy-Riemann conditions. These conditions are however not unique. Consider first the translationally invariant form of CRF conditions.
Second form of CRF conditions
Second form of CRF conditions proposed in the second reference is tailored in order to realize the almost obvious manner to realize quaternion analyticity.
Generalization of CRF conditions?
Could the proposed forms of CRF conditions be special cases of much more general CRF conditions as CR conditions are?
Geometric formulation of the CRF conditions
The previous naive generalization of CRF conditions treats imaginary units without trying to understand their geometric content. This leads to difficulties when when tries to formulate these conditions for maps between quaternionic and hyper-quaternionic spaces using purely algebraic representation of imaginary units since it is not clear how these units relate to each other.
In the first article the CRF conditions are formulated in terms of the antisymmetric (1,1) type tensors representing the imaginary units: they exist for almost quaternionic structure and presumably also for almost hyper-quaternionic structure needed in Minkowskian signature.
The generalization of CRF conditions is proposed in terms of the Jacobian J of the map mapping tangent space TM to TN and antisymmetric tensors Ju and Ju representing the quaternionic imaginary units of N and M. The generalization of CRF conditions reads as
J- ∑u Ju J ju=0 .
For N=M it reduces to the translationally invariant algebraic form of the conditions discussed above. These conditions seem to be well-defined also when one maps quaternionic to hyper-quaternionic space or vice versa. These conditions are not unique. One can perform an SO(3) rotation (quaternion automorphism) of the imaginary units mediated by matrix Λuv to obtain
J- Λuv JuJ jv=0 .
The matrix Λ can depend on point so that one has a kind of gauge symmetry. The most general triholomorphic map allows the presence of Λ Note that these conditions make sense on any coordinates and complex analytic maps generate new forms of these conditions.
Covariant forms of structure constant tensors reduce to octonionic structure constants and this allows to write the conditions explicitly. The index raising of the second index of the structure constants is however needed using the metrics of M and N. This complicates the situation and spoils linearity: in particular, for surfaces induced metric is needed. Whether local SO(3) rotation can eliminate the dependence on induced metric is an interesting question. Minkowskian imaginary units differ only by multiplication by i from Euclidian so that Minkowskian structure constants differ only by sign from those for Euclidian ones.
In the octonionic case the geometric generalization of CRF conditions does not seem to make sense. By non-associativity of octonion product it is not possible to have a matrix representation for the matrices so that a faithful representation of octonionic imaginary units as antisymmetric 1-1 forms does not make sense. If this representation exists it it must map octonionic associators to zero. Note however that CRF conditions do not involve products of three octonion units so that they make sense as algebraic conditions at least.
Does residue calculus generalize?
CRF conditions allow to generalize Cauchy formula allowing to express value of analytic function in terms of its boundary values. This would give a concrete realization of the holography in the sense that the physical variables in the interior could be expressed in terms of the data at the light-like partonic orbits and and the ends of the space-time surface. Triholomorphic function satisfies d'Alembert/Laplace equations - in induced metric in TGD framework- so that the maximum modulus principle holds true. The general ansatz for a preferred extremals involving Hamilton-Jacobi structure leads to d'Alembert type equations for preferred extremals.
Could the analog of residue calculus exist? Line integral would become 3-D integral reducing to a sum over poles and possible cuts inside the 3-D contour. The space-like 3-surfaces at the ends of CDs could define natural integration contours, and the freedom to choose contour rather freely would reflect General Coordinate Invariance. A possible choice for the integration contour would be the closed 3-surface defined by the union of space-like surfaces at the ends of CD and by the light-like partonic orbits.
Poles and cuts would be in the interior of the space-time surface. Poles have co-dimension 2 and cuts co-dimension 1. Strong form of holography suggests that partonic 2-surfaces and perhaps also string world sheets serve as candidates for poles. Light-like 3-surfaces (partonic orbits) defining the boundaries between Euclidian and Minkowskian regions are singular objects and could serve as cuts. The discontinuity would be due to the change of the signature of the induced metric. There are CDs inside CDs and one can also consider the possibility that sub-CDs define cuts, which in turn reduce to cuts associated with sub-CDs.
Could one understand the preferred extremals in terms of quaternion-analyticity?
Could one understand the preferred extremals in terms of quaternion-analyticity or its possible generalization to an analytic representation for co-quaternionicity expected in space-time regions with Euclidian signature? What is the generalization of the CRF conditions for the counterparts of quaternion-analytic maps from hyper-quaternionic X4 to quaternionic CP2 and from quaternionic X4 to hyper-quaternionic M4? It has already become clear that this problem can be probably solved by using the the geometric representation for quaternionic imaginary units.
The best thing to do is to look whether this is possible for the known extremals: CP2 type vacuum extremals, vacuum extremals expressible as graph of map from M4 to a Lagrangian sub-manifold of CP2, cosmic strings of form X2× Y2⊂ M4 × CP2 such that X2 is string world sheet (minimal surface) and Y2 complex sub-manifold of CP2. One can also check whether Hamilton-Jacobi structure of M4 and of Minkowskian space-time regions and complex structure of CP2 have natural counterparts in the quaternion-analytic framework.
To sum up, connections between different conjectures related to the preferred extremals - M8-H duality, Hamilton-Jacobi structure, induced twistor space structure, quaternion-Kähler property and its Minkowskian counterpart, and even quaternion analyticity, are clearly emerging. The underlying reason is strong form of GCI forced by the construction of WCW geometry and implying strong from of holography posing extremely powerful quantization conditions on the extremals of Kähler action in ZEO. Without the conformal gauge conditions the mutual inconsistency of these conjectures looks rather infeasible.
In blog comments Anonymous gave a link to an article about construction of 4-D quaternion-Kähler metrics with an isometry: they are determined by so called SU(∞) Toda equation. I tried to see whether quaternion-Kähler manifolds could be relevant for TGD.
From Wikipedia one can learn that QK is characterized by its holonomy, which is a subgroup of Sp(n)×Sp(1): Sp(n) acts as linear symplectic transformations of 2n-dimensional space (now real). In 4-D case tangent space contains 3-D sub-manifold identifiable as imaginary quaternions. CP2 is one example of QK manifold for which the subgroup in question is SU(2)× U(1) and which has non-vanishing constant curvature: the components of Weyl tensor represent the quaternionic imaginary units. QKs are Einstein manifolds: Einstein tensor is proportional to metric.
What is really interesting from TGD point of view is that twistorial considerations show that one can assign to QK a special kind of twistor space (twistor space in the mildest sense requires only orientability). Wiki tells that if Ricci curvature is positive, this (6-D) twistor space is what is known as projective Fano manifold with a holomorphic contact structure. Fano variety has the nice property that as (complex) line bundle it has enough sections to define the imbedding of its base space to a projective variety. Fano variety is also complete: this is algebraic geometric analogy of topological property known as compactness.
QK manifolds and twistorial formulation of TGD
How the QKs could relate to the twistorial formulation of TGD?
Parallellizability is a very special property of 3-manifolds allowing to choose quaternionic imaginary units: global choice of one of them gives rise to twistor structure.
The relationship to quaternionicity conjecture and M8-H duality
One of the basic conjectures of TGD is that preferred extremals consist of quaternionic/ co-quaternionic (associative/co-associative) regions (see this). Second closely related conjecture is M8-H duality allowing to map quaternionic/co-quaternionic surfaces of M8 to those of M4× CP2. Are these conjectures consistent with QK in Euclidian regions and Hamilton-Jacobi property in Minkowskian regions? Consider first the definition of quaternionic and co-quaternionic space-time regions.
In Thinking Allowed Original there was a link to a talk by James Sully and having the title Geometry of Compression. I must admit that I understood very little about the talk. My not so educated guess is however that information is compressed: UV or IR cutoff eliminating entanglement in short length scales and describing its presence in terms of density matrix - that is thermodynamically - is another manner to say it. The TGD inspired proposal for the interpretation of the inclusions of hyper-finite factors of type II1 (HFFs) is in spirit with this.
The space-time counterpart for the compression would be in TGD framework discretization. Discretizations using rational points (or points in algebraic extensions of rationals) make sense also p-adically and thus satisfy number theoretic universality. Discretization would be defined in terms of intersection (rational or in algebraic extension of rationals) of real and p-adic surfaces. At the level of "world of classical worlds" the discretization would correspond to - say - surfaces defined in terms of polynomials, whose coefficients are rational or in some algebraic extension of rationals. Pinary UV and IR cutoffs are involved too. The notion of p-adic manifold allows to interpret rthe p-adic variants of space-time surfaces as cognitive representations of real space-time surfaces.
Finite measurement resolution does not allow state function reduction reducing entanglement totally. In TGD framework also negentropic entanglement stable under Negentropy Maximixation Principle (NMP) is possible. For HFFs the projection into single ray of Hilbert space is indeed impossible: the reduction takes always to infinite-D sub-space.
The visit to the URL was however not in vain. There was a link to an article discussing the geometrization of entanglement entropy inspired by the AdS/CFT hypothesis.
Quantum classical correspondence is basic guiding principle of TGD and suggests that entanglement entropy should indeed have space-time correlate, which would be the analog of Hawking-Bekenstein entropy.
Generalization of AdS/CFT to TGD context
AdS/CFT generalizes to TGD context in non-trivial manner. There are two alternative interpretations, which both could make sense. These interpretations are not mutually exclusive. The first interpretation makes sense at the level of "world of classical worlds" (WCW) with symplectic algebra and extended conformal algebra associated with δ M4+/- replacing ordinary conformal and Kac-Moody algebras. Second interpretation at the level of space-time surface with the extended conformal algebras of the light-likes orbits of partonic 2-surfaces replacing the conformal algebra of boundary of AdSn.
1. First interpretation
For the first interpretation 2-D conformal invariance is generalised to 4-D conformal invariance relying crucially on the 4-dimensionality of space-time surfaces and Minkowski space.
2. Second interpretation
For the second interpretation relies on the observation that string world sheets as carriers of induced spinor fields emerge in TGD framework from the condition that electromagnetic charge is well-defined for the modes of induced spinor field.
With what kind of systems 3-surfaces can entangle?
With what system X3 is entangled/can entangle? There are several options to consider and they could correspond to the two TGD variants for the AdS/CFT correspondence.
Minimal surface property is not favored in TGD framework
Minimal surface property for the 3-surfaces X3 at the ends of space-time surface looks at first glance strange but a proper generalization of this condition makes sense if one assumes strong form of holography. Strong form of holography realizes General Coordinate Invariance (GCI) in strong sense meaning that light-like parton orbits and space-like 3-surfaces at the ends of space-time surfaces are equivalent physically. As a consequence, partonic 2-surfaces and their 4-D tangent space data must code for the quantum dynamics.
The mathematical realization is in terms of conformal symmetries accompanying the symplectic symmetries of δ M4+/-× CP2 and conformal transformations of the light-like partonic orbit. The generalizations of ordinary conformal algebras correspond to conformal algebra, Kac-Moody algebra at the light-like parton orbits and to symplectic transformations δ M4× CP2 acting as isometries of WCW and having conformal structure with respect to the light-like radial coordinate plus conformal transformations of δ M4+/-, which is metrically 2-dimensional and allows extended conformal symmetries.
The generalisation of the conjecture about surface area proportionality of entropy to TGD context looks rather straightforward but is physically highly non-trivial. There are however some technicalities involved.
p-Adic variant of Bekenstein-Hawking law
When the 3-surface corresponds to elementary particle, a direct connection with p-adic thermodynamics suggests itself and allows to answer the questions above. p-Adic thermodynamics could be interpreted as a description of the entanglement with environment. In ZEO the entanglement could also correspond to time-like entanglement between the 3-D ends of the space-time surface at opposite light-like boundaries of CD. M-matrix, which can be seen as the analog of thermal S-matrix, decomposes to a product of hermitian square root of density matrix and unitary S-matrix and this hermitian matrix could also define p-adic thermodynamics.
What is the space-time correlate for negentropic entanglement?
The new element brought in by TGD framework is that number theoretic entanglement entropy is negative for negentropic entanglement assignable to unitary entanglement and NMP states that this negentropy increases. Since entropy is essentially number of energy degenerate states, a good guess is that the number n=heff/h of space-time sheets associated with heff defines the negentropy. An attractive space-time correlate for the negentropic entanglement is braiding. Braiding defines unitary S-matrix between the states at the ends of braid and this entanglement is negentropic. This entanglement gives also rise to topological quantum computation.
The following represents an update view about construction of scattering amplitudes at the level of "world of classical worlds" (WCW).
In order to facilitate the challenge of the reader I summarize basic ideas behind the construction of scattering amplitudes.
1. Zero energy ontology
In Zero Energy Ontology (ZEO) quantum theory as hermitian square root of thermodynamics, which leads to a generalization of the notion of S-matrix to a unitary U-matrix between zero energy states having as rows M-matrices which are products of hermitian square roots of density matrices with a common unitary matrix S for given CD. Number theoretical considerations suggests that CD size comes as integer multiples of CP2 size so that one obtain a hierarchy U-matrices having interpretation in terms of length scale evolution. For given CD also sub-CDs contribute down to some minimal scale defining UV scale. The largest CD defines the IR cutoff.
Scattering amplitudes would characterize the modes of WCW spinor field as time-like entanglement coefficients between positive and negative energy states associated with zero energy states.
The construction of scattering amplitudes - or M-matrix elements in ZEO - reduces at basic level to the construction of the Feynman diagram like entities for fundamental fermions, which serve basic building bricks of elementary particles.
2. Construction of scattering amplitudes as functional integrals in WCW
The decomposition of space-time surface to Minkowskian and Eucldian regions is the basic distinction from ordinary quantum field theories since it replaces path integral with mathematically well-defined functional integral over WCW.
3. Why it might work?
There are many reasons encouraging the hopes about calculable theory.
One must have a view about what elementary particles - as opposed to fundamental fermions - are, how the ordinary view about scattering based on exchanges of elementary particles emerges from this picture and how say BFF vertex reduces to a diagram at for fundamental fermions involving only 2-fermion vertices.
Elementary particles in TGD framework
The notion of elementary particle involves two aspects: elementary particles as space-time surfaces and elementary particles as many-fermion states with fundamental fermions localized at the wormhole throats and defining elementary particles as their bound states (including physical fermions).
1. Elementary particles as space-time surfaces
Let us first summarize what kind of picture ZEO suggests about elementary particles.
Localization of the induced spinor fields at string world sheets and fermionic propagators
The localization of induced spinors to string world sheets emerges from the condition that electromagnetic charge is well-defined for the modes of induced spinor fields. There is however an exception: covariantly constant right handed neutrino spinor νR: it can be de-localized along entire space-time surface. Right handed neutrino has no couplings to electroweak fields. It couples however to left handed neutrino by induced gamma matrices except when it is covariantly constant. Note that standard model does not predict νR but its existence is necessary if neutrinos develop Dirac mass. νR is indeed something which must be considered carefully in any generalization of standard model.
It has turned out that covariantly constant right-handed neutrino very probably corresponds to a pure gauge degree of freedom. Non-covariantly constant right-handed neutrino however mixes with left handed neutrino since the modified gamma matrices involve both M4 and CP2 gamma matrices and latter mix M4 chiralities. These right-handed neutrinos localized to partonic 2-surfaces would generate broken SUSY. There are however good reasons to expect that the mass scale for SUSY breaking corresponds to that for the mixing of right and left handed neutrinos inducing neutrino massivation and that the p-adic mass scale of particle and sparticle are same. The only manner to avoid conflict with experimental facts is that sparticles are dark in TGD sense that is having Planck constant heff=n× h. This would conform with the idea that the hierarchy of Planck constants corresponds to a hierarchy of breakings of conformal invariance, which is indeed behind the massivation.
The localization has powerful consequences since it gives in fermionic degrees of freedom what looks like ordinary Feynman diagrams but with only 2-fermion vertex. Space-time topology describes the vertices, say BFF vertex.
Fermion lines correspond to boundaries of string world sheets at the parton orbits. At these lines one must pose a boundary condition and the boundary condition is that the action of the modified Dirac operator associated with the boundary equals to the action of massless Dirac operator in momentum space representation. This reduces the fermionic propagator to massless Dirac propagator and simplify the construction decisively. If residue integral applied in twistor approach makes sense for the fermionic virtual momenta, this in turn gives inverse of Dirac propagator contracted between massless spinors with non-physical helicities.
What is the correct choice for the modified Dirac operator?
Vertices can be considered at both space-time level and fermionic level.
As already noticed, the definition of Γt might be problematic. For Chern-Simons term Γt is finite but in this case one looses the justification for M4 propagator of fermion as coming from the boundary condition. For Kähler-Dirac action there are hopes of obtaining a finite limiting value for Γt and therefore of ΓtDt if the induced Kähler form satisfies condition Jti=0 in the case that one has gtα=0. Furthermore, the continuity of Γt could be posed as a condition for vertices. Note that one obtains also the non-integrable phase factor allowing Wilson line interpretation.
Nothing has been said about the discontinuity of the modified Dirac operator through the wormhole contact as one traverses from Euclidian to Minkowskian side. This discontinuity might be also relevant.One could consider also the difference of the above discussed discontinuity between Euclidian and Minkowskian regions.
What one should obtain at QFT limit?
After functional integration over WCW of one should obtain a scattering amplitude in which the fermionic 2- vertices defined as discontinuities of the modified Dirac operator at partonic 2-surfaces should boil down to a contraction of an M8 vector with gamma matrices of M8. This vector has dimension of mass. This basic parameter should characterize many different physical situations. Consider only the description of massivation of elementary particles regarded as bound states of fundamental massless fermions and the mixing of left and right-handed fermions. Also CKM mixing should involve this parameter. These vectors should also appear in Higgs couplings, which in QFT description contain Higgs vacuum expectation as a factor.
In twistor approach virtual particles have complex light-like momenta. Fundamental fermions have most naturally real and light-like momenta. N=4 SUSY describes gauge bosons which correspond to bound states of fundamental fermions in TGD. This suggests that the four-momenta of bound states of massless fermions - be they hadrons, leptons, or gauge bosons - can be taken to be complex.
There is an intriguing connection with TGD based notion of space-time. In TGD one obtains at space-time level complexified four-momenta since the four-momentum from Minkowskian/ Euclidian region is real/imaginary. In the case of physical particle necessary involving two wormhole contacts and two flux tubes connecting them the total complexifies four momentum would be sum of two real and two imaginary contributions. Every elementary particle should have also imaginary part in its four-momentum and would be massless in complexified sense allowing mass in real sense given by the length of the imaginary four-momentum.
TGD predicts Higgs field although Higgs expectation does not have any role in quantum TGD proper. Higgs vacuum expectation is however a necessary part of QFT limit (Higgs decays to WW pairs require that vacuum expectation is non-vanishing). Higgs vacuum expectation must correspond in TGD framework to a quantity with dimensions of mass. In TGD Higgs cannot be scalar but a vector in CP2 degrees of freedom. The problem is that CP2 does not allow covariantly constant vectors. The imaginary part of classical four-momentum gives a parameter which has interpretation as a vector in the tangent space of which is same as that of M4× CP2. Could M8-H duality be realized at the level of tangent space and for relate four-momentum and color quantum numbers to 8-momentum?
Elementary particles of course need not be eigenstates of the imaginary part of four-momentum. For a fixed mass one can have wave functions in the space of imaginary four-momentum analogous to S3 spherical harmonics at the sphere of E4 with radius defined by the length of imaginary four-momentum (mass). These harmonics are characterized by SO(4) quantum numbers. Could one interpret this complexification in terms of M8-H duality and say that SO(4) defines the symmetries for the low energy dual of WCW defining high energy description of QCD based on SU(3) symmetry. SO(4) would correspond to the symmetry group assigned to hadrons in the approach based on conserved vector currents and partially conserved axial currents. SO(4) would be much more general and associated also with leptons.
The anomalous color hyper-charge of leptonic spinors would imply that one can have also in the case of leptons a wave function in S3. Higher harmonics would correspond to color excitations of leptons and quarks. If one considers gamma matrices, complexification of M4 means introduction of gamma matrix algebra of complexified M4 requiring 8 gamma matrices. This suggests a connection with M8-H duality. All elementary particles have also imaginary part of four-momentum and the 8-momentum can be interpreted as M8-momentum combining the four-momentum and color quantum numbers together.
Supersymmetry is very beautiful generalization of the ordinary symmetry concept by generalizing Lie-algebra by allowing grading such that ordinary Lie algebra generators are accompanied by super-generators transforming in some representation of the Lie algebra for which Lie-algebra commutators are replaced with anti-commutators. In the case of Poincare group the super-generators would transform like spinors. Clifford algebras are actually super-algebras. Gamma matrices anti-commute to metric tensor and transform like vectors under the vielbein group (SO(n) in Euclidian signature). In supersymmetric gauge theories one introduced super translations anti-commuting to ordinary translations.
Supersymmetry algebras defined in this manner are characterized by the number of super-generators and in the simplest situation their number is one: one speaks about N=1 SUSY and minimal super-symmetric extension of standard model (MSSM) in this case. These models are most studied because they are the simplest ones. They have however the strange property that the spinors generating SUSY are Majorana spinors- real in well-defined sense unlike Dirac spinors. This implies that fermion number is conserved only modulo two: this has not been observed experimentally. A second problem is that the proposed mechanisms for the breaking of SUSY do not look feasible.
LHC results suggest MSSM does not become visible at LHC energies. This does not exclude more complex scenarios hiding simplest N=1 to higher energies but the number of real believers is decreasing. Something is definitely wrong and one must be ready to consider more complex options or totally new view abot SUSY.
What is the situation in TGD? Here I must admit that I am still fighting to gain understanding of SUSY in TGD framework. That I can still imagine several scenarios shows that I have not yet completely understood the problem and am working hardly to avoid falling to the sin of sloppying myself. In the following I summarize the situation as it seems just now.
Could covariantly constant right handed neutrinos generate SUSY?
Could covariantly constant right-handed spinors generate exact N=2 SUSY? There are two spin directions for them meaning the analog N=2 Poincare SUSY. Could these spin directions correspond to right-handed neutrino and antineutrino. This SUSY would not look like Poincare SUSY for which anticommutator of super generators would be proportional to four-momentum. The problem is that four-momentum vanishes for covariantly constant spinors! Does this mean that the sparticles generated by covariantly constant νR are zero norm states and represent super gauge degrees of freedom? This might well be the case although I have considered also alternative scenarios.
What about non-covariantly constant right-handed neutrinos?
Both imbedding space spinor harmonics and the modified Dirac equation have also right-handed neutrino spinor modes not constant in M4. If these are responsible for SUSY then SUSY is broken.
What one can say about the masses of sparticles?
The simplest form of massivation would be that all members of the super- multiplet obey the same mass formula but that the p-adic length scales associated with them are different. This could allow very heavy sparticles. What fixes the p-adic mass scales of sparticles? If this scale is CP2 mass scale SUSY would be experimentally unreachable. The estimate below does not support this option.
One can even consider the possibility that SUSY breaking makes sparticles unstable against phase transition to their dark variants with heff =n× h. Sparticles could have same mass but be non-observable as dark matter not appearing in same vertices as ordinary matter! Geometrically the addition of right-handed neutrino to the state would induce many-sheeted covering in this case with right handed neutrino perhaps associated with different space-time sheet of the covering.
This idea need not be so outlandish at it looks first.
The mixing of right- and left handed neutrinos would be the basic mechanism in the decays of sfermions. The mixing mechanism is mystery in standard model framework but in TGD it is implied by both induced and modified gamma matrices. The following argument tries to capture what is essential in this process.
See the chapter Does the QFT Limit of TGD Have Space-Time Super-Symmetry? or the article What went wrong with symmetries?.
The vanishing of conformal charges as a gauge conditions selecting preferred extremals of Kähler action
Classical TGD involves several key questions waiting for clearcut answers.
While answering the questions I made what I immediately dare to call a breakthrough discovery in the mathematical understanding of TGD. To put it concisely: one can assume that the variations at the light-like boundaries of CD vanish for all conformal variations which are not isometries. For isometries the contributions from the ends of CD cancel each other so that the corresponding variations need not vanish separately at boundaries of CD! This is extremely simple and profound fact. This would be nothing but the realisation of the analogs of conformal symmetries classically and give precise content for the notion of preferred external, Bohr orbitology, and strong form of holography. And the condition makes sense only in ZEO!
I attach below the answers to the questions of Hamed almost as such apart from slight editing and little additions, re-organization, and correction of typos.
The physical interpretation of the canonical momentum current
Hamed asked about the physical meaning of Tnk== ∂ L/∂(∂n hk) - normal components of canonical momentum labelled by the label k of imbedding space coordinates - it is good to start from the physical meaning of a more general vector field
Tαk == ∂ L/∂(∂α hk)
with both imbedding space indices k and space-time indices α - canonical momentum currents. L refers to Kähler action.
Note that in TGD field equations reduce to the conservation of isometry currents as in hydrodynamics where basic equations are just conservation laws.
The basic steps in the derivation of field equations
First a general recipe for deriving field equations from Kähler action - or any action as a matter of fact.
Tαk ∂αδ hk + (∂ L/∂ hk) δ hk .
The latter term comes only from the dependence of the imbedding space metric and Kähler form on imbedding space coordinates. One can use a simple trick. Assume that they do not depend at all on imbedding space coordinates, derive field equations, and replaced partial derivatives by covariant derivatives at the end. Covariant derivative means covariance with respect to both space-time and imbedding space vector indices for the tensorial quantities involved. The trick works because imbedding space metric and Kähler form are covariantly constant quantities.
The integral of Tαk ∂αδ hk decomposes to two parts.
In the sequel the boundary terms are discussed explicitly and it will be found that their treatment indeed involves highly non-trivial physics.
Boundary conditions at boundaries of CD
In positive energy ontology one would formulate boundary conditions as initial conditions by fixing both the 3-surface and associated canonical momentum densities at either end of CD (positions and momenta of particles in mechanics). This would bring asymmetry between boundaries of CD.
In TGD framework one must carefully consider the boundary conditions at the boundaries of CDs. What is clear that the time-like boundary contributions from the boundaries of CD to the variation must vanish.
Needless to say, the modification of this approach could make sense also at partonic orbits.
Isometry charges are complex
One must be careful also at the light-like 3-surfaces (orbits of wormhole throats) at which the induced metric changes its signature.
Boundary conditions at the wormhole throat orbits and connection with quantum criticality and hierarchy of Planck constants defining dark matter hierarchy
The contributions from the orbits of wormhole throats are singular since the contravariant form of the induced metric develops components which are infinite (det(g4)=0). The contributions are real at Euclidian side of throat orbit and imaginary at the Minkowskian side so that they must be treated as independently.
As should have become clear, the derivation of field equations in TGD framework is not just an application of a formal recipe as in field theories and a lot of non-trivial physics is involved!
See the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article The vanishing of conformal charges as a gauge conditions selecting preferred extremals of Kähler action.
Lubos wrote a commentary about a new article of Nima Arkani Hamed et al about the positivity of the amplitudes in in the amplituhedron. Nima et al argue using explicit calculations that positivity could be a quite general property: as if the amplitudes were analogous to generalized volumes of higher-dimensional polyhedra.
The Grassmannian twistor approach considers also positive Grassmannians. This is not the same thing but most be closely related to the positivity of amplitudes. I recall that the core idea is however that one considers higher-D analogs of polygons. Simple representative for positive space polygon would be a triangle bounded by positive x- and y-axis and the line y=-x. x and y coordinates are positive.
What is of course certain is that the entire scattering amplitude cannot be positive since it is complex number. Rather, it must decompose into a product of "trivial" part determined by symmetries and non-trivial part which for some reason must be positive. What does this mean? Lubos considers this question in his posting: the idea is roughly that the real amplitude in question is an exponential and this guarantees positivity. Bundle theorist might speak about global everywhere non-vanishing section of vector bundle having geometric and topological meaning in algebraic geometry. Below I try to interpret positivity in terms of number theoretic arguments.
Does number theoretical universality require positivity?
Number theoretical universality of physics is one of the key principles of TGD and states that real physics must allow algebraic continuation to p-adic physics and vice versa. I suggest that number theoretical universality requires the positivity.
How should one p-adicize?
The p-adicization of various spaces and amplitudes is needed. p-Adicization means that one assigns to a real (or complex) number a p-adic variant by some rule. In the case of trigonometric and hyperbolic angles one can use discretization and algebraic extension but what about other kinds of coordinates? There are two guesses.
p-Adic space-time surfaces as cognitive maps of real ones and real space-time surfaces as correlates for realized intentions
One wants to talk about real topological invariants also in p-adic context: p-adic space-time surface should be a kind of cognitive representation of real space-time surface.
To sum up, number theoretic universality condition for scattering amplitudes could help to understand the success of twistor Grassmann approach. The existence of p-adic variants of the amplitudes in finite algebraic extensions is a powerful constraint and I have argued that they are satisfied for the polylogarithms used. Positive Grassmannians and positivity of the amplitudes might be also seen as a manner to satisfy these constraints.
For background see the chapter Some fresh ideas about twistorialization of TGD or the article Positivity of N=4 scattering amplitudes from number theoretical universality?.