Wednesday, June 28, 2006

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 Dphys=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.

Tuesday, June 20, 2006

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 D2n+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 Ng-1 at the large N limit. The interpretation in terms of closed partonic 2-surfaces is highly suggestive and the Ng-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 Ga× Gb in SL(2,C)× SU(2) defining double Jones inclusion and implying the scalings hbar(M4)→ n(Gb)× hbar(M4) and hbar(CP2)→ n(Ga)× hbar(CP2). These space-time surfaces define n(Ga)-fold multiple coverings of CP2 and n(Gb)-fold multiple coverings of M4. In CP2 degrees of freedom the collection of Gb-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 (An,D2n 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.

Monday, June 12, 2006


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.

Saturday, June 10, 2006

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 Ln, 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 rM=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.

Wednesday, June 07, 2006


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.

Sunday, June 04, 2006

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(SS)= 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!).