DNA as topological quantum computer: XIV

I have worked hardly in refining the chapters about quantum biology in TGD Universe. DNA as topological quantum computer was the original idea that has generalized considerably and led to a beautiful unification of various basic ideas. I attach below the abstract of the chapter DNA as topological quantum computer contained in the book "Genes and Memes".

This chapter represents a vision about how DNA might act as a topological quantum computer (tqc). Tqc means that the braidings of braid strands define tqc programs and M-matrix (generalization of S-matrix in zero energy ontology) defining the entanglement between states assignable to the end points of strands define the tqc usually coded as unitary time evolution for Schrödinger equation. One can ends up to the model in the following manner.

1. Darwinian selection for which the standard theory of self-organization provides a model, should apply also to tqc programs. Tqc programs should correspond to asymptotic self-organization patterns selected by dissipation in the presence of metabolic energy feed. The spatial and temporal pattern of the metabolic energy feed characterizes the tqc program - or equivalently - sub-program call.
2. Since braiding characterizes the tqc program, the self-organization pattern should correspond to a hydrodynamical flow or a pattern of magnetic field inducing the braiding. Braid strands must correspond to magnetic flux tubes of the magnetic body of DNA. If each nucleotide is transversal magnetic dipole it gives rise to transversal flux tubes, which can also connect to the genome of another cell. As a matter fact, the flux tubes would correspond to what I call wormhole magnetic fields having pairs of space-time sheets carrying opposite magnetic fluxes.
3. The output of tqc sub-program is probability distribution for the outcomes of state function reduction so that the sub-program must be repeated very many times. It is represented as four-dimensional patterns for various rates (chemical rates, nerve pulse patterns, EEG power distributions,...) having also identification as temporal densities of zero energy states in various scales. By the fractality of TGD Universe there is a hierarchy of tqcs corresponding to p-adic and dark matter hierarchies. Programs (space-time sheets defining coherence regions) call programs in shorter scale. If the self-organizing system has a periodic behavior each tqc module defines a large number of almost copies of itself asymptotically. Generalized EEG could naturally define this periodic pattern and each period of EEG would correspond to an initiation and halting of tqc. This brings in mind the periodically occurring sol-gel phase transition inside cell near the cell membrane. There is also a connection with hologram idea: EEG rhythm corresponds to reference wave and nerve pulse patters to the wave carrying the information and interfering with the reference wave.
4. Fluid flow must induce the braiding which requires that the ends of braid strands must be anchored to the fluid flow. Recalling that lipid mono-layers of the cell membrane are liquid crystals and lipids of interior mono-layer have hydrophilic ends pointing towards cell interior, it is easy to guess that DNA nucleotides are connected to lipids by magnetic flux tubes and hydrophilic lipid ends are stuck to the flow.
5. The topology of the braid traversing cell membrane cannot be affected by the hydrodynamical flow. Hence braid strands must be split during tqc. This also induces the desired magnetic isolation from the environment. Halting of tqc reconnects them and make possible the communication of the outcome of tqc.

There are several problems related to the details of the realization.

1. How nucleotides A,T,C,G are coded to the strand color and what this color corresponds to physically? There are two options which could be characterized as fermionic and bosonic.
   i) Magnetic flux tubes having quark and anti-quark at their ends with u,d and u_c, d_c coding for A,G and T,C. CP conjugation would correspond to conjugation for DNA nucleotides.
   ii) Wormhole magnetic flux tubes having wormhole contact and its CP conjugate at its ends with wormhole contact carrying quark and anti-quark at its throats. The latter are predicted to appear in all length scales in TGD Universe.
2. How to split the braid strands in a controlled manner? High T_c super conductivity provides a possible mechanism: braid strand can be split only if the supra current flowing through it vanishes. A suitable voltage pulse induces the supra-current and its negative cancels it. The conformation of the lipid controls whether it it can follow the flow or not.
3. How magnetic flux tubes can be cut without breaking the conservation of the magnetic flux? The notion of wormhole magnetic field could save the situation now: after the splitting the flux returns back along the second space-time sheet of wormhole magnetic field. An alternative solution is based on reconnection of flux tubes. Since only flux tubes of same color can reconnect this process can induce transfer of strand color: "color inheritance": when applied at the level of amino-acids this leads to a successful model of protein folding. Reconnection makes possible breaking of flux tube connection for both the ordinary magnetic flux tubes and wormhole magnetic flux tubes.
4. How magnetic flux tubes are realized? The interpretation of flux tubes as correlates of directed attention at molecular level leads to concrete picture. Hydrogen bonds are by their asymmetry natural correlates for a directed attention at molecular level. Also flux tubes between acceptors of hydrogen bonds must be allowed and acceptors can be seen as the subjects of directed attention and donors as objects. Examples of acceptors are aromatic rings of nucleotides, O= atoms of phosphates, etc.. A connection with metabolism is obtained if it is assumed that various phosphates XMP,XDP,XTP, X=A,T,G,C act as fundamental acceptors and plugs in the connection lines. The basic metabolic process ATP® ADP+P_i allows an interpretation as a reconnection splitting flux tube connection, and the basic function of phosphorylating enzymes would be to build flux tube connections as also of breathing and photosynthesis.

The model makes several testable predictions about DNA itself. In particular, matter-antimatter asymmetry and slightly broken isospin symmetry have counterparts at DNA level induced from the breaking of these symmetries for quarks and antiquarks associated with the flux tubes. DNA cell membrane system is not the only possible system that could perform tqc like activities and store memories in braidings: flux tubes could connect biomolecules and the braiding could provide an almost definition for what it is to be living. Even water memory might reduce to braidings.

The model leads also to an improved understanding of other roles of the magnetic flux tubes containing dark matter. Phase transitions changing the value of Planck constant for the magnetic flux tubes could be key element of bio-catalysis and electromagnetic long distance communications in living matter. For instance, one ends up to what might be called code for protein folding and bio-catalysis. There is also a fascinating connection with Peter Gariaev's work suggesting that the phase transitions changing Planck constant have been observed and wormhole magnetic flux tubes containing dark matter have been photographed in his experiments.

A code for protein folding and bio-catalysis

The TGD inspired model for the evolution of genetic code leads to the idea that the folding of proteins obeys a folding code inherited from the genetic code. After some trials one ends up with a general conceptualization of the situation with the identification of wormhole magnetic flux tubes as correlates of attention at molecular level so that a direct connection with TGD inspired theory of consciousness emerges at quantitative level. This allows a far reaching generalization of the DNA as topological quantum computer paradigm and makes it much more detailed. By their asymmetric character hydrogen bonds are excellent candidates for magnetic flux tubes serving as correlates of attention at molecular level.

The constant part of free amino-acid containing O-H, O=, and NH₂ would correspond to the codon XYZ in the sense that the flux tubes would carry the "color" representing the four nucleotides in terms of quark pairs. Color inheritance by flux tube reconnection makes this possible. For the amino-adics inside protein O= and N-H would correspond to YZ. Also flux tubes connecting the acceptor atoms of hydrogen bonds are required by the model of DNA as topological quantum computer. The long flux tubes between O= atoms and their length reduction in a phase transition reducing Planck constant could be essential in protein-ligand interaction.

The model predicts a code for protein folding: depending on whether also =O-O= flux tubes are allowed or not, Y=Z or Y=Z_c condition is satisfied by the amino-acids having N-H-O= hydrogen bond. For =O-O= bonds Y-Y_c pairing holds true. Y=Z_c option predicts the average length of alpha bonds correctly. Y=Z rule is favored by the study of alpha helices for four enzymes: the possible average length of alpha helix is considerably longer than the average length of alpha helix if gene is the unique gene allowing to satisfy Y=Z rule. The explicit study of alpha helices for four enzymes demonstrates that the failure to satisfy the condition for the existence of hydrogen bond fails rarely and at most for two amino-acids (for 2 amino-acids in single case only). For beta sheets there ar no failures for Y=Z option.

The information apparently lost in the many-to-one character of the codon-amino-acid correspondence would code for the folding of the protein and similar amino-acid sequences could give rise to different foldings. Also catalyst action would reduce to effective base pairing and one can speak about catalyst code. The DNA sequences associated with alpha helices and beta sheets are completely predictable unless one assumes a quantum counterpart of wobble base pairing meaning that N-H flux tubes are before hydrogen bonding in quantum superpositions of braid colors associated with the third nucleotides Z of codons XYZ coding for amino-acid. Only the latter option works. The outcome is very simple quantitative model for folding and catalyst action based on minimization of energy and predicting as its solutions alpha helices and beta sheets.

I want to express my gratitude for Dale Trenary for interesting discussions, for suggesting proteins which could allow to test the model, as well as providing concrete help in loading data help from protein data bank. Also I want to thank Timo Immonen for loaning the excellent book "Proteins: Structures and Molecular Properties" of Creighton. Also Pekka Rapinoja for writing the program transforming protein data file to a form readable by MATLAB.

For details see the new chapter A Model for Protein Folding and Bio-catalysis of "Genes and Memes".

Is N=8 super-gravity finite and why this would be the case?

There is an interesting discussion in the blog of Peter Woit about the recent evidence for the finiteness of N=8 super-gravity and perhaps even N<8 super-gravities. I have nothing to say about horrible technicalities of these calculations and I can only admire from distance what the people involved are able to do. Nothing however prevents me to make some uneducated comments from TGD point of view.

Theoretical physicist as a thinker or as a computational virtuoso?

Certainly the cancelation of infinities would be a proof for the magic mathematical power of deep principles like General Coordinate Invariance, Equivalence Principle, and Super Symmetry. That Einstein with his not so fantastic calculational skills discovered the first two principles should make clear that it is conceptual thinking that matters most even in theoretical physics. The return to the study of N=8 super-gravity after 24 years of super-string models and the possible reduction of super string models to a mere auxiliary calculational tool would be a clear-cut answer to the question "Theoretical physics as the art of conceptualization or as mere computational methodology?".

What could be the symmetry behind cancellations of infinities in N=8 supergravities?

One question represented in discussions concerns the underlying symmetry causing the cancelation of the perturbative infinities.

1. In the latest postings (see this, this,and this) I have told about how the notion symplectic quantum field theory emerges from TGD. These theories are highly analogous to conformal field theories. Fusion rules generalize and N-point functions are proportional to symplectic invariants assignable to the set of points defined by N arguments. The basic symplectic invariants are the symplectic areas of the geodesic triangles defined by 3-point subsets of the arguments of N-point function. Under very general assumptions these invariants vanish when two of N arguments co-incide because the areas of the triangles containing these two points vanish. The behavior of N-point functions is smoothed out so that the artificial UV cutoff is replaced with a dynamical one. Infrared cutoff comes in turn from the size of the causal diamond serving as imbedding space correlate for the zero energy state. One can thus get rid of local divergences as very general arguments based on the absence of local interaction vertices and Kähler geometry of the "world of classical worlds" indeed predicted long time ago.
2. Very naively, the quantum field theory limit of TGD obtained by replacing the light-like 3-surfaces appearing as "lines" and 2-D partonic surfaces appearing as the vertices of generalized Feynman diagrams are taken to lines and points. The limiting theory could still have symplectic invariance as a hidden symmetry of N-point functions.

The obvious question is whether the N-point functions of N≤8 super-gravities have this kind of symplectic symmetry as a hidden symmetry? Note that N=4 super-conformal symmetry is the most natural candidate for the conformal super-symmetry of TGD and that TGD does not predict super-symmetry at the level of Poincare algebra so that no sparticles are predicted.

Why TGD is needed?

It is very difficult to understand how N=8 super-gravity could be consistent with standard model and my strong conviction is that one cannot avoid TGD if one is interested in physics.

1. My belief is that it is the generalization of super-conformal invariance provided by the replacement of strings with light-like 3-surfaces as fundamental objects in H=M⁴×CP₂ which is needed. The maximal conformal symmetries are obtained only for 4-D space-time and 4-D Minkowski space.
2. "Number theoretical compactification" -or more precisely, the duality of pictures based on the identification of 8-D imbedding space as hyper-octonionic space-time HO= M⁸ or as H=M⁴×CP₂ is absolutely essential for the recent formulation of quantum TGD and most "must-be-trues" of quantum TGD follow from the basic prerequisite of this duality. More precisely: space-time surfaces identified as hyper-quaternionic surfaces of HO contain preferred hyper-complex (commutative) plane HC=M² defining the plane of non-physical polarizations in their hyper-quaternionic tangent space M⁴: this kind of tangent spaces are parametrized by CP₂ which implies the duality.
3. One implication of the existence of preferred plane M² is justification for what I call generalized coset construction (see this) providing an elegant generalization of Einstein's equations replacing them with the condition that the super Virasoro algebras associated with the symplectic analog of Kac-Moody algebra and the ordinary Kac-Moody algebra defined by a symplectic sub-group of isometries of the imbedding space act in the same manner on physical states (the isometric Super Virasoro corresponds to the gravitational four-momentum and symplectic Super Virasoro to the inertial four-momentum). This provides the long sought-for explicit realization of Equivalence Principle in TGD framework.

I dare safely say that there is only a single possible choice of the imbedding space and this choice is consistent with the standard model. The geometry of classical spinor fields in the world of classical worlds and thus also physics is unique just from its mathematical existence, and physics as a generalized number theory vision allows to identify this unique and very special world of classical worlds (as does also the inspection of Particle Data Tables;-)). This motivates my humble suggestion for colleagues: why not to accept the facts and start to transform TGD to mathematical physics. A good place to start would be construction and classification of symplectic quantum field theories.

For a summary of quantum TGD see the article Topological Geometrodynamics: What Might Be the First Principles?.

The relationship between super-canonical and Super Kac-Moody algebras, Equivalence Principle, and justification of p-adic thermodynamics

The relationship between super-canonical algebra (SC) acting at light-cone boundary and Super Kac-Moody algebra (SKM) acting on light-like 3-surfaces has remained somewhat enigmatic due to the lack of physical insights. This is not the only problem. The question to precisely what extent Equivalence Principle (EP) remains true in TGD framework and what might be the precise mathematical realization of EP is waiting for an answer. Also the justification of p-adic thermodynamics for the scaling generator L₀ of Virasoro algebra -in obvious conflict with the basic wisdom that this generator should annihilate physical states- is lacking. It seems that these three problems could have a common solution.

Before going to describe the proposed solution, some background is necessary. The latest proposal for SC-SKM relationship relies on non-standard and therefore somewhat questionable assumptions.

1. SKM Virasoro algebra (SKMV) and SC Virasoro algebra (SCV) (anti)commute for physical states.
2. SC algebra generates states of negative conformal weight annihilated by SCV generators L_n, n < 0, and serving as ground states from which SKM generators create states with non-negative conformal weight.

This picture could make sense for elementary particles. On other hand, the recent model for hadrons [see this] assumes that SC degrees of freedom contribute about 70 per cent to the mass of hadron but at space-time sheet different from those assignable to quarks. The contribution of SC degrees of freedom to the thermal average of the conformal weight would be positive. A contradiction results unless one assumes that there exists also SCV ground states with positive conformal weight annihilated by SCV elements L_n, n < 0, but also this seems implausible.

1. New vision about the relationship between SCV and SKMV

Consider now the new vision about the relationship between SCV and SKMV.

1. The isometries of H assignable with SKM are also symplectic transformations [see this] (note that I have used the term canonical instead of symplectic previously). Hence might consider the possibility that SKM could be identified as a subalgebra of SC. If this makes sense, a generalization of the coset construction obtained by replacing finite-dimensional Lie group with infinite-dimensional symplectic group suggests itself. The differences of SCV and SKMV elements would annihilate physical states and (anti)commute with SKMV. Also the generators O_n, n > 0, for both algebras would annihilate the physical states so that the differences of the elements would annihilate automatically physical states for n > 0.
2. The super-generator G₀ contains the Dirac operator D of H. If the action of SCV and SKMV Dirac operators on physical states are identical then cm of degrees of freedom disappear from the differences G₀(SCV)-G₀(SKMV) and L₀(SCV)-L₀(SKMV). One could interpret the identical action of the Dirac operators as the long sought-for precise realization of Equivalence Principle (EP) in TGD framework. EP would state that the total inertial four-momentum and color quantum numbers assignable to SC (imbedding space level) are equal to the gravitational four-momentum and color quantum numbers assignable to SKM (space-time level). Note that since super-canonical transformations correspond to the isometries of the "world of classical worlds" the assignment of the attribute "inertial" to them is natural.
3. The analog of coset construction applies also to SKM and SC algebras which means that physical states can be thought of as being created by an operator of SKM carrying the conformal weight and by a genuine SC operator with vanishing conformal weight. Therefore the situation does not reduce to that encountered in super-string models.

This picture provides also a justification for p-adic thermodynamics.

1. In physical states the p-adic thermal expectation value of the SKM and SC conformal weights would be non-vanishing and identical and mass squared could be identified to the expectation value of SKM scaling generator L₀. There would be no need to give up Super Virasoro conditions for SCV-SKMV.
2. There is consistency with p-adic mass calculations for hadrons [see this] since the non-perturbative SC contributions and perturbative SKM contributions to the mass correspond to space-time sheets labeled by different p-adic primes. The earlier statement that SC is responsible for the dominating non-perturbative contributions to the hadron mass transforms to a statement reflecting SC-SKM duality. The perturbative quark contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SKM whereas non-perturbative contributions to hadron masses can be calculated most conveniently by using p-adic thermodynamics for SC. Also the proposal that the exotic analogs of baryons resulting when baryon looses its valence quarks [see this] remains intact in this framework.
3. The results of p-adic mass calculations depend crucially on the number N of tensor factors contributing to the Super-Virasoro algebra. The required number is N=5 and during years I have proposed several explanations for this number. It seems that holonomic contributions that is electro-weak and spin contributions must be regarded as contributions separate from those coming from isometries. SKM algebras in electro-weak degrees and spin degrees of of freedom, would give 2+1=3 tensor factors corresponding to U(2)_ew×SU(2). SU(3) and SO(3) (or SO(2) Ì SO(3) leaving the intersection of light-like ray with S² invariant) would give 2 additional tensor factors. Altogether one would indeed have 5 tensor factors.

2. Can SKM be lifted to a sub-algebra of SC?

A picture introducing only a generalization of coset construction as a new element, realizing mathematically Equivalence Principle, and justifying p-adic thermodynamics is highly attractive but there is a problem. SKM is defined at light-like 3-surfaces X³ whereas SC acts at light-cone boundary dH_±=dM⁴_±×CP₂. One should be able to lift SKM to imbedding space level somehow. Also SC should be lifted to entire H. This problem was the reason why I gave up the idea about coset construction and SC-SKM duality as it appeared for the first time.

A possible solution of the lifting problem comes from the observation making possible a more rigorous formulation of HO-H duality stating that one can regard space-time surfaces either as surfaces in hyper-octonionic space HO=M⁸ or in H=M⁴×CP₂ [see this]. Consider first the formulation of HO-H duality.

1. Associativity also in the number theoretical sense becomes the fundamental dynamical principle if HO-H duality holds true [see this]. For a space-time surface X⁴ Ì HO=M⁸ associativity is satisfied at space-time level if the tangent space at each point of X⁴ is some hyper-quaternionic sub-space HQ=M⁴ Ì M⁸. Also partonic 2-surfaces at the boundaries of causal diamonds formed by pairs of future and past directed light-cones defining the basic imbedding space correlate of zero energy state in zero energy ontology and light-like 3-surfaces are assumed to belong to HQ=M⁴ Ì HO.
2. HO-H duality requires something more. If the tangent spaces contain the same preferred commutative and thus hyper-complex plane HC=M², the tangent spaces of X⁴ are parameterized by the points s of CP₂ and X⁴ Ì HO can be mapped to X⁴ Ì M⁴×CP₂ by assigning to a point of X⁴ regarded as point (m,e) of M⁴₀×E⁴=M⁸ the point (m,s). Note that one must also fix a preferred global hyper-quaternionic subspace M⁴₀ Ì M⁸ containing M² to be not confused with the local tangent planes M⁴.
3. The preferred plane M² can be interpreted as the plane of non-physical polarizations so that the interpretation as a number theoretic analog of gauge conditions posed in both quantum field theories and string models is possible.
4. An open question is whether the resulting surface in H is a preferred extremal of Kähler action. This is possible since the tangent spaces at light-like partonic 3-surfaces are fixed to contain M² so that the boundary values of the normal derivatives of H coordinates are fixed and field equations fix in the ideal case X⁴ uniquely and one obtains space-time surface as the analog of Bohr orbit.
5. The light-like "Higgs term" proportional to O=g_kt^k appearing in the generalized eigenvalue equation for the modified Dirac operator [see this] is an essential element of TGD based description of Higgs mechanism. This term can cause complications unless t is a covariantly constant light-like vector. Covariant constancy is achieved if t is constant light-like vector in M². The interpretation as a space-time correlate for the light-like 4-momentum assignable to the parton might be considered.
6. Associativity requires that the hyper-octonionic arguments of N-point functions in HO description are restricted to a hyperquaternionic plane HQ=M⁴ Ì HO required also by the HO-H correspondence. The intersection M⁴Çint(X⁴) consists of a discrete set of points in the generic case. Partonic 3-surfaces are assumed to be associative and belong to M⁴. The set of commutative points at the partonic 2-surface X² is discrete in the generic case whereas the intersection X³ÇM² consists of 1-D curves so that the notion of number theoretical braid crucial for the p-adicization of the theory as almost topological QFT is uniquely defined.
7. The preferred plane M² Ì M⁴ Ì HO can be assigned also to the definition of N-point functions in HO picture. It is not clear whether it must be same as the preferred planes assigned to the partonic 2-surfaces. If not, the interpretation would be that it corresponds to a plane containing the over all cm four-momentum whereas partonic planes M²_i would contain the partonic four-momenta. M² is expected to change at wormhole contacts having Euclidian signature of the induced metric representing horizons and connecting space-time sheets with Minkowskian signature of the induced metric.

The presence of globally defined plane M² and the flexibility provided by the hyper-complex conformal invariance raise the hopes of achieving the lifting of SC and SKM to H. At the light-cone boundary the light-like radial coordinate can be lifted to a hyper-complex coordinate defining coordinate for M². At X³ one can fix the light-like coordinate varying along the braid strands can be lifted to some hyper-complex coordinate of M² defined in the interior of X⁴. The total four-momenta and color quantum numbers assignable to the SC and SKM degrees of freedom are naturally identical since they can be identified as the four-momentum of the partonic 2-surface X² Ì X³ÇdM⁴_±×CP₂. Equivalence Principle would emerge as an identity.

3. Questions

There are still several open questions.

1. Is it possible to define hyper-quaternionic variants of the superconformal algebras in both H and HO or perhaps only in HO. A positive answer to this question would conform with the conjecture that the geometry of "world of classical worlds" allows Hyper-Kähler property in either or both pictures [see this].
2. How this picture relates to what is known about the extremals of field equations [see this] characterized by generalized Hamilton-Jacobi structure bringing in mind the selection of preferred M²?
3. Is this picture consistent with the views about Equivalence Principle and its possible breaking based on the identification of gravitational four-momentum in terms of Einstein tensor is interesting [see this] ?

For more details see the chapter Massless particles and particle massivation.

Still more detailed view about the construction of M-matrix elements

After three decades there are excellent hopes of building an explicit recipe for constructing M-matrix elements but the devil is in the details.

1. Elimination of infinities and coupling constant evolution

The elimination of infinities would follow from the symplectic QFT part of the theory. The symplectic contribution to n-point functions vanishes when two arguments co-incide. The UV cancellation mechanism has nothing to do with the finite measurement resolution which corresponds to the size of the causal diamonds inside which the space-time sheets representing radiative corrections are. There is also IR cutoff due to the presence of largest causal diamond.

On can decompose the radiative corrections two two types. First kind of corrections appear both at the level of positive/and negative energy parts of zero energy states. Second kind of corrections appear at the level of interactions between them. This decomposition is standard in quantum field theories and corresponds to the renormalization constants of fields resp. renormalization of coupling constants. The corrections due to the increase of measurement resolution in time comes as very specific corrections to positive and negative energy states involving gluing of smaller causal diamonds to the upper and lower boundaries of causal diamonds along any radial light-like ray. The radiative corresponds to the interactions correspond to the addition of smaller causal diamonds in the interior of the larger causal diamond. Scales for the corrections come as scalings in powers of 2 rather than as continuous scaling of measurement resolution.

2. Conformal symmetries

The basic questions are the following ones. How hyper-octonionic/-quaternionic/-complex super-conformal symmetry relates to the super-canonical conformal symmetry at the imbedding space level and the super Kac-Moody symmetry associated with the light-like 3-surfaces? How do the dual HO=M⁸ and H=M⁴×CP₂ descriptions (number theoretic compactifcation) relate?

Concerning the understanding of these issues, the earlier construction of physical states poses strong constraints.

1. The state construction utilizes both super-canonical and super Kac-Moody algebras. Super-canonical algebra has negative conformal weights and creates tachyonic ground states from which Super Kac-Moody algebra generates states with non-negative conformal weight determining the mass squared value of the state. The commutator of these two algebras annihilates the physical states. This requires that both super conformal algebras must allow continuation to hyper-octonionic algebras, which are independent.
2. The light-like radial coordinate at dM⁴_± can be continued to a hyper-complex coordinate in M²_± defined the preferred commutative plane of non-physical polarizations, and also to a hyper-quaternionic coordinate in M⁴_±. Hence it would seem that super-canonical algebra can be continued to an algebra in M²_± or perhaps in the entire M⁴_±. This would allow to continue also the operators G, L and other super-canonical operators to operators in hyper-quaternionic M⁴_± needed in stringy perturbation theory.
3. Also the super KM algebra associated with the light-like 3-surfaces should be continuable to hyper-quaternionic M⁴_±. Here HO-H duality comes in rescue. It requires that the preferred hyper-complex plane M² is contained in the tangent plane of the space-time sheet at each point, in particular at light-like 3-surfaces. We already know that this allows to assign a unique space-time surface to a given collection of light-like 3-surfaces as hyper-quaternionic 4-surface of HO hypothesized to correspond to (an obviously preferred) extremal of Kähler action. An equally important implication is that the light-like coordinate of X³ can be continued to hyper-complex coordinate M² coordinate and thus also to hyperquaternionic M⁴ coordinate.
4. The four-momentum appears in super generators G_n and L_n. It seems that the formal Fourier transform of four-momentum components to gradient operators to M⁴_± is needed and defines these operators as particular elements of the CH Clifford algebra elements extended to fields in imbedding space.

3. What about stringy perturbation theory?

The analog of stringy perturbation theory does not seems only a highly attractive but also an unavoidable outcome since a generalization of massless fermionic propagator is needed. The inverse for the sum of super Kac-Moody and super-canonical super-Virasoro generators G (L) extended to an operator acting on the difference of the M⁴ coordinates of the end points of the propagator line connecting two partonic 2-surfaces should appear as fermionic (bosonic) propagator in stringy perturbation theory. Virasoro conditions imply that only G₀ and L₀ appear as propagators. Momentum eigenstates are not strictly speaking possible since since discretization is present due to the finite measurement resolution. One can however represent these states using Fourier transform as a superposition of momentum eigenstates so that standard formalism can be applied.

Symplectic QFT gives an additional multiplicative contribution to n-point functions and there would be also braiding S-matrices involved with the propagator lines in the case that partonic 2-surface carriers more than 1 point. This leaves still modular degrees of freedom of the partonic 2-surfaces describable in terms of elementary particle vacuum functionals and the proper treatment of these degrees of freedom remains a challenge.

4. What about non-hermiticity of the CH super-generators carrying fermion number?

TGD represents also a rather special challenge, which actually represents the fundamental difference between quantum TGD and super string models. The assignment of fermion number to CH gamma matrices and thus also to the super-generator G is unavoidable. Also M⁴ and H gamma matrices carry fermion number. This has been a long-standing interpretational problem in quantum TGD and I have been even ready to give up the interpretation of four-momentum operator appearing in G_n and L_n as actual four-momenta. The manner to get rid of this problem would be the assumption of Majorana property but this would force to give up the interpretation of different imbedding space chiralities in terms of conserved lepton and quark numbers and would also lead to super-string theory with critical dimension 10 or 11. A further problem is how to obtain amplitudes which respect fermion number conservation using string perturbation theory if 1/G=G₀^f/L₀ carries fermion number.

The recent picture does not leave many choices so that I was forced to face the truth and see how everything falls down to this single nasty detail! It became as a total surprise that gamma matrices carrying fermion number do not cause any difficulties in zero energy ontology and make sense even in the ordinary Feynman diagrammatics.

1. Non-hermiticity of G means that the center of mass terms CH gamma matrices must be distinguished from their Hermitian conjugates. In particular, one has g₀ ¹ g₀^f. One can interpret the fermion number carrying M⁴ gamma matrices of the complexified quaternion space appearing naturally in number theoretical framework.
2. One might think that M⁴×CP₂ gamma matrices carrying fermion number is a catastrophe but this is not the case in a massless theory. Massless momentum eigen states can be created by the operator p^kg_k^f from a vacuum annihilated by gamma matrices and satisfying massless Dirac equation. For instance, the conserved fermion number defined by the integral of \overline{Y}g⁰Y over 3-space gives just its standard value. A further experimentation shows that Feynman diagrams with non-hermitian gamma matrices give just the standard results since fermionic propagator and boson-emission vertices give compensating fermion numbers.
3. If the theory would contain massive fermions or a coupling to a scalar Higgs, a catastrophe would result. Hence ordinary Higgs mechanism is not possible in this framework. Of course, also the quantization of fermions is totally different. In TGD fermion mass is not a scalar in H. Part of it is given by CP₂ Dirac operator, part by p-adic thermodynamics for L₀, and part by Higgs field which behaves like vector field in CP₂ degrees of freedom, so that the catastrophe is avoided.
4. In zero energy ontology zero energy states are characterized by M-matrix elements constructed by applying the combination of stringy and symplectic Feynman rules and fermionic propagator is replaced with its super-conformal generalization reducing to an ordinary fermionic propagator for massless states. The norm of a single fermion state is given by a propagator connecting positive energy state and its conjugate with the propagator G₀/L₀ and the standard value of the norm is obtained by using Dirac equation and the fact that Dirac operator appears also in G₀.
5. The hermiticity of super-generators G would require Majorana property and one would end up with superstring theory with critical dimension D=10 or D=11 for the imbedding space. Hence the new interpretation of gamma matrices, proposed already years ago, has very profound consequences and convincingly demonstrates that TGD approach is indeed internally consistent.

For more details see the previous posting, the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix", and the article Topological Geometrodynamics: What Might Be the First Principles?.

What goes wrong with string models?

Something certainly goes wrong with super-string models and M-theory. But what this something is? One could of course make the usual list involving spontaneous compactification, landscape, non-predictivity, and all that. The point I however want to make relates to the relationship of string models to quantum field theories.

The basic wisdom has been that when Feynman graphs of quantum field theories are replaced by their stringy variants everything is nice and finite. The problem is that stringy diagrams do not describe what elementary particles are doing and quantum field theory limit is required at low energies. This non-renormalizable QFT limit is obtained by the ad hoc procedure called spontaneous compactification, and leads to all this misery that spoils the quality of our life nowadays.

My intention is not to ridicule or accuse string theorists. I feel also myself very very stupid since I realized only now what the relationship between super-conformal and super-symplectic QFTs and generalized Feynman diagrams is in TGD framework: I described this already in the previous posting but did not want to make noise of my stupidity. This discovery (discovery only at level of my own subjective experience) was just becoming aware about something which should have been absolutely obvious for anyone with IQ above 20;-).

A very brief summary goes as follows.

1. M-matrix elements characterize the time like entanglement between positive and negative energy parts of zero energy states. The positive/negative energy part can be localized to the boundary of past/future directed light-cone and these light-cones form a causal diamond.
2. M-matrix elements can be expressed in terms of generalized Feynman diagrams with the lines of Feynman diagrams replaced with light-like 3-surfaces glued together along their 2-D ends representing vertices. For a given Feynman diagram of this kind one assigns an n-point function with additional intermediate points coming from the generalized vertices. In hyper-octonionic conformal field theory approach these vertices are fixed uniquely.
3. The amplitude associated with a given generalized Feynman diagram is calculated by a recursive procedure using the fusion rules of a combination of conformal and symplectic QFT:s as described in the previous posting.

What this means that a fusion of generalizations of stringy conformal QFT to conformal-symplectic QFT and of ordinary QFT gives M-matrix elements. Feynman diagrams are not given up! Only the manner how they are computed is completely new: instead of the iterative approach one uses recursive approach based on fusion rules and involving automatically the cutoff which has interpretation in terms of finite measurement resolution.

For more details see the previous posting, the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix", and the article Topological Geometrodynamics: What Might Be the First Principles?.

Could a symplectic analog of conformal field theory be relevant for quantum TGD?

Symplectic (or canonical as I have called them) symmetries of dM⁴₊×CP₂ (light-cone boundary briefly) act as isometries of the "world of classical worlds". One can see these symmetries as analogs of Kac-Moody type symmetries with symplectic transformations of S²×CP₂, where S² is r_M=constant sphere of lightcone boundary, made local with respect to the light-like radial coordinate r_M taking the role of complex coordinate. Thus finite-dimensional Lie group G is replaced with infinite-dimensional group of symplectic transformations. This inspires the question whether a symplectic analog of conformal field theory at dM⁴₊×CP₂ could be relevant for the construction of n-point functions in quantum TGD and what general properties these n-point functions would have.

1 Symplectic QFT at sphere

Actually the notion of symplectic QFT emerged as I tried to understand the properties of cosmic microwave background which comes from the sphere of last scattering which corresponds roughly to the age of 5×10⁵ years. In this situation vacuum extremals of Kähler action around almost unique critical Robertson-Walker cosmology imbeddable in M⁴×S², where there is homologically trivial geodesic sphere of CP₂. Vacuum extremal property is satisfied for any space-time surface which is surface in M⁴×Y², Y² a Lagrangian sub-manifold of CP₂ with vanishing induced Kähler form. Symplectic transformations of CP₂ and general coordinate transformations of M⁴ are dynamical symmetries of the vacuum extremals so that the idea of symplectic QFT emerges natural. Therefore I shall consider first symplectic QFT at the sphere S² of last scattering with temperature fluctuation DT/T proportional to the fluctuation of the metric component g_aa in Robertson-Walker coordinates.

1. In quantum TGD the symplectic transformation of the light-cone boundary would induce action in the "world of classical worlds" (light-like 3-surfaces). In the recent situation it is convenient to regard perturbations of CP₂ coordinates as fields at the sphere of last scattering (call it S²) so that symplectic transformations of CP₂ would act in the field space whereas those of S² would act in the coordinate space just like conformal transformations. The deformation of the metric would be a symplectic field in S². The symplectic dimension would be induced by the tensor properties of R-W metric in R-W coordinates: every S² coordinate index would correspond to one unit of symplectic dimension. The symplectic invariance in CP₂ degrees of freedom is guaranteed if the integration measure over the vacuum deformations is symplectic invariant. This symmetry does not play any role in the sequel.
2. For a symplectic scalar field n ³ 3-point functions with a vanishing anomalous dimension would be functions of the symplectic invariants defined by the areas of geodesic polygons defined by subsets of the arguments as points of S². Since n-polygon can be constructed from 3-polygons these invariants can be expressed as sums of the areas of 3-polygons expressible in terms of symplectic form. n-point functions would be constant if arguments are along geodesic circle since the areas of all sub-polygons would vanish in this case. The decomposition of n-polygon to 3-polygons brings in mind the decomposition of the n-point function of conformal field theory to products of 2-point functions by using the fusion algebra of conformal fields (very symbolically F_kF_l = c_kl^mF_m). This intuition seems to be correct.
3. Fusion rules stating the associativity of the products of fields at different points should generalize. In the recent case it is natural to assume a non-local form of fusion rules given in the case of symplectic scalars by the equation
   
   

   Fk(s1)Fl(s2) = ó õ cklmf(A(s1,s2,s3))Fm(s)dms

   Here the coefficients c_kl^m are constants and A(s₁,s₂,s₃) is the area of the geodesic triangle of S² defined by the sympletic measure and integration is over S² with symplectically invariant measure dm_s defined by symplectic form of S². Fusion rules pose powerful conditions on n-point functions and one can hope that the coefficients are fixed completely.
4. The application of fusion rules gives at the last step an expectation value of 1-point function of the product of the fields involves unit operator term òc_klf(A(s₁,s₂,s))Id dm_s so that one has
   
   

   áFk(s1)Fl(s2)ñ = ó õ cklf(A(s1,s2,s))dms.

   Hence 2-point function is average of a 3-point function over the third argument. The absence of non-trivial symplectic invariants for 1-point function means that n=1- an are constant, most naturally vanishing, unless some kind of spontaneous symmetry breaking occurs. Since the function f(A(s₁,s₂,s₃)) is arbitrary, 2-point correlation function can have both signs. 2-point correlation function is invariant under rotations and reflections.

2 Symplectic QFT with spontaneous breaking of rotational and reflection symmetries

CMB data suggest breaking of rotational and reflection symmetries of S². A possible mechanism of spontaneous symmetry breaking is based on the observation that in TGD framework the hierarchy of Planck constants assigns to each sector of the generalized imbedding space a preferred quantization axes. The selection of the quantization axis is coded also to the geometry of "world of classical worlds", and to the quantum fluctuations of the metric in particular. Clearly, symplectic QFT with spontaneous symmetry breaking would provide the sought-for really deep reason for the quantization of Planck constant in the proposed manner.

1. The coding of angular momentum quantization axis to the generalized imbedding space geometry allows to select South and North poles as preferred points of S². To the three arguments s₁,s₂,s₃ of the 3-point function one can assign two squares with the added point being either North or South pole. The difference
   
   

   DA(s1,s2,s3) º A(s1,s2,s3,N)-A(s1,s2,s3,S)

   of the corresponding areas defines a simple symplectic invariant breaking the reflection symmetry with respect to the equatorial plane. Note that DA vanishes if arguments lie along a geodesic line or if any two arguments co-incide. Quite generally, symplectic QFT differs from conformal QFT in that correlation functions do not possess singularities.
2. The reduction to 2-point correlation function gives a consistency conditions on the 3-point functions
   
   

   á(Fk(s1)Fl(s2))Fm(s3)ñ

   
   

   = cklr ó õ f(DA(s1,s2,s))áFr(s)Fm(s3)ñdms

   
   

   =cklrcrm ó õ f(DA(s1,s2,s)) f(DA(s,s3,t))dmsdmt.

   Associativity requires that this expression equals to áF_k(s₁)(F_l(s₂)F_m(s₃))ñ and this gives additional conditions. Associativity conditions apply to f(DA) and could fix it highly uniquely.
3. 2-point correlation function would be given by
   
   

   áFk(s1)Fl(s2)ñ = ckl ó õ f(DA(s1,s2,s)) dms
4. There is a clear difference between n > 3 and n=3 cases: for n > 3 also non-convex polygons are possible: this means that the interior angle associated with some vertices of the polygon is larger than p. n=4 theory is certainly well-defined, but one can argue that so are also n > 4 theories and skeptic would argue that this leads to an inflation of theories. TGD however allows only finite number of preferred points and fusion rules could eliminate the hierarchy of theories.
5. To sum up, the general predictions are following. Quite generally, for f(0)=0 n-point correlation functions vanish if any two arguments co-incide which conforms with the spectrum of temperature fluctuations. It also implies that symplectic QFT is free of the usual singularities. For symmetry breaking scenario 3-point functions and thus also 2-point functions vanish also if s₁ and s₂ are at equator. All these are testable predictions using ensemble of CMB spectra.

3 Generalization to quantum TGD

Since number theoretic braids are the basic objects of quantum TGD, one can hope that the n-point functions assignable to them could code the properties of ground states and that one could separate from n-point functions the parts which correspond to the symplectic degrees of freedom acting as symmetries of vacuum extremals and isometries of the 'world of classical worlds'.

1. This approach indeed seems to generalize also to quantum TGD proper and the n-point functions associated with partonic 2-surfaces can be decomposed in such a manner that one obtains coefficients which are symplectic invariants associated with both S² and CP₂ Kähler form.
2. Fusion rules imply that the gauge fluxes of respective Kähler forms over geodesic triangles associated with the S² and CP₂ projections of the arguments of 3-point function serve basic building blocks of the correlation functions. The North and South poles of S² and three poles of CP₂ can be used to construct symmetry breaking n-point functions as symplectic invariants. Non-trivial 1-point functions vanish also now.
3. The important implication is that n-point functions vanish when some of the arguments co-incide. This might play a crucial role in taming of the singularities: the basic general prediction of TGD is that standard infinities of local field theories should be absent and this mechanism might realize this expectation.

Next some more technical but elementary first guesses about what might be involved.

1. It is natural to introduce the moduli space for n-tuples of points of the symplectic manifold as the space of symplectic equivalence classes of n-tuples.

   i) In the case of sphere S² convex n-polygon allows n+1 3-sub-polygons and the areas of these provide symplectically invariant coordinates for the moduli space of symplectic equivalence classes of n-polygons (2ⁿ-D space of polygons is reduced to n+1-D space). For non-convex polygons the number of 3-sub-polygons is reduced so that they seem to correspond to lower-dimensional sub-space.

   ii) In the case of CP₂ n-polygon allows besides the areas of 3-polygons also 4-volumes of 5-polygons as fundamental symplectic invariants. The number of independent 5-polygons for n-polygon can be obtained by using induction: once the numbers N(k,n) of independent k £ n-simplices are known for n-simplex, the numbers of k £ n+1-simplices for n+1-polygon are obtained by adding one vertex so that by little visual gymnastics the numbers N(k,n+1) are given by N(k,n+1) = N(k-1,n)+N(k,n). In the case of CP₂ the allowance of 3 analogs {N,S,T} of North and South poles of S² means that besides the areas of polygons (s₁,s₂,s₃), (s₁,s₂,s₃,X), (s₁,s₂,s₃,X,Y), and (s₁,s₂,s₃,N,S,T) also the 4-volumes of 5-polygons (s₁,s₂,s₃,X,Y), and of 6-polygon (s₁,s₂,s₃,N,S,T), X,Y Î {N,S,T} can appear as additional arguments in the definition of 3-point function.
2. What one really means with symplectic tensor is not clear since the naive first guess for the n-point function of tensor fields is not manifestly general coordinate invariant. For instance, in the model of CMB, the components of the metric deformation involving S² indices would be symplectic tensors. Tensorial n-point functions could be reduced to those for scalars obtained as inner products of tensors with Killing vector fields of SO(3) at S². Again a preferred choice of quantization axis would be introduced and special points would correspond to the singularities of the Killing vector fields.
   The decomposition of Hamiltonians of the "world of classical worlds" expressible in terms of Hamiltonians of S²×CP₂ to irreps of SO(3) and SU(3) could define the notion of symplectic tensor as the analog of spherical harmonic at the level of configuration space. Spin and gluon color would have natural interpretation as symplectic spin and color. The infinitesimal action of various Hamiltonians on n-point functions defined by Hamiltonians and their super counterparts is well-defined and group theoretical arguments allow to deduce general form of n-point functions in terms of symplectic invariants.
3. The need to unify p-adic and real physics by requiring them to be completions of rational physics, and the notion of finite measurement resolution suggest that discretization of also fusion algebra is necessary. The set of points appearing as arguments of n-point functions could be finite in a given resolution so that the p-adically troublesome integrals in the formulas for the fusion rules would be replaced with sums. Perhaps rational/algebraic variants of S²×CP₂=SO(3)/SO(2)×SU(3)/U(2) obtained by replacing these groups with their rational/algebraic variants are involved. Tedrahedra, octahedra, and dodecahedra suggest themselves as simplest candidates for these discretized spaces.

   Also the symplectic moduli space would be discretized to contain only n-tuples for which the symplectic invariants are numbers in the allowed algebraic extension of rationals. This would provide an abstract looking but actually very concrete operational approach to the discretization involving only areas of n-tuples as internal coordinates of symplectic equivalence classes of n-tuples. The best that one could achieve would be a formulation involving nothing below measurement resolution.
4. This picture based on elementary geometry might make sense also in the case of conformal symmetries. The angles associated with the vertices of the S² projection of n-polygon could define conformal invariants appearing in n-point functions and the algebraization of the corresponding phases would be an operational manner to introduce the space-time correlates for the roots of unity introduced at quantum level. In CP₂ degrees of freedom the projections of n-tuples to the homologically trivial geodesic sphere S² associated with the particular sector of CH would allow to define similar conformal invariants. This framework gives dimensionless areas (unit sphere is considered). p-Adic length scale hypothesis and hierarchy of Planck constants would bring in the fundamental units of length and time in terms of CP₂ length.

The recent view about M-matrix described in is something almost unique determined by Connes tensor product providing a formal realization for the statement that complex rays of state space are replaced with N rays where N defines the hyper-finite sub-factor of type II₁ defining the measurement resolution. M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state and need not be unitary. It is identified as square root of density matrix with real expressible as product of of real and positive square root and unitary S-matrix. This S-matrix is what is measured in laboratory. There is also a general vision about how vertices are realized: they correspond to light-like partonic 3-surfaces obtained by gluing incoming and outgoing partonic 3-surfaces along their ends together just like lines of Feynman diagrams. Note that in string models string world sheets are non-singular as 2-manifolds whereas 1-dimensional vertices are singular as 1-manifolds. These ingredients we should be able to fuse together. So we try once again!

1. Iteration starting from vertices and propagators is the basic approach in the construction of n-point function in standard QFT. This approach does not work in quantum TGD. Symplectic and conformal field theories suggest that recursion replaces iteration in the construction of given generalized Feynman diagram. One starts from an n-point function and reduces it step by step to a vacuum expectation value of a 2-point function using fusion rules. Associativity becomes the fundamental dynamical principle in this process. Associativity in the sense of classical number fields has already shown its power and led to a hyper-octoninic formulation of quantum TGD promising a unification of various visions about quantum TGD .
2. Let us start from the representation of a zero energy state in terms of a causal diamond defined by future and past directed light-cones. Zero energy state corresponds to a quantum superposition of light-like partonic 3-surfaces each of them representing possible particle reaction. These 3-surfaces are very much like generalized Feynman diagrams with lines replaced by light-like 3-surfaces coming from the upper and lower light-cone boundaries and glued together along their ends at smooth 2-dimensional surfaces defining the generalized vertices.
3. It must be emphasized that the generalization of ordinary Feynman diagrammatics arises and conformal and symplectic QFTs appear only in the calculation of single generalized Feynman diagram. Therefore one could still worry about loop corrections. The fact that no integration over loop momenta is involved and there is always finite cutoff due to discretization together with recursive instead of iterative approach gives however good hopes that everything works.
4. One can actually simplify things by identifying generalized Feynman diagrams as maxima of Kähler function with functional integration carried over perturbations around it. Thus one would have conformal field theory in both fermionic and configuration space degrees of freedom. The light-like time coordinate along light-like 3-surface is analogous to the complex coordinate of conformal field theories restricted to some curve. If it is possible continue the light-like time coordinate to a hyper-complex coordinate in the interior of 4-D space-time sheet, the correspondence with conformal field theories becomes rather concrete. Same applies to the light-like radial coordinates associated with the light-cone boundaries. At light-cone boundaries one can apply fusion rules of a symplectic QFT to the remaining coordinates. Conformal fusion rules are applied only to point pairs which are at different ends of the partonic surface and there are no conformal singularities since arguments of n-point functions do not co-incide. By applying the conformal and symplectic fusion rules one can eventually reduce the n-point function defined by the various fermionic and bosonic operators appearing at the ends of the generalized Feynman diagram to something calculable.
5. Finite measurement resolution defining the Connes tensor product is realized by the discretization applied to the choice of the arguments of n-point functions so that discretion is not only a space-time correlate of finite resolution but actually defines it. No explicit realization of the measurement resolution algebra N seems to be needed. Everything should boil down to the fusion rules and integration measure over different 3-surfaces defined by exponent of Kähler function and by imaginary exponent of Chern-Simons action. The continuation of the configuration space Clifford algebra for 3-surfaces with cm degrees of freedom fixed to a hyper-octonionic variant of gamma matrix field of super-string models defined in M⁸ (hyper-octonionic space) and M⁸« M⁴×CP₂ duality leads to a unique choice of the points, which can contribute to n-point functions as intersection of M⁴ subspace of M⁸ with the counterparts of partonic 2-surfaces at the boundaries of light-cones of M⁸. Therefore there are hopes that the resulting theory is highly unique. Symplectic fusion algebra reduces to a finite algebra for each space-time surface if this picture is correct.
6. Consider next some of the details of how the light-like 3-surface codes for the fusion rules associated with it. The intermediate partonic 2- surfaces must be involved since otherwise the construction would carry no information about the properties of the light-like 3-surface, and one would not obtain perturbation series in terms of the relevant coupling constants. The natural assumption is that partonic 2-surfaces belong to future/past directed light-cone boundary depending on whether they are on lower/upper half of the causal diamond. Hyper-octonionic conformal field approach fixes the n_int points at intermediate partonic two-sphere for a given light-like 3-surface representing generalized Feynman diagram, and this means that the contribution is just N-point function with N=n_out+n_int+n_in calculable by the basic fusion rules. Coupling constant strengths would emerge through the fusion coefficients, and at least in the case of gauge interactions they must be proportional to Kähler coupling strength since n-point functions are obtained by averaging over small deformations with vacuum functional given by the exponent of Kähler function. The first guess is that one can identify the spheres S² Ì dM⁴_± associated with initial, final and, and intermediate states so that symplectic n-point functions could be calculated using single sphere.

These findings raise the hope that quantum TGD is indeed a solvable theory. Even if one is not willing to swallow any bit of TGD, the classification of the symplectic QFTs remains a fascinating mathematical challenge in itself. A further challenge is the fusion of conformal QFT and symplectic QFT in the construction of n-point functions. One might hope that conformal and symplectic fusion rules can be treated separately.

For details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix" and the article Topological Geometrodynamics: What Might Be the First Principles?.

Could symplectic QFT allow to understand the fluctuations of CMB?

Depending on one's attitudes, the anomalies of the fluctuation spectrum of the cosmic microwave background (CMB) can be seen as a challenge for people analyzing the experiments or that of the inflationary scenario. I do not pretend to be deeply involved with CMB but as I read about one these anomalies in Sean Carrol's blog and next day in Lubos's blog, I felt that I could spend some days by clarifying myself what is involved.

What interests me whether the replacement of inflation with quantum criticality and hbar changing phase transitions could provide fresh insights about fluctuations and anomalies of CMB. In the following I try first to explain to myself what the anomalies are and after that I will consider some TGD inspired crazy (as always) ideas. My motivations to communicate are indeed strong: the consideration of the anomalies led to a generalization of the notion of conformal QFT to what might be called symplectic QFT having very natural place also in quantum TGD proper.

A brief summary about my views is as follows.

1. There are several types of anomalies (with respect to the expectations motivated by inflation theory). Fluctuation spectrum shows hot and cold spots; there is so called hemispherical asymmetry in the spectrum; rotationally averaged two-point correlation function C(q) is vanishing for angle separations above 60 degrees; the so called cosmic axis of evil means that 3 of the multipole area vectors assignable to the l=2 and l=3 spherical harmonics in the expansion of C(q) are aligned and in the galactic plane (very near to ecliptic) and in roughly the same direction as the dipole corresponding to the motion of the Milky Way with respect to cosmic frame of reference; there is also evidence for non-Gaussianity meaning non-vanishing 3-point functions. Especially strange finding is that the features of the local geometry seems to reflect themselves in CMB at the surface of last scattering. If these findings are not artifacts of the analysis or pure accidents, the consequences for our understanding of the Cosmos would be dramatic.
2. In TGD framework quantum criticality replaces inflation. This means that the fluctuations of CMB do not correspond to primordial fluctuations of inflaton field evolved into large scale fluctuations during rapid expansion but to long range fluctuations involved with a phase transition increasing Planck constant and occurring at the time of decoupling. The p-adic length scale involved with the sphere of last scattering and the amplitude of the fluctuations provide two dimensionless couplings and allow to make estimates for the scaling of Planck constant in this transition.
3. I have suggested earlier a conformal field theory defined at the sphere of last scattering as a TGD based model for the anomalies. The analysis of the situation however demonstrates that a more natural approach is based on symplectic variant of conformal QFT at the sphere of last scattering. By combining generalized fusion rules with the knowledge about symplectic invariants associated with 3 or 4 points of sphere, one can deduce surprisingly detailed information about the n-point functions of symplectic QFTs. There are two variants of the theory: the first one is rotation- and reflection symmetric. The fact that the generalization of the notion of imbedding space allows to identify preferred quantization axis, allows to formulate also a variant theory breaking these symmetries. Fusion rules determine n-point functions highly uniquely in terms of 3-point functions expressible as functions of simple symplectic invariants. Symplectic QFT makes special predicions distinguishing it from inflationary models: a sizable non-Gaussianity is predicted and correlation functions vanish when any two arguments are very near to each other. It is certainly possible to reproduce the vanishing of C(q) for large values of q but it is not clear whether fusion rules allow this.
4. Quite generally, symplectic QFT provides a long sought-for manner to describe the vacuum degeneracy of TGD in terms of n-point functions. What is of special importance is that the n-point functions have no singularities at the limit when some arguments co-incide. This means a profound distinction from quantum field theories and something like this is required by general arguments demonstrating that quantum TGD is free of the standard divergences due to the non-locality of Kähler function as a functional of 3-surface. The classification of symplectic QFTs should be a fascinating challenge for a mathematician. The basic challenge is to determine how uniquely fusion rules determine the 3-point functions generating all other n-point functions.
5. The possibility of having gigantic values of gravitational Planck constant and zero energy ontology suggests that quantum measurements in cosmological scales are possible. This would mean that time-like entanglement between positive and negative energy parts of zero energy state could correlate galactic geometry with the geometry of fluctuation spectrum so that the hemispherical asymmetry with respect to galactic plane could be produced by this kind of quantum measurement. This would mean a dramatic proof of the notion of participatory Universe introduced by Wheeler.

For details see the chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Space-Time" or the article Could symplectic quantum field theory allow to model the fluctuations cosmic microwave background?.

About the arrow of psychological time and the notion of self

Quantum classical correspondence predicts that the arrow of subjective time is somehow mapped to that for the geometric time. The detailed mechanism for how the arrow of psychological time emerges has however remained open and I have discussed a handful of alternative explanations. Zero energy ontology and the identification of the space-time correlate of self as causal diamond of imbedding space rather than space-time sheet leads however to a resolution of the problem and precise quantitative predictions.

Also the notion of self has been problematic: the original view identified self as a subsystem able to remain unentangled. It was assumed that the sequence of quantum jumps experienced by self integrates to a stream of consciousness in this kind of situation: self would be for quantum jump what atom is for elementary particles. A more ambitious idea inspired by dark matter hierarchy was that the notion of self hierarchy reduces to a fractal hierarchy of quantum jumps within quantum jumps. It indeed seems that this reduction is consistent with the original definition of self. In the following I shall discuss the new view briefly.

1. About the arrow of psychological time

The explanation for the arrow of psychological time based on zero energy ontology is favored by Occam's razor since it uses only the assumption that space-time sheets are replaced by more evolved ones in each quantum jump. Also the model of topological quantum computation favors it.

1. In standard picture the attention would gradually shift towards geometric future and space-time in 4-D sense would remain fixed. Now however the fact that quantum state is quantum superposition of space-time surfaces allows to assume that the attention of the conscious observer is directed to a fixed volume of 8-D imbedding space. Quantum classical correspondence is achieved if the evolution in a reasonable approximation means shifting of the space-time sheets and corresponding field patterns backwards backwards in geometric time by some amount per quantum jump so that the perceiver finds the geometric future in 4-D sense to enter to the perceptive field. This makes sense since the shift with respect to M⁴ time coordinate is an exact symmetry of extremals of Kähler action. It is also an excellent approximate symmetry for the preferred extremals of Kähler action and thus for maxima of Kähler function spoiled only by the presence of light-cone boundaries. This shift occurs for both the space-time sheet that perceiver identifies itself and perceived space-time sheet representing external world: both perceiver and percept change.
2. Both the landscape and observer space-time sheet remain in the same position in imbedding space but both are modified by this shift in each quantum jump. The perceiver experiences this as a motion in 4-D landscape. Perceiver (Mohammed) would not drift to the geometric future (the mountain) but geometric future (the mountain) would effectively come to the perceiver (Mohammed)!
3. There is an obvious analogy with Turing machine: what is however new is that the tape effectively comes from the geometric future and Turing machine can modify the entire incoming tape by intentional action. This analogy might be more than accidental and could provide a model for quantum Turing machine operating in TGD Universe. This Turing machine would be able to change its own program as a whole by using the outcomes of the computation already performed.
4. The concentration of the sensory input and the effects of conscious motor action to a narrow interval of time (.1 seconds typically, secondary p-adic time scale associated with the largest Mersenne M₁₂₇ defining p-adic length scale which is not completely super-astronomical) can be understood as a concentration of sensory/motor attention to an interval with this duration: the space-time sheet representing sensory "me" would have this temporal length and "me" definitely corresponds to a zero energy state.
5. The fractal view about topological quantum computation strongly suggests an ensemble of almost copies of sensory "me" scattered along my entire life cycle and each of them experiencing my life as a separate almost copy. My childhood is still sensorily lived but has moved about 57 years backwards in geometric time and would live the year 1897 but enjoy all techno conveniences of the year 1950!
6. The model of geometric and subjective memories would not be modified in an essential manner: memories would result when "me" is connected with my almost copy in the geometric past by braid strands or massless extremals (MEs) or their combinations (ME parallel to magnetic flux tube is the analog of Alfwen wave in TGD).

This argument leaves many questions open. What is the precise definition for the volume of attention? Is the attention of self doomed to be directed to a fixed volume or can quantum jumps change the volume of attention? What distinguishes between geometric future and past as far as contents of conscious experience are considered? How this picture relates to p-adic and dark matter hierarchies? Does this framework allow to formulate more precisely the notion of self? Zero energy ontology allows to give tentative answers to these questions.

2. About the notion of self

I have considered two different notions of "self" and it is interesting to see whether the new view about time might allow to choose between them or to show that they are actually equivalent.

1. In the original variant of the theory "self" corresponds to a sequence of quantum jumps. "Self" would result through a binding of quantum jumps to single "selftring" in close analogy and actually in a concrete correspondence with the formation of bound states. Each quantum jump has a fractal structure: unitary process is followed by a sequence of state function reductions and preparations proceeding from long to short scales. Selves can have sub-selves and one has self hierarchy. The questionable assumption is that self remains conscious only as long as it is able to avoid entanglement with environment.
   Even slightest entanglement would destroy self unless on introduces the notion of finite measurement resolution applying also to entanglement. This notion is indeed central for entire quantum TGD also leads to the notion of sharing of mental images: selves unentangled in the given measurement resolution can experience shared mental images resulting as fusion of sub-selves by entanglement not visible in the resolution used.
2. According to the newer variant of theory, quantum jump has a fractal structure so that there are quantum jumps within quantum jumps: this hierarchy of quantum jumps within quantum jumps would correspond to the hierarchy of dark matters labeled by the values of Planck constant. Each fractal structure of this kind would have highest level (largest Planck constant) and this level would corresponds to the self. What might be called irreducible self would corresponds to a quantum jump without any sub-quantum jumps (no mental images). The quantum jump sequence for lower levels of dark matter hierarchy would create the experience of flow of subjective time.

It would be nice to reduce the original notion of self hierarchy to the hierarchy defined by quantum jumps. There are some objections against this idea. One can argue that fractality is a purely geometric notion and since subjective experience does not reduce to the geometry it might be that the notion of fractal quantum jump does not make sense. It is also not quite clear whether the reasonable looking idea about the role of entanglement as destroyer of self can be kept in the fractal picture.

These objections fail if one can construct a well-defined mathematical scheme allowing to understand what fractality of quantum jump at the level of space-time correlates means and showing that the two views about self are equivalent. The following argument represents such a proposal. Let us start from the causal diamond model as a lowest approximation for a model of zero energy states and for the space-time region defining the contents of sensory experience.

Let us make the following assumptions.

1. Assume the hierarchy of causal diamonds within causal diamonds in a sense to be specified more precisely below. Causal diamonds would represent the volumes of attention. Assume that the highest level in this hierarchy defines the quantum jump containing sequences of lower level quantum jumps in some sense to be specified. Assume that these quantum jumps integrate to single continuous stream of consciousness as long as the sub...-sub-self in question remains unentangled and that entangling means loss of consciousness or at least that it is not possible to remember anything about contents of consciousness during entangled state.
2. Assume that the contents of conscious experience come from the interior of the causal diamond. A stronger condition would be that the contents come from the boundaries of the two light-cones involved since physical states are defined at these in the simplest picture. In this case one could identify the lower light-cone boundary as giving rise to memory.
3. The time span characterizing the contents of conscious experience associated with a given quantum jump would correspond to the temporal distance T between the tips of the causal diamond. T would also characterize the average and approximate shift of the superposition of space-time surfaces backwards in geometric time in single quantum jump at a given level of hierarchy. This time scale naturally scales as T_n=2ⁿT_CP2 so that p-adic length scale hypothesis follows as a consequence. T would be essentially the secondary p-adic time scale T_2,p=ÖpT_p for p @ 2^k. This assumption - absolutely essential for the hierarchy of quantum jumps within quantum jumps - would differentiate the model from the model in which T corresponds to either CP₂ time scale or p-adic time scale T_p. One would have hierarchy of quantum jumps with increasingly longer time span for memory and with increasing duration of geometric chronon at the highest level of fractal quantum jump. Without additional restrictions, the quantum jump at n^th level would contain 2ⁿ quantum jumps at the lowest level of hierarchy. Note that in the case of sub-self - and without further assumptions which will be discussed next - one would have just two quantum jumps: mental image appears, disappears or exists all the time. At the level of sub-sub-selves 4 quantum jumps and so on. Maybe this kind of simple predictions might be testable.
4. We know that that the contents of sensory experience comes from a rather narrow time interval of duration about .1 seconds, which corresponds to the time scale T₁₂₇ associated with electron. We also know that there is asymmetry between positive and negative energy parts of zero energy states both physically and at the level of conscious experience. This asymmetry must have some space-time correlate. The simplest correlate for the asymmetry between positive and negative energy states would be that the upper light-like boundaries in the structure formed by light-cones within light-cones intersect along light-like radial geodesic. No condition of this kind would be posed on lower light-cone boundaries. The scaling invariance of this condition makes it attractive mathematically and would mean that arbitrarily long time scales T_n can be present in the fractal hierarchy of light cones. At all levels of the hierarchy all contribution from upper boundary of the causal diamond to the conscious experience would come from boundary of same past directed light-cone so that the conscious experience would be sharply localized in time in the manner as we know it to be. The new element would be that content of conscious experience would come from arbitrarily large region of Universe and seing Milky Way would mean direct sensory contact with it.
5. These assumptions relate the hierarchy of quantum jumps to p-adic hierarchy. One can also include also dark matter hierarchy into the picture. For dark matter hierarchy the time scale hierarchy {T_n} is scaled by the factor r=hbar/hbar₀ which can be also rational number. For r=2^k the hierarchy of causal diamonds generalizes without difficulty and there is a kind of resonance involved which might relate to the fact that the model of EEG favors the values of k=11n, where k=11 also corresponds in good approximation to proton-electron mass ratio. For more general values of r the generalization is possible assuming that the position of the upper tip of causal diamond is chosen in such a manner that their positions are always the same whereas the position of the lower light-cone boundary would correspond to {rT_n} for given value of Planck constant. Geometrically this picture generalizes the original idea about fractal hierarchy of quantum jumps so that it contains both p-adic hierarchy and hierarchy of Planck constants.

The contributions from lower the boundaries identifiable in terms of memories would correspond to different time scales and for a given value of time scale T the net contribution to conscious experience would be much weaker than the sensory input in general. The asymmetry between geometric now and geometric past would be present for all contributions to conscious experience, not only sensory ones. What is nice that the contents of conscious experience would rather literally come from the boundary of the past directed light-cone along which the classical signals arrive. Hence the mystic feeling about telepathic connection with a distant object at distance of billions of light years expressed by an astrophysicist, whose name I have unfortunately forgotten, would not be romantic self deception.

This framework explains also the sharp distinction between geometric future and past (not surprisingly since energy and time are dual): this distinction has also been a long standing problem of TGD inspired theory of consciousness. Precognition is not possible unless one assumes that communications and sharing of mental images between selves inside disjoint causal diamonds is possible. Physically there seems to be no good reason to exclude the interaction between zero energy states associated with disjoint causal diamonds.

3. Feedback to Quantum TGD

The mathematical formulation of this intuition is however a non-trivial challenge and can be used to articulate more precisely the views about what configuration space and configurations space spinor fields actually are mathematically.

1. Suppose that the causal diamonds with tips at different points of H=M⁴×CP₂ and characterized by distance between tips T define sectors CH_i of the full configuration space CH ("world of classical worlds"). Precognition would represent an interaction between zero energy states associated with different sectors CH_i in this scheme and tensor factor description is required.
2. Inside given sector CH_i it is not possible to speak about second quantization since every quantum state correspond to a single mode of a classical spinor field defined in that sector.
3. The question is thus whether the Clifford algebras and zero energy states associated with different sectors CH_i combine to form a tensor product so that these zero energy states can interact. Tensor product is required by the vision about zero energy insertions assignable to CH_i which correspond to causal diamonds inside causal diamonds. Also the assumption that zero energy states form an ensemble in 4-D sense - crucial for the deduction of scattering rates from M-matrix - requires tensor product.
4. The argument unifying the two definitions of self requires that the tensor product is restricted when CH_i correspond to causal diamonds inside each other. The tensor factors in shorter time scales are restricted to the causal diamonds hanging from a light-like radial ray at the upper end of the common past directed light-cone. If the causal diamonds are disjoint there is no obvious restriction to be posed, and this would mean the possibility of also precognition and sharing of mental images.

This scenario allows also to answers the questions related to a more precise definition of volume of attention. Causal diamond - or rather - the associated light-like boundaries containing positive and negative energy states define the primitive volume of attention. The obvious question whether the attention of a given self is doomed to be fixed to a fixed volume can be also answered. This is not the case. Selves can delocalize in the sense that there is a wave function associated with the position of the causal diamond and quantum jumps changing this position are possible. Also many-particle states assignable to a union of several causal diamonds are possible. Note that the identification of magnetic flux tubes as space-time correlates of directed attention in TGD inspired quantum biology makes sense if these flux tubes connect different causal diamonds. The directedness of attention in this sense should be also understood: it could be induced from the ordering of p-adic primes and Planck constant: directed attention would be always from longer to shorter scale. For a more background see for instance, the chapter Matter,Mind, Quantum of "TGD Inspired Theory of Consciousness".