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.