N=4 SCA as basic symmetry of TGD and the basic mistake of M-theory

N=4 super-conformal algebra (SCA) emerges naturally in TGD framework and is basically due to the covariantly constant right handed neutrinos and super Kac-Moody and super-canonical algebras defining the generalized coset representations. N=4 SCA is the maximal associative SCA and has an interpretation in terms of super-affinization of a complexified quaternion algebra. The (4,4) signature of target space metric characterizing N=4 SCA topological field theory is not a problem since in TGD framework the target space becomes a fictive concept defined by the Cartan algebra. This picture allows to assign the critical dimensions of super string models and M-theory with the fictive target spaces associated with the vertex operator construction.

N=4 super-conformal topological field theory defines in TGD framework a non-trivial physical theory since classical interactions induce correlations between partonic 2-surfaces and CP₂ type extremals provide a space-time correlate for virtual particles. Both M⁴× CP₂ decomposition of the imbedding space and space-time dimension are crucial for the 2+2+2+2 structure of the Cartan algebra, which together with the notion of the configuration space guarantees N=4 super-conformal invariance. Therefore it is not exaggeration to say that the basic structure of TGD is uniquely fixed also by N=4 super-conformal invariance. In the following the interpretation of the critical dimension and critical signature of metric is discussed in some detail.

The basic problem is that the signature of the induced space-time metric cannot be (2,2) which is essential for obtaining the cancellation for N=2 SCA imbedded to N=4 SCA with critical dimension D=8 and signature (4,4). Neither can the metric of imbedding space correspond to the signature (4,4). The (4,4) signature of the target space metric is not so serious limitation as it looks if one is ready to consider the target space appearing in the calculation of N-point functions as a fictive notion.

The resolution of the problems relies on two observations.

1. The super Kac-Moody and super-canonical Cartan algebras have dimension D=2 in both M⁴=M²×E² and CP₂ degrees of freedom giving total effective dimension D=2+2+2+2=8.
   • Super Kac-Moody algebra acts as deformations of partonic 2-surfaces X². It consists of supersymmetrized X²-local E² translations completely analogous to transversal deformations of string in M⁴ and supersymmetrized electro-weak Kac-Moody algebra U(2)_ew acting on quantum variants of spinors identifiable as super-counterparts of super-symmetrized complexified quaternions. Both Cartan algebras have obviously dimension D=2.

   • Super-canonical algebra acts as deformations of 3-surfaces and is generated by Hamiltonians in δ M⁴_+/-× CP₂ with degenerate symplectic and complex structures made possible by the metric 2-dimensionality of the boundary of the four-dimensional light-cone. It consists of canonical transformations of CP₂ and (with a suitable choice of gauge) of the canonical transformations of the tangent plane E² of sphere having origin at the tip of δ M⁴_+/-. Also now both Cartan algebras have dimensions D=2.

2. The generalized coset construction discussed in detailhere allows to assign opposite signatures of metric to super Kac-Moody Cartan algebra and corresponding super-canonical Cartan algebra so that the desired signature (4,4) results. Altogether one has 8-D effective target space with signature (4,4) characterizing N=4 super-conformal topological strings. Hence the number of physical degrees of freedom is D_phys=8 as in super-string theory. Including the non-physical M² degrees of freedom, one has critical dimension D=10. If also the radial degree of freedom associated with δ M⁴_+/- is taken into account, one obtains D=11 as in M-theory.

From TGD point view perhaps the most fatal blunder of super-string approach and M-theory is the identification of the flat target space defined by the Cartan algebra in vertex operator construction with the physical space-time. The attempt to feed some physics into the obviously unphysical theory leads to the ad hoc notion of spontaneous compactification. Branes represent a further desperate attempt to get something sensible out of the theory. This in turn leads to landscape and to the anthropic principle as the last straw to save the theory (or at least its continual funding;-)).

The last section of chapter Construction of Quantum Theory of "Towards S-Matrix" represents detailed form of the argument.

Jones inclusions, the large N limit of SU(N) gauge theories, and AdS/CFT correspondence

The framework based on Jones inclusions has obvious resemblance with larger N limit of SU(N) gauge theories and also with the celebrated AdS/CFT correspondence so that a more detailed comparison is in order.

1. Large N limit of gauge theories and series of Jones inclusions

The large N limit of SU(N) gauge field theories has as definite resemblance with the series of Jones inclusions with the integer n≥ 3 characterizing the quantum phase q=exp(iπ/n) and the order of the maximal cyclic subgroup of the subgroup of SU(2) defining the inclusion. Recall that all ADE groups except D_2n+1 and E₇ are allowed (SU(2) is excluded since it would correspond to n=2).

The limiting procedure keeps the value of g²N fixed. Rather remarkably, this is equivalent with keeping α N constant but assuming hbar to scale as n=N. Thus the quantization of Planck constants would provide a physical laboratory for the testing of large N limit.

The observation suggesting a description of YM theories in terms of closed strings is that Feynman diagrams can be interpreted as being imbedded at closed 2-surfaces of minimal genus guaranteing that the internal lines meet except in vertices. The contribution of genus g diagrams is proportional to N^g-1 at the large N limit. The interpretation in terms of closed partonic 2-surfaces is highly suggestive and the N^g-1 should come from the multiple covering property of CP₂ by N M⁴-points (or vice versa) with the finite subgroup of G of SU(2) defining the Jones inclusion and acting as symmetries of the surface.

2. Analogy between stacks of branes and multiple coverings of M⁴ and CP₂

An important aspect of AdS/CFT dualities is a prediction of an infinite hierarchy of gauge groups, which as such is as interesting as the claimed dualities. The prediction relies on the notion Dp-branes. Dp-branes are p+1-dimensional surfaces of the target space at which the ends of open strings can end. In the simplest situation one considers N parallel p-branes at the limit when the distances between branes characterized by an expectation value of Higgs fields approach zero to obtain what is called N-stack of branes. There are N² different strings connecting the branes and the heuristic idea is that they correspond to gauge bosons of U(N) gauge theory. Note that the requirement that AdS/CFT dualities exist forces the introduction of branes and the optimistic interpretation is that a non-perturbative effect of still unknown M-theory is in question. In the limit of an ideal stack one assumes that U(N) gauge theory at the brane representing the stack is obtained. The branes must also carry a p-form defining gauge potential for a closed p+1-form. This Ramond charge is quantized and its value equals to N.

Consider now the group G_a× G_b in SL(2,C)× SU(2) defining double Jones inclusion and implying the scalings hbar(M⁴)→ n(G_b)× hbar(M⁴) and hbar(CP₂)→ n(G_a)× hbar(CP₂). These space-time surfaces define n(G_a)-fold multiple coverings of CP₂ and n(G_b)-fold multiple coverings of M⁴. In CP₂ degrees of freedom the collection of G_b-related partonic 2-surfaces (/3-surfaces/4-surfaces) is highly analogous to the stack of branes. In M⁴ degrees of freedom the stack of copies of surface typically correspond to along a circle (A_n,D_2n or at vertices of tedrahedron or isosahedron.

In TGD framework strings are not needed to define gauge fields. The group algebra of G realized as discrete plane waves at G-orbit gives rise to representations of G. The hypothesis supported by few examples is that these additional degrees of freedom allow to construct multiplets of the gauge group assignable to the ADE diagram characterizing the inclusion.

The detailed comparison with AdS/CFT corresponds provides non-trivial insights. For instance for G=SU(2) inclusions which correspond to Kac-Moody group associated with extended ADE diagram defining the inclusion, Ramond charge corresponds to nontrivial homology (magnetic charge) in CP₂ or in δ M⁴_+/-.

The last section of chapter Construction of Quantum Theory of "Towards S-Matrix" represents the detailed construction in its recent form.

Confession

Lubos Motl represented his view about science and democracy in his posting Science vs. democracy and in the same occarion continued his campaign against those miserable human beings known as quantum gravity crackpots (all those inhabitants of this planet who do not believe in M-theory). Since I cannot believe in M-theory, I cannot but confess that I am quantum gravity crackpot. To get rid of my horrible feelings of guilt I took the decisive step and made a public confession and glue it also below.

Lubos said:

I think that every reader who has at least a vague idea what physics is must know that such a method of determining the critical dimension as well as the result are completely absurd. M-theory as we know it just can't work in the total number of 8 dimensions.

The number D=8 mentioned here makes me feel terribly guilty of being quantum gravity crackpot.

First confession: I (yes,it is me, I cannot blame anyone else) regard the target space dimension of string theories as a purely formal notion. The construction of stringy vertex operators in terms of ordered exponentials of bosonic free fields brings in formal space-time dimension equal to that of Cartan algebra of the Kac-Moody algebra defining symmetries of the theory. This stringy dimension is roughly the dimension of space-time as it is determined in string theories. And no it comes: I have used almost 28 years of my life to construct a "theory" in which space-times are 4-D surfaces in 8-D space-time with stringy dimension equal to 10 and cannot but continue arguing that D=8 is the physical dimension.

Second confession: This D=8 imbedding space is fixed and non-dynamical apart from the scaling factors of metric but leaves us without landscape.

Third confession: D=4 for space-time surfaces is critical dimension for D=2 super-conformal closed strings with metric signature (2,2) or (0,4): this algebra actually extends to D=4 superconformal algebra which is the symmetry algebra of my 8-D theory. Very conservatively (even condensed matter physicist would agree), the dynamical spacetime surfaces in my own theory have the same role as 10-D dynamical target space in super string models. Hence I believe that gravity is at classical level four-dimensional phenomenon, not 10- or 11-D one.

Fourth confession: I dare to regard also strings as purely formal objects. In my own "theory" particles correspond to 2-D partonic 2-surfaces, submanifolds of space-time surface. N-point functions of conformal field theory at partonic 2-surfaces give stringy scattering amplitudes using the usual formulas. No fundamental strings in the "theory". The 4-D classical dynamics creates correlations between partons and even N=4 topological string could be physically realistic in this framework as a description of perturbative phase.

I have now confessed all that I can in the limitations posed by the limit the length of posting and I feel immense relief. Am I a quantum-gravity-crackpot? Jury can decide.

Jury can find a short summary about the most recent activities of the accused in the last section of chapter Construction of Quantum Theory.

To sum up

During last one and half months the progress in the understanding of the basic mathematical structure of TGD has been really impressibve thanks to the realization of the role of von Neumann algebras and these diaries reflect also the unavoidable side tracks in this process. Hence it is perhaps time to try to give some kind of overall view about what has happened and make confessions about the deviations to the side tracks. At least following crucial steps can be distinguished.

1. TGD emerges from the localization of infinite-dimensional Clifford algebra

The first step of progress was the realization that TGD emerges from the mere idea that a local version of hyper-finite factor of type II₁ represented as an infinite-dimensional Clifford algebra must exist (as analog of say local gauge groups). This implies a connection with the classical number fields. Quantum version of complexified octonions defining the coordinate with respect to which one localizes is unique by its non-associativity allowing to uniquely separate the powers of octonionic coordinate from the associative infinite-dimensional Clifford algebra elements appearing as Taylor coefficients in the expansion of Clifford algebra valued field.

Associativity condition implies the classical and quantum dynamics of TGD. Space-time surfaces are hyper-quaternionic of co-hyper-quatenionic sub-manifolds of hyper-octonionic imbedding space HO. Also the interpretation as a four-surface in H=M₄ emerges and implies HO=H duality. What is also nice that Minkowski spaces correspond to the spectra for the eigenvalues of maximal set of commuting quantium coordinates of suitably defined quantum spaces. Thus Minkowski signature has quantal explanation.

2. Quantization of Planck constants

The geometric and topological interpretation of Jones inclusions led to the understanding of the quantization of Planck constants assignable to M⁴ and CP₂ degrees of freedom (identical in "ground state"). The Planck constants are scaled up by the integer n defining the quantum phase q=exp(iπ/n) characterizing the Jones inclusion, which in turn corresponds to subgroup G of SL(2,C)× SU(2)_L× U(1) in the simplest situation. The quantum phase can be assigned also to q=1 inclusions in which case second quantum phase can be associated with the monodromies of the corresponding conformal field theory.

The scaling of M⁴ Planck constant means scaling of M⁴ metric. The space-time sheets are n(G)-fold coverings of M⁴ by points of CP₂. Analogous statement applies in CP₂ degrees of freedom. CP₂ can therefore have arbitrarily large size: hyper space travel might not be unrealistic after all;-)! For infinite subgroups such as G=SU(2) in SU(3) the situation is somewhat different. The variants of imbedding space can meet each other if either M⁴ or CP₂ factors have same value of Planck constant so that a fan (or rather tree-) like structure results. Analogous picture emerged already earlier from the gluing of the p-adic variants of imbedding space along common rationals (or algebraics in more general case). The phase transitions changing the Planck constant have purely topological description.

An important outcome was the interpretation of McKay correspondence: one can assign to the ADE diagram of q≠1 Jones inclusion the corresponding gauge group. The n(G)-fold covering of M points by a finite number of CP₂ points makes possible to realize the multiplets of gauge group purely geometrically in terms of G group algebra. In the case of extended ADE diagrams assignable to q=1 Jones inclusions the group is Kac-Moody group. This picture applies both in M⁴ and CP₂ degrees of freedom.

1. In CP₂ degrees of freedom this framework allows to understand anyonic charge fractionization and raises the question whether fractional Hall effect corresponds to the integer valued quantum Hall effect with scaled up Planck constant and whether free quarks could be integer charged and have fractional charges only inside hadrons.

2. In M⁴ degrees of fredom this picture has fascinating cosmological consequences and leads to a possible explanation for the quantization of cosmic recession velocities in terms of lattice like structures (tesslations) of lightcone proper time constant hyperboloid defined by infinite subgroups of Lorentz group and consisting of dark matter in macroscopically quantum coherent phase.

3. S-matrix from Connes tensor product, naster formula for S-matrix, and interpretation of quantum measurement in terms of Jones inclusions

The key idea was that S-matrix can be constructed by replacing the tensor product for free fields with Connes tensor product and this tensor product is essentially the product of conformal fields defined at the partonic 2-surfaces and obtained by fusion rules. This means enormous simplification and implies a reduction of S-matrix elements to stringy n-point functions.

The next realization was that physical states are zero energy states generated from vacuum combined with quantum classical correspondences allows to develop a general master formula for S-matrix in which classical physics at space-time level neatly combines with the quantal aspects of TGD. The entanglement between interior space-time degrees of freedom (representing zero modes of the configuration space geometry and classical observables) and 2-D partonic degrees of freedom characterizes the measurement. Hence the effective 2-dimensionality implying generalized super-conformal invariance finds a beautiful physical interpretation in terms of quantum measurement theory.

Classical conserved quantities correspond to the Cartan algebra of commuting quantum observables so that the quantum states at the partonic boundary components dictate the classical state in the interior of the space-time sheet. All quantum measurements give information about a finite number of observables and Jones inclusions allow to describe this: the inclusion N subset M characterizes the degrees of freedom about which the measurement gives no information and the quantum space N/M characterizes the observables. A nice generalization of the notion unitarity and hermiticity in which N plays the role of ring of complex numbers emerges. For instance, the eigenvalues of Hermitian operators are now replaced with Hermitian operators!

The master formula also justifies the earlier discovery that ordinary stringy diagrams cannot describe particle reactions in TGD and that generalizations of Feynman diagrams obtained by glueing space-time sheets together along their ends is the only reasonable option. As a matter fact, glueing is possible only along 2-D partonic surfaces since one must leave the interior degrees of freedom free so that quantum classical correspondences at the level of commuting observables can be realized. This structure is only a fictive notion allowing the representation of S-matrix elements and does not mean that actual space-time surfaces would have this character. In this framework the stringy view about particle reactions is wrong: stringy diagrams would only describe what happens when particle travels simultaneously along several different paths (double slit experiment).

4. Factorizing S-matrices and zero energy ontology

The next step was a partial sidetrack but very fruitful.

1. The idea was that the factorizing S-matrices of integrable 2-D quantum field theories might be enough to construct S-matrix. The number theoretic idea was that quantum field theories restricted to at most 2-D sub-manifolds M² and E² of M⁴ =M² × E² (analogous statement applies in CP₂ degrees of freedom) could define 2-dimensional factorizing S-matrices whose tensor products could be used as basic building blocks of the S-matrix of TGD. The almost triviality of these S-matrices however killed this idea but led to the realization that these S-matrices can be assigned to the scattering zero energy states and the realization that in the construction of S-matrix physical states must be explicitly treated as states of this kind.

2. This led to a detailed articulation of what I call zero energy ontology at quantum level (this ontology has been applied for years in cosmology). The almost non-triviality of factorizing S-matrices turns into a blessing in this framework and allows to understand why our western ontology assuming that the energy density of the universe is non-vanishing, works so well and why the world around us seems to be rather stable rather than being a sequence of uncorrelated flashes of zero energy states. Large values of Planck constant meaning long geometric durations of quantum jumps (roughly temporal distances between positive and negative energy components of the state) are absolutely essential for this. The most important implications relate to a more profound understanding of consciousness theory and the description of intentional actions as p-adic-to-real transitions.

3. In zero energy ontology the most general option does not assume unitarity and replaces it with a "thermal average" of the unitarity conditions for the S-matrix describing the scattering of zero energy states. An essential role is played by the Tr(Id)=1 condition of hyper-finite factors of type II₁. Ordinary S-matrix can be however unitary in hyper-finite sense and would characterize the unitary entanglement between positive and zero energy components of zero energy states being thus basically a property of physical states rather than that of dynamics.

   Jones inclusions N subset M code for different physical measurements. What this means is that subfactor N represents the degrees of freedom which are not measured and the quantum Clifford algebra (space) M/N characterizes the measured degrees of freedom. The measurement leads to a new state represented by a unitary entanglement matrix and even infinite number of measurements fails reduce the state to a one-dimensional ray of state space as in ordinary measurement theory. Physical state is like a hologram.

5. A side track with R-matrices of Kac Moody algebras

The next step was an unsuccessful attempt to understand S-matrix in super-conformal degrees of freedom as a product of R-matrices characterizing the braiding properties of Kac-Moody representations. The resulting R-matrix is either trivial or non-unitary for infinite-dimensional representations of Kac-Moody algebra but unitary for finite-dimensional representations of Kac-Moody algebra having vanishing central extension (factorizing S-matrices correspond to this situation). R-matrix does not possess stringy character as I believed for few days.

This identification of S-matrix could make sense if the M⁴ local Clifford algebra of the configuration space would generate the physical states. In this case the S-matrix would characterize the braiding of fermions associated with the partonic boundary components. The motivation for this was following. In TGD Universe all particles consist of fermions and antifermions so that fermion-antifermion pair cannot disappear to or pop from vacuum: this reincarnation of the Zweig rule would make possible to assign braiding to any zero energy state. The reason why this picture does not work is that physical states are generated by operators analogous to local composites of quantum fields.

6. The reduction of S-matrix to stringy amplitudes of rational/critical conformal field theories

The next step was based on the realization that ordinary stringy amplitudes can be assigned to partonic 2-surfaces having formal interpretation as orbits of closed strings in string models.

1. The construction of zero energy states led to their identification as states N> ×-N> having a vanishing net conformal weight. The state N> is annihilated by all generators L_n, n≥ 0 and this leads to the familiar formulas for Kac determinant characterizing the allowed ground state conformal weights Δ(m,n) for a given value of central extension parameter c in terms of two integers m and m': N= mm' holds true.

2. The requirement of p-adicizability implies that the conformal field theories in question must be rational and thus contain a finite number of primary fields except in exceptional cases. Unitarity and modular invariance pose additional conditions.

3. The discretization of the integral appearing in the standard formulas in terms of vertex operators allows to define the p-adic variants of string scattering amplitudes (p₁→p₂, real-to-padic, etc.. amplitudes). The discrete points correspond to rational points common to real and p-adic partonic 2-surfaces.

4. In the generic case one obtains a finite number of primary fields and there is both IR and UV cutoff on the conformal weights for rational conformal field theories and the integer n≥3 characterizing the Jones inclusion appears explicitly in the formulas for conformal weights and c so that a connection with ADE diagrams and groups emerges. Massless states are absent from these theories. Critical theories which correspond to the q=1 and c=0 or c=1 can contain massless states.

5. The calculation of the amplitudes for generating zero energy states from vacuum reduces to a construction of appropriate conformal field theories defined at partonic 2-surface and to the calculation of N-point functions for these theories. Fictive string picture emerges with target space being defined by the free fields appearing in the ordered exponentials representing various primary fields. The dimension of the Cartan algebra defines the number of transversal dimensions of the imbedding space: it is 8 in TGD and thus same as in M-theory! Super-string folks have not been completely wrong;-)!

7. From N=0 to N=2 to N=4

The basic observation was that closed N=2 super-conformal strings are almost topological and their symmetry algebra extends to the so called small N=4 SCA defining a topological field theory with target space having critical dimension 8 and metric signature (4,4). This discovery was made by Vafa and Berkovits. This raises the hope that in the critical phase corresponding to q=1 Jones inclusion the theory would become exactly solvable and only n<4-point>2 type extremals represents exchanges of virtual particles.

N=4 SCA is the maximal associative SCA. N=4 SCA allows actually several variants but the so called small and large SCA are the most interesting ones from the point of view of TGD. Small N=4 SCA contains SU(2) Kac-Moody algebra and has an interpretation in terms of super-affinization of a complexified quaternion algebra. Large N=4 SCA contains SU(2)₊× SU(2)_-× U(1) Kac-Moody algebra plus super-generators consisting of 4 spin 1/2 spinors.

SU(2)_-× U(1) would have a natural interpretation as electro-weak gauge algebra. The two spin states of covariantly constant right handed neutrino and their charge conjugates allow to realize N=4 super-conformal algebra allowing also an interpretation as quaternionic super-conformal algebra with the associated SU(2)₊ Kac-Moody having interpretation as rotations of quaternion units (spin direction of right handed neutrinos). SU(2)₊ acts also as the isotropy group of the r_M=constant sphere at d M⁴_+/-} defining generalize Kähler and symplectic structures at light-cone boundary. This group can be identified as the isotropy group of the total 4-momentum of the 3-surface containing the partonic 2-surfaces.

N=4 SCA could be an exact symmetry of the leptonic sector of TGD at least in some phases. For quarks fractional charges are expected to spoil this symmetry since covariantly constant quark spinors are not possible. If the solutions of the modified Dirac equation generate super-conformal symmetries then also quark sector could allow at least N=2 super-conformal symmetries.

All possible unitary and rational N=4 and N=2 super-conformal field theories are in principle allowed.

1. For large N=4 SCA there are two Kac-Moody central extension parameters k₊ and k_- and one has c= 6k₊k_-/(k₊+k_-). All positive rational values of c are possible but unitarity probably poses restrictions on the values of c. There are two quantum phases q_+/-=exp(ip/n_+/-) corresponding to n_+/-=k_+/-+2 and they would correspond to Jones inclusions in M⁴ and CP₂ degrees of freedom classified by pairs of ADE diagrams. In these phases Lorentz invariance could be broken for all c? 0 representations: this breaking would have interpretation as an outcome of quantum measurement.

2. c=0 representations result for k₊=0 or k_-=0. These representations correspond in TGD framework to non-trivial Jones inclusion only in the second SU(2) factor. For this representation all states have vanishing Super Virasoro norm and p-adic thermodynamics applies to these states. The number of super Virasoro tensor factors is predicted to be five as required by the p-adic mass calculations.

3. Small N=4 SCA results as a limit k_+/-? 8y from the small SCA and allows only representations with c= 6k. c=6 corresponds to the critical representation with q=1 and G=SU(2) and extended ADE diagrams classify these inclusions. c=6 corresponds to a topological string theory but defines in TGD framework a non-trivial physical theory since classical interactions induce correlations between partonic 2-surfaces and CP₂ type extremals provide a space-time correlate for virtual particles.

4. For N=2 super-conformal symmetry single central extension parameter k associated with U(1) Kac-Moody algebra classifies the representations and unitary representations result for c= 3k/(k+2). The interpretation in terms of Jones inclusions with n=k+2 is possible. The critical theory with c=3 corresponds to G=SU(2) for Jones inclusion.

8. Questions

A priori one can consider 3 different options concerning the identification of quarks and leptons.

1. Could also quarks define $N=4$ superconformal symmetry?

One can ask, whether the construction could be extended by allowing H-spinors of opposite chirality to have leptonic quantum numbers so that free quarks would have integer charge. The construction does not work. The direct sum of N=4 SCAs can be realized but N=8 algebra would require SO(7) rotations mixing states with different fermion numbers: for N=4 SCA this is not needed. Furthermore, only N=4 super-conformal algebras allow an associative realization and N=8 non-associative realization discovered first by Englert exists only at the limit when Kac-Moody central extension parameter k becomes infinite (this corresponds to a critical phase formally and q=1 Jones inclusion). This is not enough for the purposes of TGD and number theoretic vision strongly supports the N=4 restriction.

2. Integer charged leptons and fractionally charged quarks?

Second option would be leptons and fractionally charged quarks with N=4 SCA in leptonic sector. Also quark could give rise to super-conformal algebra if solutions of the modified Dirac equation define generators of SCA (possibly N=2 algebra). It is indeed possible to realize both quark and lepton spinors as super generators of super affinized quaternion algebras (a generalization of super-Kac Moody algebras) so that the fundamental spectrum generating algebra is obtained. Quarks with their natural charges can appear only in n=3,k=1 phase together with fractionally charged leptons. Leptons in this phase would have strong interactions with quarks. The penetration of lepton into hadron would give rise to this kind of situation. Leptons can indeed move in triality 1 states since 3-fold covering of CP₂ points by M⁴ points means that 3 full rotations for the phase angle of CP₂ complex coordinate corresponds to single 2p rotation for M⁴ point.

Hadron like states would correspond to the lowest possible Jones inclusion characterized by n=3 and the sugroup A₂ of SU(2). The work with quantization of Planck constant had already earlier led to the realization that ADE Dynkin diagrams assignable to Jones inclusions indeed correspond to gauge groups: in particular, A₂ corresponds to color group SU(3). Infinite hierarchy of hadron like states with n=3,4,5... quarks or leptons is predicted corresponding to the hierarchy of Jones inclusions, and I have already earlier proposed that this hierarchy should be crucial for the understanding of living matter. For states containing quarks n would be multiple of 3.

One can understand color confinement of quarks as absolute if one accepts the generalization of the notion of imbedding space forced by the quantization of Planck constant. Ordinary gauge bosons come in two varieties depending on whether their couplings are H-vectorial or H-axial. Strong interactions inside hadrons could be also interpreted as H-axial electroweak interactions which have become strong (presumably because corresponding gauge bosons are massless) as is clear from the fact that arbitrary high n-point functions are non-vanishing in the phases with q? 1. Already earlier the so called HO-H duality inspired by the number theoretical vision led to the same proposal but for ordinary electroweak interactions which can be also imagined in the scenario in which only leptons are fundamental fermions.

3. Quarks as fractionally charged leptons?

For the third option only leptons would appear as free fermions. The dramatic prediction would be that quarks would be fractionally charged leptons. It is however not clear whether proton can decay to positron plus something (recall the original erratic interpretation of positron as proton by Dirac!): lepton number fractionization meaning that baryon consists of three positrons with fermion number 1/3 might allow this. If not, then only the interactions mediated by the exchanges of gauge bosons (vanishing lepton number is essential) between worlds corresponding to different Jones inclusions are possible and proton would be stable.

There are however also objections. In particular, the resulting states are not identical with color partial waves assignable to quarks and the nice predictions of p-adic mass calculations for quark and hadron masses might be lost. I really would not like to loose these fruits of labor;-)!

9. Conclusion

This is the situation as it is now. Just at this moment I would tend to believe in the original scenario with both leptons and quarks appearing as fundamental particles but I cannot predict what I believe tomorrow. Of course I know that details cannot be never precisely correct and with a deep frustration I must admit that at this age I cannot hope of being ever able to cope with the horribly technical computational machinery of conformal field theories. What is however clear that everything is now ready for a collective effort making possible to deduce the predictions of quantum TGD in full detail.

What I have done during these almost 28 years (perfect number;-)) is however not total trivia. For instance, I have demonstrated that TGD

1. is consistent with everything that I have learned about physics between standard model length scales and cosmology,
2. makes an impressive number of detailed predictions (say p-adic mass calculations),
3. explains a long list of existing anomalies, in particular provides an explanation for dark matter as macroscopic quantum phases with values of large Planck constants and predicts it to be crucial for understanding of living matter,
4. implies a totally new vision about various aspects of reality including the nature of conscious experience.

The last section of the chapter Construction of Quantum Theory: Symmetries and the new chapter Construction of Quantum Theory: S-matrix of the book "Towards S-matrix" represents the detailed construction in its recent form.

From N=0 to N=2 to N=4

Below a somewhat hastily written letter that I sent to Istvan and Carlos. I have done some editing of typos.

The development have been really fast: super-conformal N has increased in two units per day during the last three days!

1. N=0 meant the realization that rational conformal field theories and c=1 and c=0 theories would give stringy representations for scattering amplitudes.

2. The basic observation of Vafa and Berkovitz is that critical N=2 super-conformal theories extend to N=4 super-conformal theories at criticality. N=2 super-conformal symmetry is implied in leptonic sector by complex covariantly constant right handed neutrinos: quarks are problem. For N=2 critical theories all n-particle scattering amplitudes vanish for particle number n>3 and this has extremely nice interpretation in TGD framework although the result cannot hold true for c≤3 phases described by rational N=2 super-conformal field theories.

3. N=4 super-conformal algebras in turn allow an exceptional variant found by Rasmussen for five years ago: this I learned today. The N=4 super-conformal algebra of Rasmussen contains besides Virasoro, 4 super generators G, U(2) Kac-Moody, the counterparts of right handed neutrino spinors of both H=M⁴× CP₂-chirality, and scalar which cannot however correspond to Higgs as I noticed later. This is a perfect fit in TGD framework. Stringy amplitudes with this symmetry algebra would allow to calculate the predictions of TGD and one expects enormous symmetries in critical phase.

N=4 superconformal symmetry would however mean that both M⁴×CP₂ spinor chiralities would carry leptonic charge assignments.

1. Both quarks and leptons could exist in integer charged free phase and in fractionally charged anyonic phases with quantum phase q different from one.

2. Usual fractional quark charges would correspond to the lowest possible Jones inclusion for which fermion number is fractionized to 1/n= 1/3 and em charge becomes fractional and color corresponds to 3-fold covering of M⁴ points by three CP₂ points.

3. Both leptons and quarks could form hadron like states and n could have all values n≥ 3 meaning entire hierarchy of color confined states with various SU(n):s and other ADE groups. The extremely beautiful topological understanding of color confinement using the generalized view about imbedding space is perhaps the best justificication for this option.

The basic hypothesis hitherto has been that different H-chiralities (quarks and leptons) correspond to fractional and integer em charges. The ontological question is whether I accept that both quarks and leptons can exist in integer charged phase and infinite number of anyonic phases with fractional charges proportional to 1/n, n=3,4,5,.... The quantum biological model already assumes large values of n coming as powers of 2¹¹ for leptons so that internal consistency favors this picture. And most theoreticians would say that it would be idiotic to break these gigantic symmetries just because of standard model based prejudices about what can exist. Even more so after I have populated the universe with quantum coherent dark matter with varying values of hbar!

I have been trying to find whether this interpretation is consistent with the existing wisdom.

1. One should understand why the free integer charged quarks interact so weakly with ordinary leptons. This is easy to understand: as a matter fact, gauge boson exchanges and even graviton exchange would interfere to zero if H-axial and vectorial bosons have same masses and couplings (already for 10 years ago I realized that gauge bosons can couple to quarks and leptons with same or opposite -signs so that one as H-axial or H-vectorial bosons and the question was why only vectorial would be visible).

2. Also the interactions of leptons with genuine quarks in fractionally charged phase come out correctly under very simple assumption.

3. If H-axial (say) vector bosons inside hadrons are massless or light, they could give rise to strong interactions so that the explanation for why isospin and hypercharge have interpretation both as weak and strong isospin would emerge. This I realized for a couple years ago but had not yet realized the possibility of H-axial and -vectorial electroweak gauge bosons.

4. According to the earlier dualities this description of strong interactions would be dual to a description in terms of SU(3) gluon exchanges.

Putting all this together, I feel that I have right to say that it is more or less done now. The rest is just filling up the details and getting the results to general awareness. The technical side is not problem anymore.

The last section of chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is just now.

Finally?

The development of the ideas related to the S-matrix has proceeded essentially by enthusiastic trial and disappointing error, the best scientific method that exists. It however seems that the big picture is now clear and stringy form for the S-matrix, or rather amplitudes for the generation of zero energy states from vacuum emerges although the interpretation is quite different as compared to that in string models.

Here I list just the main points.

1. The concrete application of Virasoro conditions on zero energy states combined with the requirement that the conformal parameters are rational numbers (required by p-adicization) implies that rational conformal field theories characterize the particle multiplets.

2. Massless sector cannot be understood in this manner since the conformal weights of all states are non-vanishing. Massless states would naturally correspond to c=0 representation without breaking of conformal symmetry and all states being null norm states with respect to the Virasoro norm. p-Adic thermodynamics determining mass squared values could be interpreted as thermodynamics describing massless zero energy states with massive ones.

3. Finite number for primary fields conforms with the vision that Jones inclusions N subset of M reduce the number of state space degrees of freedom to a finite number. An essential assumption is quantum criticality which allows to assign to physical states conformal fields expressible in terms of ordered exponentials of string fields.

4. Under very general conditions the amplitides for the generation of zero energy states from vacuum reduce to Lorentz invariant stringy amplitudes defined at partonic 2-surfaces taking the role of Euclidianized stringy world sheet. The reason is that the fields of minimal models are expressible in terms of ordered exponentials of stringy fields plus additional factors taking care of internal quantum numbers.

5. The analog of stringy perturbation theory could follow from the unitarity condition for the entanglement matrix between positive and negative energy states playing the role of S-matrix.

To sum up, just at this moment I cannot see any inconsistencies in the picture. But so I have believed also in previous summaries of the situation so that do not take me too seriously;-)!

The chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is just now.

An email to Carlos

Below an email response to Carlos Castro. Actually a detypoed, edited, and decorated version of the original.

Dear Carlos, thank you for telling about the journal.

The enthusiastic burst that I sent to you was probably still quite far away from the recent situation. Many painful drawbacks have occurred during last two weeks but the situation seems to have settled down now.

It turned out that 2-D factorizable S-matrices cannot give rise to realistic S-matrices in ordinary sense. The are however tailor made for describing the scattering of zero energy states (all states in TGD are predicted to possess vanishing net conserved charges: only now I realized that this must be the starting point in the construction of quantum states and S-matrix). The almost triviality of this S-matrix guarantees almost equivalence of zero energy ontology with the ordinary one although there are important and fascinating differences too. One nice thing is that zero energy states can be localized to a finite portion of Minkowski space and one can assign to them densities,etc. This allows also to replace the horribly ugly routed from S-matrix to scattering rates with a real understanding.

Amusingly, this Buddhistic ontology (what you perceive represents different forms of emptiness) is also consistent with the creation and doomsday myths of religions. That perceived world looks stable continuum rather than like sequence of uncorrelated little bangs followed by little crunches tells that the temporal distances between positive and negative energy components are much longer than the time scale of human perception: dark matter and large hbars. This of course applies only on what we can sensorily perceive.

Zero energy states could code for the counterpart of ordinary S-matrix as entanglement coefficients between negative and positive energy components of the states. This is however not necessary if one accepts that theory predicts only scattering probabilities and unitarity results as a kind of thermal average. This kind of coding of S-matrix is already familiar from spherically symmetric potential scattering in wave mechanics. It is absolutely crucial that the condition Tr(S^†S)= Tr(Id)=1 (instead of infinity) applies in the case of hyperfinite type II₁ factors and implies that unitarity condition can be interpreted as a normalization of states formed by entangling negative energy states with positive energy states.

Quantum measurement can induce projections to subfactors of II₁ factor M for both positive and negative energy factors but the result is always a state with S-matrix entangled state in sub-factor so that it there is no hope of achieving a one-dimensional ray in the state space. Quantum state is like a hologram representing quantum dynamics in its structure and containing infinite amounts of information.

Super Virasoro and Super Kac-Moody conditions for zero energy states are extremely natural and dictate the entanglement coefficients in super conformal degrees of freedom to a high degree. k and c for positive and negative energy states are of opposite sign and compensate each other so that k=c=0 results. The entanglement matrix turns out to be a generalized braiding matrix determined by the universal R-matrix associated with the Super Kac Moody algebras: braiding operation emerges naturally in the calculation of the amplitude for creating zero energy state from Fock vacuum. The product of R-matrices describing the braiding emerges as one moves an operator creating negative energy fermion past negative and positive energy fermions until it anticommutes to c-number with some positive energy fermion. This R-matrix is highly analogous to the exponent of L₀ but more complicated so that stringy picture emerges naturally for the scattering amplitudes.

An absolutely essential point is that in TGD framework all particles consist of fermions. Therefore fermion and antifermion numbers are separately conserved although they can arrange themselves to fermions and bosons, even Higgs particles with fermions associated with the throats of wormhole contact. This makes possible braiding connecting positive energy fermions with negative energy antifermions and R is associated with this braiding and there is no need to generalize braiding by allowing branchings as was the original proposal. The Zweig rule of pre QCD era would be much deeper truth than ever thought.

There remains many things to be understood but I think that TGD S-matrix, or actually an infinite hierarchy (or hierarchies) of S-matrices, is now understood at the level of physical interpretation and what comes to the basic mathematical building blocks. The book of Chari and Pressley turned out to be surprisingly easy to read although at first it looks rather unfriendly acquaintance. I wish I had their mathematical brains.

The chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is now (it could still change!).

Physical states are zero energy states: an ontology consistent with the existing world view?

The factorizing S-matrices seem unavoidably as basic building blocks of S-matrix in TGD framework and it might be that the S-matrix, which depends on von Neumann inclusion characterizing the limitations of quantum measurer, could quite generally reduce to a tensor product of these S-matrices in partonic degrees of freedom. The basic problem has been the fact that these S-matrices are essentially trivial as far as scattering in momentum degrees of freedom is considered. After many wrong guesses an incredibly simple solution to the puzzle emerged and the almost triviality of factorizing S-matrices transformed from a curse to blessing.

1. Zero energy states as the ultimate building block of matter

The idea is to modify the original vision about particle reactions somewhat and in complete consistency with the prediction that in TGD Universe all quantum states have vanishing total quantum numbers. Instead of thinking scattering of positive energy particles as creation of zero energy state from vacuum, the original vision, one considers scattering of zero energy states. I have proposed already earlier that this scattering and also higher level scatterings occur but that this process corresponds to higher level process in the cognitive hierarchy than the scattering that we detect in laboratory. What is nice is that one can deduce scattering rates for the illusory scattering of positive energy particles from this higher level S-matrix as thermal expectation values in a sense von Neumann would have defined them. One could of course assume that also positive energy states are there but scattering for them would be rather trivial and they would not correspond to observed particles.

In the new rather Buddhistic ontology zero energy states are identified as experienced events and objective reality in the conventional sense becomes only an illusion. Before the new view can be taken seriously one must demonstrate how the illusion about positive energy reality is created and why it is so stable.

1. The very fact that the factorizing S-matrices are trivial apart from the changes in the internal degrees of freedom means that the event pairs are extremely stable once they are generated (how they are generated is an unavoidable question to be addressed below). Infinite sequences of transition between states with same positive energies and same initial energies occur. What is nice that this makes it possible to test the predictions of the theory by experiencing the transition again and again.

2. Statistical physics becomes statistical physics for an an ensemble consisting of zero energy states |m₊n_-> including also their time reversals |n₊,m_->. In the usual kinetics one deduces equilibrium values for various particle densities as ratios for the rates for transitions m₊→ n₊ and their reversals n₊→ m₊ so that the densities are given by n(n₊)/n(m₊)= ∑_n+Γ(m₊→ n₊)/∑_n+Γ/(n₊→ m₊). In the recent situation the same formula can be used to define the particle number densities in kinetic equilibrium using the proposed identification of the transition probabilities.

3. Because of the stability of the zero energy states, one can construct many particle systems consisting of zero energy states and can speak about the density of zero energy states per volume. Also the densities n_+,i (n_-,i) of initial (final) states of given type can be defined and densitites of positive energy states can be identified as densities assignable to the ordinary matter. Also densities of particles contained by these states can be defined. It would seem that the new ontology can reproduce the standard ontology as something which is not necessary but to which we are accustomed and which does not produce too much harm.

4. The sequence of quantum jumps between negative energy states defines also a sequence between initial (final) states of quantum jumps and, as far as momentum and color degrees of freedom are considered, this sequence represents rather immutable reality if S-matrix is factorizing.

2. How does the quantum measurement theory generalize?

There are also important questions related to the quantum measurement theory. The zero modes associated with the interior degrees of freedom of space-time surface represent classical observables entangled with partonic observables and this entanglement is reduced in quantum jump. Negentropy Maximization Principle is the TGD based proposal for the variational principle governing the statistical dynamics of quantum jumps. NMP states that entanglement negentropy tends to be maximized in the reduction of entanglement. Number theoretic variants of Shannon entropy making sense for rationally or even algebraically entangled states can be positive so that NMP can also lead to generation of this kind of entanglement and gives rise to a highly stable bound state entanglement.

Does this picture generalize to the new framework in which zero energy states become physical states? Factorizing S-matrices describe partonic dynamics and should be responsible for generating entanglement in the partonic degrees of freedom. One should understand also the S-matrix generating entanglement between zero modes and partonic degrees of freedom and quantum classical correspondence is the only guideline in the recent situation.

3. Understanding quantum computation in the new ontology?

The understanding of what really happens in quantum computation, in particular topological quantum computation, is a challenge for the recent framework since the theory of quantum computation relies heavily on the Hamiltonian time evolution which cannot be an exact description in the new ontology. The basic element is entanglement between positive and negative energy states. It is generated by time evolution in standard framework whereas in the recent framework the creation of the quantum computer program and its realization reduces to the creation of a zero energy state realizing this entanglement. Note that also entanglement between positive energy states can be used for quantum computational purposes.

The problem is obvious: the creation of quantum computer program requires a creation of a zero energy state realizing the program. Can one allow quantum jumps creating zero energy states representing the desired program? The extreme stability of the zero energy states against evolution defined by a factorizing S-matrix does not allow the popping up of zero energy states from vacuum since four-momenta are in this case vanishing. Must be accept that we are passive spectators who just observe the already existing zero energy states representing quantum computer programs as we drift towards geometric future along larger space-time sheet?

It seems that this is not necessary. p-Adic physics as a physics of intentionality and cognition however suggests how the obstacle could be overcome at the level of principle. For zero energy states, p-adic-to-real transitions and vice versa are in principle possible and I have in fact proposed a general quantum model for how intentions might be transformed to actions in this manner. In the second direction the process corresponds to formation of cognitive representation of a zero energy physical state.

In the degrees of freedom corresponding to configuration space spinors situation is very much like for reals. Rational, and more generally algebraic number based physics applies in both cases. p-Adic space-time sheets however differ dramatically from their real counterparts since they have only rational (algebraic) points in common with real space-time sheets and p-adic transcendentals are infinite as real numbers. The S-matrix elements for p-adic-to-real transitions can be formulated using n-point functions restricted to these rational points common to matter and mind stuff. If this picture is not terribly wrong, it would be possible to generate zero energy states from vacuum and the construction of quantum computer programs would be basically a long and tedious process involving very many intentional acts.

One can of course, make a further question. What about the generation of intentions: can p-adic space-time sheets and quantum numbers pop up spontaneously from vacuum? What kind of p-adic space-time sheets and quantum numbers assignable to their partonic 2-surfaces can do so spontaneously?

Here an interesting aspect of the p-adic conservation laws passes a helping hand. p-Adic integration constants are pseudo constants in the sense that a quantity having vanishing (say) time derivative can depend on a finite number of pinary digits t_n of the time coordinate t=∑_nt_np^-n. Could one think that quantum jumps can generate from vacuum exact vacuum states as vacuum tensor factors of the configuration space spinor, and that in subsequent quantum jumps factorizing p-adic S-matrix conserving quantum numbers only in p-adic sense transforms this state into a non-trivial zero energy state which then transforms to real state in intentional action? Note that if conserved quantum numbers are integers they are automatically pseudo constants. p-Adic conservation laws could allow also the p-adic zero energy states to pop up directly from vacuum.

Real-to-p-adic transitions would represent transformation of reality to cognition and would be also possible and mean destruction of zero energy states universe. The characteristic and perhaps the defining feature of living matter would be its highly developed ability to reconstruct reality by performing p-adic-to-real transitions and their reversals.

The chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is now (it could change!).

S-matrix for the scattering of zero energy states representing ordinary scattering events as a solution to the problems?

The properties of factorizing S-matrices are extremely beautiful but it seems that we have been trying to make sense of them in a wrong interpretational framework. The proper understanding of the situation might require a radically new interpretation relying on the idea that particle reactions correspond to creation of negative energy states from vacuum in TGD framework. This idea has not been used in any manner hitherto.

1. Fundamental scattering as a scattering zero energy states representing particle reactions

In TGD framework that initial and final states of particle reaction form zero energy states and it is scattering of these states representing particles reactions that we actually observe in TGD Universe.

1. What is observed are not particles in the initial and final states but particle reactions identified as zero energy states consisting of positive and negative energy particles which we interpret as initial and final states of the particle reaction. This interpretation might well require an appropriate generalization of the notion of S-matrix and to modify the notion of unitarity which must be in some form be still there.

   What comes in mind is the construction of factorizing S-matrix for zero energy states consisting of the particles of initial and finals states having arbitrary momenta. This scattering would describe scattering between zero energy states interpreted in terms of ordinary particle reactions.

2. The basic property of factorizing S-matrices is that they affect only the internal degrees of freedom: therefore the scattering between zero energy states does not affect the initial and final momenta of positive and negative energy states so that the curses of factorizing S-matrices become blessings. The quantum jumps describing scattering makes it possible to experience these reactions consciously and affects only internal degrees of freedom which are not detected.

3. One should be even ready to give up the cherished Lorentz invariance and color symmetries since Jones inclusions associated with the scattering experiment could mean symmetry breaking caused by the selection of subgroups SO(1,1)× SO(2) SO(1)× SO(3), and U(2)subset of SU(3). In the recent picture these choices affect even the geometry and topology of imbedding space and they reflect directly the effect of experimenter to the measurement situation, which simply cannot neglected as is usually done axiomatically. One could also speak about number theoretic breaking o symmetries induced by the requirement that fundamental commutative sub-manifolds of imbedding space are 2-dimensional.

4. One can generalize the construction of factorizing S-matrices without difficulties to the case in which incoming and outgoing states are zero energy states representing particle reactions. Pass-by is possible also for negative and positive energy states if pass by is considered for the projections of rapidities on iπ-&eta_i to real plane. The construction goes through as such for the other tensors. Also the crossing symmetry and other symmetries of factorizing S-matrices generalized in obvious manner.

2. Generalization of unitarity conditions

Consider now the unitarity conditions for the scattering between zero energy states observed as particle reactions. The aim is to derive ordinary unitarity conditions from the unitarity in the scattering of zero energy states as what might be regarded as thermal averages with motivation for averaging coming from the fact that the internal degrees of freedom affected by thescattering between zero energy states are not detected.

The unitarity conditions for the scattering of zero energy states read formally as

∑ _{m+ n-} S_{m+n-→ m + n-} S^* _{r+s- → m+ n-} =δ_m+,r+ δ _n-,s- .

Note that the summed final states are expressed using italic. The sum over the final zero energy states can be also written as a trace for the product of matrices labelled by incoming zero energy states.

Tr(S _m+n- S^*_r+s-) =δ_m+,r+δ_n-,s- .

One can put s_-=n_- on both sides and perform the sum over n_- to get

∑ _n-Tr(S_{m+n- S^*_r+n-) =δ_m+,r+∑_n-δ_n-,n- .

For the factors of type II₁ the sum

∑_n-δ_n-,n- is equal to the trace Tr(Id)=1 of the identity matrix so that one obtains

∑_n-Tr(S_m+n- S^*_r+n-) =δ_m+,r+ .

One could also divide the left hand side by Tr(Id) to get average as one understands it usually.

The usual unitarity condition would read ∑_n-S_{m+n-→ m+n-} S^*_{r+n-→ r+n-} =δ_m+,r+ .

This condition would be replaced with an average over the final states in the scattering described by a factorizing S-matrix.

The interpretation of the result would be as a thermal expectation value of the unitarity condition in the sense of hyper-finite factors of type II₁. This averaging is necessary if we do not have any control over the scattering between zero energy states: this scattering is just a means to become conscious about the existence of the state we usually interpret as change of state. What looks a very non-physical feature of factorizing S-matrices turns into a victory since the trace is only over final states which are characterized by the same collection of momenta and same particle number and uncertainties relate only to the internal degrees of freedom which we cannot measure and whose basic function is to make it possible to consciously perceive the particle reaction as a zero energy state.

3. Should one accept spontaneous breaking of Lorentz symmetry?

The proposed picture does not seem to provide any way to achieve unitarity by summing over all choices of (M²,E²) and (S²,S¹) pairs since in a more general scattering situation the crucial transversal or longitudinal momentum exchange can occur only under very special situations. Hence a statistical averaging of the probabilities over Lorentz group seems to be the only manner to achieve Lorentz invariance.

An interesting question, is whether breaking of Lorentz symmetry is already encountered in the hadronic scattering in quark model description, which involves the reduction of Lorentz group to SO(1,1)× SO(2), and longitudinal and transverse momenta.

4. Summary

To sum up, quantum classical correspondence combined with the number theoretical view about conformal invariance could fix highly uniquely the dependence of S-matrix on cm degrees of freedom and on net momenta and color quantum numbers associated with various lightcones whose tips define the arguments of n-point function. If one is ready to accept the new view about scattering event as a scattering between two zero energy states between representing initial resp. final states as positive resp. negative energy particles of the state one obtains physically highly non-trivial counterpart of S-matrix. One must also accept the generalization of the unitarity condition to what might be regarded as a thermal average stating explicitly the assumption that experimenter does not have control over the scattering between zero energy states whose basic function is to make the zero energy state representing ordinary scattering consciously observed. The properties of factorizing S-matrices are ideal for this purpose.

The price being paid is the breaking of the full Lorentz and color invariances, which at the level of Jones inclusion means a change of the geometry topology of imbedding space and space-time. This kind of breaking of course happens in realistic experimental situation. Lorentz invariance is obtained only in statistical sense.

The most fascinating aspect of the new interpretation is that the factorizing and physically almost trivial S-matrices of integrable 2-D systems generalize to S-matrices describing scattering between states with vanishing net quantum numbers could be imbedded into 4-D theory in such a manner that the resulting S-matrix could be physically completely non-trivial and perhaps even physically realistic.

The basic requirement on S-matric between zero energy states is its almost triviality. This motivates the hope that the tensor factoring of 2-D factorizing matrices could be extended also to configuration space degrees of freedom so that each complex configuration space dimension would contribute one factorizing S-matrix to the tensor product.

The chapter Construction of Quantum Theory of "Towards S-matrix" represents the detailed construction as it is now (it could change!).

Precise definition of the notion of unitarity for Connes tensor product

Connes tensor product for free fields provides an extremely promising manner to define S-matrix and I have worked out the master formula in a considerable detail. The subfactor N subset of M in Jones represents the degrees of freedom which are not measured. Hence the infinite number of degrees of freedom for M reduces to a finite number of degrees of freedom associated with the quantum Clifford algebra N/M and corresponding quantum spinor space.

The previous physical picture helps to characterize the notion of unitarity precisely for the S-matrix defined by Connes tensor product. For simplicity restrict the consideration to configuration space spin degrees of freedom.

1. Tr(Id)=1 condition implies that it is not possible to define S-matrix in the usual sense since the probabilities for individual scattering events would vanish. Connes tensor product means that in quantum measurement particles are described using finite-dimensional quantum state spaces M/N defined by the inclusion. For standard inclusions they would correspond to single Clifford algebra factor C(8). This integration over the unobserved degrees of freedom is nothing but the analog for the transitions from super-string model to effective field theory description and defines the TGD counterpart for the renormalization process.

2. The intuitive mathematical interpretation of the Connes tensor product is that N takes the role of the coefficient field of the state space instead of complex numbers. Therefore S-matrix must be replaced with N-valued S-matrix in the tensor product of finite-dimensional state spaces. The notion of N unitarity makes sense since matrix inversion is defined as S_ij→ S_ji^† and does not require division (note that i and j label states of M/N). Also the generalization of the hermiticity makes sense: the eigenvalues of a matrix with N-hermitian elements are N Hermitian matrices so that single eigenvalue is abstracted to entire spectrum of eigenvalues. Kind of quantum representation for conceptualization process is in question and might have direct relevance to TGD inspired theory of consciousness. The exponentiation of a matrix with N Hermitian elements gives unitary matrix.

3. The projective equivalence of quantum states generalizes: two states differing by a multiplication by N unitary matrix represent the same ray in the state space. By adjusting the N unitary phases of the states suitably it might be possible to reduce S-matrix elements to ordinary complex vacuum expectation values for the states created by using elements of quantum Clifford algebra M/N, which would mean the reduction of the theory to TGD variant of conformal field theory or effective quantum field theory.

4. The probabilities P_ij for the general transitions would be given by

   P_ij=N_ijN_ij^† ,

   and are in general N-valued unless one requires

   P_ij=p_ije_N ,

   where e_N is projector to N. N_ij is therefore proportional to N-unitary matrix. S-matrix is trivial in N degrees of freedom which conforms with the interpretation that N degrees of freedom remain entangled in the scattering process.

5. If S-matrix is non-trivial in N degrees of freedom, these degrees of freedom must be treated statistically by summing over probabilities for the initial states. The only mathematical expression that one can imagine for the scattering probabilities is given by

   p_ij=Tr(N_ijN_ij ^†)_N .

   The trace over N degrees of freedom means that one has probability distribution for the initial states in N degrees of freedom such that each state appears with the same probability which indeed was von Neumann's guiding idea. By the conservation of energy and momentum in the scattering this assumption reduces to the basic assumption of thermodynamics.

6. An interesting question is whether also momentum degrees of freedom should be treated as a factor of type II₁ although they do not correspond directly to configuration space spin degrees of freedom. This would allow to get rid of mathematically unattractive squares of delta functions in the scattering probabilities.

For details see the chapter Was von Neumann Right After All of "TGD: an Overview".

Tree like structure of the extended imbedding space

The quantization of hbar in multiples of integer n characterizing the quantum phase q=exp(iπ/n) in M⁴ and CP₂ degreees of freedom separately means also separate scalings of covariant metrics by n₂ in these degrees of freedom. The question is how these copies of imbedding spaces are glued together. The gluing of different p-adic variants of imbedding spaces along rationals and general physical picture suggest how the gluing operation must be carried out.

Two imbedding spaces with different scalings factors of metrics are glued directly together only if either M⁴ or CP₂ scaling factor is same and only along M⁴ or CP₂. This gives a kind of evolutionary tree (actually in rather precise sense as the quantum model for evolutionary leaps as phase transitions increasing hbar(M⁴) demonstrates!). In this tree vertices represent given M⁴ (CP₂) and lines represent CP₂:s (M⁴:s) with different values of hbar(CP₂) (hbar(M⁴)) emanating from it much like lines from from a vertex of Feynman diagram.

1. In the phase transition between different hbar(M⁴):s the projection of the 3-surface to M⁴ becomes single point so that a cross section of CP₂ type extremal representing elementary particle is in question. Elementary particles could thus leak between different M⁴:s easily and this could occur in large hbar(M⁴) phases in living matter and perhaps even in quantum Hall effect. Wormhole contacts which have point-like M⁴ projection would allow topological condensation of space-time sheets with given hbar(M⁴) at those with different hbar(M⁴) in accordance with the heuristic picture.

2. In the phase transition different between CP₂:s the CP₂ projection of 3-surface becomes point so that the transition can occur in regions of space-time sheet with 1-D CP₂ projection. The regions of a connected space-time surface corresponding to different values of hbar (CP₂) can be glued together. For instance, the gluing could take place along surface X³=S²× T (T corresponds time axis) analogous to black hole horizon. CP₂ projection would be single point at the surface. The contribution from the radial dependence of CP₂ coordinates to the induced metric giving ds²= ds²(X³)+g_rrdr² at X³ implies a radial gravitational acceleration and one can say that a gravitational flux is transferred between different imbedding spaces.

   Planetary Bohr orbitology predicting that only 6 per cent of matter in solar system is visible suggests that star and planetary interiors are regions with large value of CP₂ Planck constant and that only a small fraction of the gravitational flux flows along space-time sheets carrying visible matter. In the approximation that visible matter corresponds to layer of thickness Δ R at the outer surface of constant density star or planet of radius R, one obtains the estimate Δ R=.12R for the thickness of this layer: convective zone corresponds to Δ R=.3R. For Earth one would have Δ R≈ 70 km which corresponds to the maximal thickness of the crust. Also flux tubes connecting ordinary matter carrying gravitational flux leaving space-time sheet with a given hbar (CP₂) at three-dimensional regions and returning back at the second end are possible. These flux tubes could mediate dark gravitational force also between objects consisting of ordinary matter.

Concerning the mathematical description of this process, the selection of origin of M⁴ or CP₂ as a preferred point is somewhat disturbing. In the case of M⁴ the problem disappears since configuration space is union over the configuration spaces associated with future and past light cones of M⁴: CH= CH⁺U CH^-, CH^+/-= U_{m in M⁴} CH^+/-_m. In the case of CP₂ the same interpretation is necessary in order to not lose SU(3) invariance so that one would have CH^+/-= U_{h in H} CH^+/-_h. A somewhat analogous but simpler book like structure results in the fusion of different p-adic variants of H along common rationals (and perhaps also common algebraics in the extensions).

For details see the chapter Does TGD Predict the Spectrum of Planck Constants of "TGD: an Overview".

More precise view about S-matrix

In TGD framework the nontriviality of S-matrix would basically result by a replacement of the ordinary tensor product with Connes tensor product for free fields. Each Jones inclusion defines different Connes tensor product. According to the speculation of Jones, the fusion rules of conformal theories are equivalent with Connes tensor product for tensor products of Kac Moody representations at least. Various conformal field theories would thus represent different Jones inclusions characterizing limitations of the measurer in various kinds of quantum measument situations and characterized by ADE diagrams or extended ADE diagrams. Somewhat ironically, in ideal measurement S-matrix would be trivial! In string model context this approach does not work since one has necessarily c=h=0.

The picture about S-matrix has been developing rapidly and it is now clear that the 27 year old dream is finally realized: I understand S-matrix (or rather S-matrices) in TGD framework. Below comments about results that have emerged during last week.

1. Effective 2-dimensionality and quantum classical correspondence

The requirement that quantum measurement theory in the sense of TGD emerges as part of the construction of S-matrix allows to add details to the master formula and derive also non-trivial predictions. The effective 2-dimensionality of the configuration space metric (only the deformations of 2-D partonic surfaces appear in the line element) suggests that the incoming space-time sheets and the space-time sheet representing vertex intersect only along 2-D partonic space-time sheets. This allows incoming space-time sheets to have classical conserved charges identical with a maximal subset of commuting quantum numbers. The functional integral over 3-surfaces is very much analogous to the 2-dimensional functional integral of Euclidian string models. Vertex 3-surface in turn represents the quantum jump at space-time level.

The 3-surface representing N-vertex can be decomposed into parts representing vertices/propagators of conventional tree diagrams as space-/light like portions. No loops appear. By their strong classical non-determinism CP₂ type extremals representing elementary particles are unique as classical representations of propagation of off-mass-shell particles. For other light like causal determinants, such as those assignable to massless extremals, only massless momentum exchanges are typically possible (not however in 2-particle scattering) so that scattering becomes extremely deterministic. The prediction is that even gravitational interaction could be strong in lower-dimensional regions of the phase space corresponding to light-like momentum exchanges.

2. Reduction of the Connes tensor product to fusion in conformal field theories

There are good reasons to expect that Connes tensor product corresponds to the fusion procedure of conformal field theories so that various conformal field theories at partonic boundary components would characterize various measurement situations.

The realization that various super-conformal algebras at partonic space-time sheets have dual hyper-quaterionic representations and that the restriction of these representations to commuting hyper-complex planes or real lines has strong mathematical and physical justification allows to write formulas for S-matrix in terms of n-point functions of the conformal field theory characterizing Jones inclusion applying to the quantum measurement situation by just replacing complex argument with hyper-complex argument.

The differences between TGD and string models become clear and one can formulate precisely where string models go wrong. In TGD framework breaking of conformal symmetry can occur and mass squared corresponds to a genuine conformal weight rather than a contribution causing the vanishing of conformal weight. Hence the vanishing of conformal parameters (c,h), which was originally believed to make string models unique, is not necessary. Furthermore, four-momentum does not appear in Super Virasoro generators as in string models so that super generators can carry fermion number since the Majorana character of super-generators becomes un-necessary. Hence dimension D=10 or 11 for imbedding space is not necessary.

One can understand why gravitational constant is so weak as compared to gauge couplings. Also particle massivation and p-adic mass scale are coded into the statistical properties of the zitterbewegung of CP₂ type extremals (the projection to Minkowski space is random light-like curve).

3. What went wrong with string models?

Things went wrong with string models at many levels. With my personal background it is not difficult to see what went wrong with string models at the level of mathematical and physical understanding. Basically the wrong view about conformal invariance implied that theory made sense only in critical dimensions. Because this is so important I want to recapitulate: the basic mistake was the idea that mass squared compensates the contributions to conformal weight obtained in the ordinary conformal theory. This led to c=h=0 and to wrong realization of super generators and Majorana conditions and the rest we known. It is not surprising that so powerful constraints led to what looked a highly unique theory.

p-Adic mass calculations made it obvious for me that this view about conformal invariance is wrong. Thermal mass squared in string model sense means breaking of Lorentz invariance: mass squared must result as thermal average of conformal weight. This requires that physical states have non-vanishing conformal weights and that mass squared corresponds to this weight and that momentum does not appear in Super Virasoro generators. Ordinary Euclidian conformal theory for partonic 2-surfaces and its dual for hyper-complex planes of Minkowski space become the basic mathematical tool in the construction of physical states and S-matrix.

There are also many other thins that went wrong: I shared for a long time the belief in the stringy description of particle decay in terms of trouser diagrams until physical arguments forced to interpret this process as a space-time correlate for what happens in double slit experiments. Mathematically this was of course obvious from the beginning since it is just propagation via different routes which happens for induced spinor fields: it is extremely artificial to build vertices using this framework. A direct generalization of Feynman diagrams as singular manifolds obtained by gluing surfaces together along their ends turned out to the only acceptable topological description of particle decay but one can argue that this is mathematically too singular process. It is: and this description indeed turned out to be only effective when the master formula for S-matrix emerged.

Let us return to the consequences of misunderstood conformal invariance. I can well understand the excitement when people realized that perhaps only some little work might reveal the theory of everything. The resulting physics was of course not about this world. It is quite understandable that this sparkled the dream that Kaluza-Klein might save the theory and eventually we had landscape, anthropic principle, and even the brand new vision that the physics that we experience everyday could be anything else and that even at the level of principle it is impossible to predict anything. Now the theory is sold using the exact opposites of the arguments that were used two decades ago.

What is so sad in this that immense amount of high level technical work is being carried out in attempt to get something sensible out of a dead theory.

Conformal field theories represent just the opposite to string model hype. Instead of ad hoc quantization recipes and uncritical introduction of poorly defined notions like spontaneous compactification and effective field theories, rigorous formulation and understanding of the theory has been the primary goal. The formalism generalizes as such to TGD framework and finds a new interpretation in the framework of quantum measurement theory.

Details can be found here, here, and here.

Does the quantization of Planck constant transform integer quantum Hall effect to fractional quantum Hall effect?

The TGD based model for topological quantum computation inspired the idea that Planck constant might be dynamical and quantized. The work of Nottale (astro-ph/0310036) gave a strong boost to concrete development of the idea and it took year and half to end up with a proposal about how basic quantum TGD could allow quantization Planck constant associated with M⁴ and CP₂ degrees of freedom such that the scaling factor of the metric in M⁴ degrees of freedom corresponds to the scaling of hbar in CP₂ degrees of freedom and vice versa (see the new chapter Does TGD Predict the Spectrum of Planck constants?). The dynamical character of the scaling factors of M⁴ and CP₂ metrics makes sense if space-time and imbedding space, and in fact the entire quantum TGD, emerge from a local version of an infinite-dimensional Clifford algebra existing only in dimension D=8.

The predicted scaling factors of Planck constant correspond to the integers n defining the quantum phases q=exp(iπ/n) characterizing Jones inclusions. A more precise characterization of Jones inclusion is in terms of group

G_b subset of SU(2) subset of SU(3)

in CP₂ degrees of freedom and

G_a subset of SL(2,C)

in M⁴ degrees of freedom. In quantum group phase space-time surfaces have exact symmetry such that to a given point of M⁴ corresponds an entire G_b orbit of CP₂ points and vice versa. Thus space-time sheet becomes N(G_a) fold covering of CP₂ and N(G_b)-fold covering of M⁴. This allows an elegant topological interpretation for the fractionization of quantum numbers. The integer n corresponds to the order of maximal cyclic subgroup of G.

In the scaling hbar₀→ n× hbar₀ of M⁴ Planck constant fine structure constant would scale as

α= (e²/(4πhbar c)→ α/n ,

and the formula for Hall conductance would transform to

σ_H =να → (ν/n)× α .

Fractional quantum Hall effect would be integer quantum Hall effect but with scaled down α. The apparent fractional filling fraction ν= m/n would directly code the quantum phase q=exp(iπ/n) in the case that m obtains all possible values. A complete classification for possible phase transitions yielding fractional quantum Hall effect in terms of finite subgroups G subset of SU(2) subset of SU(3) given by ADE diagrams would emerge (A_n, D_2n, E₆ and E₈ are possible). What would be also nice that CP₂ would make itself directly manifest at the level of condensed matter physics.

For more details see the chapter Topological Quantum Computation in TGD Universe, the chapter Was von Neumann Right After All?, and the chapter Does TGD predict the Spectrum of Planck Constants?.

Large values of Planck constant and coupling constant evolution

There has been intensive evolution of ideas induced by the understanding of large values of Planck constants. This motivated a separate chapter which I christened as "Does TGD Predict the Spectrum of Planck Constants?". I have commented earlier about various ideas related to this topic and comment here only the newest outcomes.

1. hbar_gr as CP₂ Planck constant

What gravitational Planck constant means has been somewhat unclear. It turned out that hbar_gr can be interpreted as Planck constant associated with CP₂ degrees of freedom and its huge value implies that also the von Neumann inclusions associated with M⁴ degrees of freedom meaning that dark matter cosmology has quantal lattice like structure with lattice cell given by H_a/G, H_a the a=constant hyperboloid of M⁴₊ and G subgroup of SL(2,C). The quantization of cosmic redshifts provides support for this prediction.

2. Is Kähler coupling strength invariant under p-adic coupling constant evolution

Kähler coupling constant is the only coupling parameter in TGD. The original great vision is that Kähler coupling constant is analogous to critical temperature and thus uniquely determined. Later I concluded that Kähler coupling strength could depend on the p-adic length scale. The reason was that the prediction for the gravitational coupling strength was otherwise non-sensible. This motivated the assumption that gravitational coupling is RG invariant in the p-adic sense.

The expression of the basic parameter v₀=2^-11 appearing in the formula of hbar_gr=GMm/v₀ in terms of basic parameters of TGD leads to the unexpected conclusion that α_K in electron length scale can be identified as electro-weak U(1) coupling strength α_U(1). This identification, or actually something slightly complex (see below), is what group theory suggests but I had given it up since the resulting evolution for gravitational coupling predicted G to be proportional to L_p² and thus completely un-physical. However, if gravitational interactions are mediated by space-time sheets characterized by Mersenne prime, the situation changes completely since M₁₂₇ is the largest non-super-astrophysical p-adic length scale.

The second key observation is that all classical gauge fields and gravitational field are expressible using only CP₂ coordinates and classical color action and U(1) action both reduce to Kähler action. Furthermore, electroweak group U(2) can be regarded as a subgroup of color SU(3) in a well-defined sense and color holonomy is abelian. Hence one expects a simple formula relating various coupling constants. Let us take α_K as a p-adic renormalization group invariant in strong sense that it does not depend on the p-adic length scale at all.

The relationship for the couplings must involve α_U(1), α_s and α_K. The formula 1/α_U(1)+1/α_s = 1/α_K states that the sum of U(1) and color actions equals to Kähler action and is consistent with the decrease of the color coupling and the increase of the U(1) coupling with energy and implies a common asymptotic value 2α_K for both. The hypothesis is consistent with the known facts about color and electroweak evolution and predicts correctly the confinement length scale as p-adic length scale assignable to gluons. The hypothesis reduces the evolution of α_s to the calculable evolution of electro-weak couplings: the importance of this result is difficult to over-estimate.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants? of "TGD: an Overview".

Could the basic parameters of TGD be fixed by a number theoretical miracle?

If the v₀ deduced to have value v₀=2^-11 appearing in the expression for gravitational Planck constant hbar_gr=GMm/v₀ is identified as the rotation velocity of distant stars in galactic plane, it is possible to express it in terms of Kähler coupling strength and string tension as v₀^-2= 2×α_KK,

α_K(p)= a/log(pK) , K= R²/G .

The value of K is fixed to a high degree by the requirement that electron mass scale comes out correctly in p-adic mass calculations. The uncertainties related to second order contributions in p-adic mass calculations however leave the precise value open. Number theoretic arguments suggest that K is expressible as a product of primes p ≤ 23: K= 2×3×5×...×23 .

If one assumes that α_K is of order fine structure constant in electron length scale, the value of the parameter a cannot be far from unity. A more precise condition would result by identifying α_K with weak U(1) coupling strength α_K = α_U(1)=α_em/cos²(θ_W)≈ 1/105.3531 ,

sin²(θ_W)≈ .23120(15),

α_em= 0.00729735253327 .

Here the values refer to electron length scale. If the formula v₀= 2^-11 is exact, it poses both quantitative and number theoretic conditions on Kähler coupling strength. One must of course remember, that exact expression for v₀ corresponds to only one particular solution and even smallest deformation of solution can change the number theoretical anatomy completely. In any case one can make following questions.

1. Could one understand why v₀≈ 2^-11 must hold true.
2. What number theoretical implications the exact formula v₀= 2^-11 has in case that it is consistent with the above listed assumptions?

1. Are the ratios π/log(q) rational?

The basic condition stating that gravitational coupling constant is renormalization group invariant dictates the dependence of the Kähler coupling strength of p-adic prime exponent of Kähler action for CP₂ type extremal is rational if K is integer as assumed: this is essential for the algebraic continuation of the rational physics to p-adic number fields. This gives a general formula α_K= a π/log(pK). Since K is integer, this means that v₀² is of form

v₀²= qlog(pK)/π, q rational.

if a is rational.

1. Since v₀² should be rational for rational value of a, the minimal conclusion would be that the number log(pK)/π should be rational for some preferred prime p=p₀ in this case. If this miracle occurs, the p-adic coupling constant evolution of Kähler coupling strength, the only coupling constant in TGD, would be completely fixed. Same would also hold true for the ratio of CP₂ to length characterized by K^1/2.

2. A more general conjecture would be that log(q)/π is rational for q rational: this conjecture turns out to be wrong as discussed in the previous posting. The rationality of π/log(q) for single q is however possible in principle and would imply that exp(π) is an algebraic number. This would indeed look extremely nice since the algebraic character of exp(π) would conform with the algebraic character of the phases exp(iπ/n). Unfortunately this is not the case. Hence one loses the extremely attractive possibility to fix the basic parameters of theory completely from number theory.

The condition for v₀=2^-m, m=11, allows to deduce the value of a as

a= (log(pK)/π) × (2^2m/K).

The condition that α_K is of order fine structure constant for p=M₁₂₇= 2¹²⁷-1 defining the p-adic length scale of electron indeed implies that m=11 is the only possible value since the value of a is scaled by a factor 4 in m→ m+1.

The value of α_K in the length scale L_p0 in which condition of the first equation holds true is given by

1/α_K= 2²¹/K≈ 106.379 .

2. What is the value of the preferred prime p₀?

The condition for v₀ can hold only for a single p-adic length scale L_p0. This correspondence would presumably mean that gravitational interaction is mediated along the space-time sheets characterized by p₀, or even that gravitons are characterized by p₀.

1. If same p₀ characterizes all ordinary gauge bosons with their dark variants included, one would have p₀=M₈₉=2⁸⁹-1.

2. One can however argue that dark gravitons and dark bosons in general can correspond to different Mersenne prime than ordinary gauge bosons. Since Mersenne primes larger than M₁₂₇ define super-astrophysical length scales, M₁₂₇ is the unique candidate. M₁₂₇ indeed defines a dark length scale in TGD inspired quantum model of living matter. This predicts 1/α_U(1)(M₁₂₇)= 106.379 to be compared with the experimental estimate 1/α_U(1)(M₁₂₇)= 105.3531 deduced above. The deviation is smaller than one percent, which indeed puts bells ringing!

This agreement seems to provide dramatic support for the general picture but one must be very cautious.

1. The identification of Kähler coupling strength as U(1) coupling strength poses strong conditions on the p-adic length scale evolution of Weinberg angle using the knowledge about the evolution of the electromagnetic coupling constant. The condition

   cos²(θ_W)(89)= [log(M₁₂₇K)/log(M₈₉K)] × [α_em(M₁₂₇)/α_em(M₈₉)]× cos²(θ_W)(127) .

   Using the experimental value 1/α_em(M₈₉)≈ 128 as predicted by standard model one obtains sin²(θ_W)(89)=.0479. There is a bad conflict with experimental facts unless the experimentally determined value of Weinberg angle corresponds to M₁₂₇ space-time sheet.

   I will leave leave the implications of this conflict to the future posting.

The reader interested in details is recommended to look previous postings and the new chapter Can TGD Predict the Spectrum of Planck Constants? of the book "TGD: an Overview" and the chapter TGD and Astrophysics of the book "Physics in Many-Sheeted Space-Time".