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Mathematical Aspects of Consciousness

Note: Newest contributions are at the top!

Year 2006

Algebraic Brahman=Atman Identity and Algebraic Holography

The TGD based view about how fermions and bosons serve as correlates of cognition and intentionality emerges from the notion of infinite primes (see this and this), which was actually the first genuinely new mathematical idea inspired by TGD inspired consciousness theorizing.

Infinite primes, integers, and rationals have a precise number theoretic anatomy. For instance, the simplest infinite primes corresponds to the numbers P+/-= X+/- 1, where X=∏kpk is the product of all finite primes. Indeed, P+/-mod p=1 holds true for all finite primes.

The construction of infinite primes at the first level of the hierarchy is structurally analogous to the quantization of super-symmetric arithmetic quantum field theory with finite primes playing the role of momenta associated with fermions and bosons. Also the counterparts of bound states emerge. This process can be iterated: at the second level the product of infinite primes constructed at the first level replaces X and so on.

The structural similarity with repeatedly second quantized quantum field theory suggests that physics might in some sense reduce to a number theory for infinite rationals M/N and that second quantization could be followed by further quantizations. As a matter fact, the hierarchy of space-time sheets could realize this endless second quantization geometrically and have also a direct connection with the hierarchy of logics labelled by their order. This could have rather breathtaking implications.

  • Could this hierarchy correspond to a hierarchy of realities for which level below corresponds in a literal sense infinitesimals and the level next above to infinity?
  • There is an infinite number of infinite rationals behaving like real units (M/N=1 in real sense) so that space-time points could have infinitely rich number theoretical anatomy not detectable at the level of real physics. Infinite integers would correspond to positive energy many particle states and their inverses (infinitesimals with number theoretic structure) to negative energy many particle states and M/N= 1 would be a counterpart for zero energy ontology to which oneness and emptiness are assigned in mysticism.

  • Single space-time point, which is usually regarded as the most primitive and completely irreducible structure of mathematics, would take the role of Platonia of mathematical ideas being able to represent in its number theoretical structure even the quantum state of entire Universe. Algebraic Brahman=Atman identity and algebraic holography would be realized in a rather literal sense .

This number theoretical anatomy should relate to mathematical consciousness in some manner. For instance, one can ask whether it makes sense to speak about quantum jumps changing the number theoretical anatomy of space-time points and whether these quantum jumps give rise to mathematical ideas. In fact, the identifications of Platonia as spinor fields in WCW (world of classical worlds) on one hand,and as the set number theoretical anatomies of point of imbedding space on the other hand, force the conclusion that configuration space spinor fields (recall also the identification as correlates for logical mind) can be realized in terms of the space for number theoretic anatomies of imbedding space points.

Therefore quantum jumps would correspond to changes in the anatomy of the space-time points. Imbedding space would be experiencing genuine number theoretical evolution. Physics would reduce to the anatomy of numbers. All mathematical notions which are more than mere human inventions would be imbeddable to the Platonia realized as the number theoretical anatomies of single imbedding space point.

This picture give also a justification for the decomposition of WCW to a union of WCW:s associated with imbedding spaces with preferred point (tip of the lightcone and point of CP2 fixing U(2) subgroup as isotropy group). Given point of space-time would provide representation for the spinors fields in WCW associated with the future and/or past light-cone at this point. The "big bang" singularity would code all the information about the quantum state of this particular sub-universe in its number theoretical anatomy.

Interestingly, this picture can be deduced by taking into extreme quantum-classical correspondence and by requiring that both configuration space and configuration space spinor fields have not only space-time correlates but representation at the level of space-time: the only reasonable identification is in terms of algebraic structure of space-time point.

To summarize my own feelings: I strongly feel that number theoretic Brahman=Atman, or number theoretic holography if you prefer western terms, is the deepest idea which I have become conscious of during these 28 years of TGD.

See the chapter Intentionality, Cognition, and Physics as Number theory or Space-Time Point as Platonia . For a brief summary of quantum TGD inspired theory of consciousness see the article TGD Inspired Theory of Consciousness.

Has dark matter been observed?

The group of G. Cantatore has reported an optical rotation of a laser beam in a magnetic field (hep-exp/0507107). The experimental arrangement involves a magnetic field of strength B=5 Tesla. Laser beam travels 22000 times forth and back in a direction orthogonal to the magnetic field travelling 1 m during each pass through the magnet. The wavelength of the laser light is 1064 nm. A rotation of (3.9+/-.5)× 10-12 rad/pass is observed.

A possible interpretation for the rotation would be that the component of photon having polarization parallel to the magnetic field mixes with QCD axion, one of the many candidates for dark matter. The mass of the axion would be about 1 meV. Mixing would imply a reduction of the corresponding polarization component and thus in the generic case induce a rotation of the polarization direction. Note that the laser beam could partially transform to axions, travel through a non-transparent wall, and appear again as ordinary photons.

The disturbing finding is that the rate for the rotation is by a factor 2.8× 104 higher than predicted. This would have catastrophic astrophysical implications since stars would rapidly lose their energy via axion radiation.

The motivation for introducing axion was the large CP breaking predicted by the standard QCD. No experimental evidence has been found has been found for this breaking. The idea is to introduce a new broken U(1) gauge symmetry such that is arranged to cancel the CP violating terms predicted by QCD. Because axions interact very weakly with the ordinary matter they have been also identified as candidates for dark matter p"../articles/.

In TGD framework there is special reason to expect large CP violation analogous to that in QCD although one cannot completely exclude it. Axions are however definitely excluded. TGD predicts a hierarchy of scaled up variants of QCD and entire standard model plus their dark variants corresponding to some preferred p-adic length scales, and these scaled up variants play a key role in TGD based view about nuclear strong force (see this and this), in the explanation of the anomalous production of e+e- pairs in heavy nucleus collisions near Coulomb wall (this), high Tc superconductivity (see this, this, and this), and also in the TGD based model of EEG (see this). Therefore a natural question is whether the particle in question could be a pion of some scaled down variant of QCD having similar coupling to electromagnetic field. Also dark variants of this pion could be considered.

What raises optimism is that the Compton length of the scaled down quarks is of the same order as cyclotron wavelength of electron in the magnetic field in question. For the ordinary value of Planck constant this option however predicts quite too high mixing rate. This suggests that dark matter has been indeed observed in the sense that the pion corresponds to a large value of Planck constant. Here the encouraging observation is that the ratio λc/λ of wavelength of cyclotron photon and laser photon is n=211, which corresponds to the lowest level of the biological dark matter hierarchy with levels characterized the value hbar= 211khbar0, k=1,2,..

The most plausible model is following.

  1. Suppose that the photon transform first to a dark cyclotron photon associated with electron at the lowest n=211 level of the biological dark matter hierarchy. Suppose that the coupling of laser photon to dark photon can be modelled as a coefficient of the usual amplitude apart from a numerical factor of order one equal to αem(n) propto 1/n.
  2. Suppose that the coupling gπNN for the scaled down hadrons is proportional to αs4(n) propto 1/n4 as suggested by a simple model for what happens for the nucleon and pion at quark level in the emission of pion.
Under these assumptions one can understand why only an exotic pion with mass of 1 meV couples to laser photons with wavelength λ= 1 μm in magnetic field B=5 Tesla. The general prediction is that λc/λ must correspond to preferred values of n characterizing Fermat polygons constructible using only ruler and compass, and that the rate for the rotation of polarization depends on photon frequency and magnetic field strength in a manner not explained by the model based on the photon-axion mixing.

The chapter Does TGD Predict the Spectrum of Planck Constants? contains the detailed calculations.

Infinite primes, cognition, and intentionality

Somehow it is obvious that infinite primes (see this) must have some very deep role to play in quantum TGD and TGD inspired theory of consciousness. What this role precisely is has remained an enigma although I have considered several detailed interpretations (see the link above).

In the following an interpretation allowing to unify the views about fermionic Fock states as a representation of Boolean cognition and p-adic space-time sheets as correlates of cognition is discussed. Very briefly, real and p-adic partonic 3-surfaces serve as space-time correlates for the bosonic super algebra generators, and pairs of real partonic 3-surfaces and their algebraically continued p-adic variants as space-time correlates for the fermionic super generators. Intentions/actions are represented by p-adic/real bosonic partons and cognitions by pairs of real partons and their p-adic variants and the geometric form of Fermi statistics guarantees the stability of cognitions against intentional action.

1. Infinite primes very briefly

Infinite primes have a decomposition to infinite and finite parts allowing an interpretation as a many-particle state of a super-symmetric arithmetic quantum field theory for which fermions and bosons are labelled by primes. There is actually an infinite hierarchy for which infinite primes of a given level define the building blocks of the infinite primes of the next level. One can map infinite primes to polynomials and these polynomials in turn could define space-time surfaces or at least light-like partonic 3-surfaces appearing as solutions of Chern-Simons action so that the classical dynamics would not pose too strong constraints.

The simplest infinite primes at the lowest level are of form mBX/sF + nBsF, X=∏i pi (product of all finite primes). mB, nB, and sF are defined as mB= ∏ipimi, nB= ∏iqini, and sF= ∏iqi, mB and nB have no common prime factors. The simplest interpretation is that X represents Dirac sea with all states filled and X/sF + sF represents a state obtained by creating holes in the Dirac sea. The integers mB and nB characterize the occupation numbers of bosons in modes labelled by pi and qi and sF= ∏iqi characterizes the non-vanishing occupation numbers of fermions.

The simplest infinite primes at all levels of the hierarchy have this form. The notion of infinite prime generalizes to hyper-quaternionic and even hyper-octonionic context and one can consider the possibility that the quaternionic components represent some quantum numbers at least in the sense that one can map these quantum numbers to the quaternionic primes.

The obvious question is whether configuration space degrees of freedom and configuration space spinor (Fock state) of the quantum state could somehow correspond to the bosonic and fermionic parts of the hyper-quaternionic generalization of the infinite prime as proposed here. That hyper-quaternionic (or possibly hyper-octonionic) primes would define as such the quantum numbers of fermionic super generators does not make sense. It is however possible to have a map from the quantum numbers labelling super-generators to the finite primes. One must also remember that the infinite primes considered are only the simplest ones at the given level of the hierarchy and that the number of levels is infinite.

2. Precise space-time correlates of cognition and intention

The best manner to end up with the proposal about how p-adic cognitive representations relate bosonic representations of intentions and actions and to fermionic cognitive representations is through the following arguments.

  1. In TGD inspired theory of consciousness Boolean cognition is assigned with fermionic states. Cognition is also assigned with p-adic space-time sheets. Hence quantum classical correspondence suggets that the decomposition of the space-time into p-adic and real space-time sheets should relate to the decomposition of the infinite prime to bosonic and fermionic parts in turn relating to the above mention decomposition of physical states to bosonic and fermionic parts.

    If infinite prime defines an association of real and p-adic space-time sheets this association could serve as a space-time correlate for the Fock state defined by configuration space spinor for given 3-surface. Also spinor field as a map from real partonic 3-surface would have as a space-time correlate a cognitive representation mapping real partonic 3-surfaces to p-adic 3-surfaces obtained by algebraic continuation.

  2. Consider first the concrete interpretation of integers mB and nB. The most natural guess is that the primes dividing mB=∏ipmi characterize the effective p-adicities possible for the real 3-surface. mi could define the numbers of disjoint partonic 3-surfaces with effective pi-adic topology and associated with with the same real space-time sheet. These boundary conditions would force the corresponding real 4-surface to have all these effective p-adicities implying multi-p-adic fractality so that particle and wave pictures about multi-p-adic fractality would be mutually consistent. It seems natural to assume that also the integer ni appearing in mB=∏iqini code for the number of real partonic 3-surfaces with effective qi-adic topology.

  3. Fermionic statistics allows only single genuinely qi-adic 3-surface possibly forming a pair with its real counterpart from which it is obtained by algebraic continuation. Pairing would conform with the fact that nF appears both in the finite and infinite parts of the infinite prime (something absolutely essential concerning the consistency of interpretation!).

    The interpretation could be as follows.

    1. Cognitive representations must be stable against intentional action and fermionic statistics guarantees this. At space-time level this means that fermionic generators correspond to pairs of real effectively qi-adic 3-surface and its algebraically continued qi-adic counterpart. The quantum jump in which qi-adic 3-surface is transformed to a real 3-surface is impossible since one would obtain two identical real 3-surfaces lying on top of each other, something very singular and not allowed by geometric exclusion principle for surfaces. The pairs of boson and fermion surfaces would thus form cognitive representations stable against intentional action.

    2. Physical states are created by products of super algebra generators Bosonic generators can have both real or p-adic partonic 3-surfaces as space-time correlates depending on whether they correspond to intention or action. More precisely, mB and nB code for collections of real and p-adic partonic 3-surfaces. What remains to be interpreted is why mB and nB cannot have common prime factors (this is possible if one allows also infinite integers obtained as products of finite integer and infinite primes).

    3. Fermionic generators to the pairs of a real partonic 3-surface and its p-adic counterpart obtained by algebraic continuation and the pictorial interpretation is as a pair of fermion and hole.

    4. This picture makes sense if the partonic 3-surfaces containing a state created by a product of super algebra generators are unstable against decay to this kind of 3-surfaces so that one could regard partonic 3-surfaces as a space-time representations for a configuration space spinor field.

  4. Are alternative interpretations possible? For instance, could q=mB/mB code for the effective q-adic topology assignable to the space-time sheet as suggested here. That q-adic numbers form a ring but not a number field casts however doubts on this interpretation as does also the general physical picture.

3. Number theoretical universality of S-matrix

The discreteness of the intersection of the real space-time sheet and its p-adic variant obtained by algebraic continuation would be a completely universal phenomenon associated with all fermionic states. This suggests that also real-to-real S-matrix elements involve instead of an integral a sum with the arguments of an n-point function running over all possible combinations of the points in the intersection. S-matrix elements would have a universal form which does not depend on the number field at all and the algebraic continuation of the real S-matrix to its p-adic counterpart would trivialize. Note that also fermionic statistics favors strongly discretization unless one allows Dirac delta functions.

The chapter Infinite Primes and Consciousness contains this piece of text too.

Tree like structure of the extended imbedding space

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

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

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

  2. In the phase transition different between CP2:s the CP2 projection of 3-surface becomes point so that the transition can occur in regions of space-time sheet with 1-D CP2 projection. The regions of a connected space-time surface corresponding to different values of hbar (CP2) can be glued together. For instance, the gluing could take place along surface X3=S2× T (T corresponds time axis) analogous to black hole horizon. CP2 projection would be single point at the surface. The contribution from the radial dependence of CP2 coordinates to the induced metric giving ds2= ds2(X3)+grrdr2 at X3 implies a radial gravitational acceleration and one can say that a gravitational flux is transferred between different imbedding spaces.

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

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

For details see the chapter Does TGD Predict the Spectrum of Planck Constants.

Precise definition of the notion of unitarity for Connes tensor product

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

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

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

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

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

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

    Pij=NijNij ,

    and are in general N-valued unless one requires

    Pij=pijeN ,

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

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

    pij=Tr(NijNij )N .

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

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

For details see the chapter Was von Neumann Right After All?.

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

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

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

Gb subset of SU(2) subset of SU(3)

in CP2 degrees of freedom and

Ga subset of SL(2,C)

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

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

α= (e2/(4πhbar c)→ α/n ,

and the formula for Hall conductance would transform to

σH =να → (ν/n)× α .

Fractional quantum Hall effect would be integer quantum Hall effect but with scaled down α. The apparent fractional filling fraction ν= m/n would directly code the quantum phase q=exp(iπ/n) in the case that m obtains all possible values. A complete classification for possible phase transitions yielding fractional quantum Hall effect in terms of finite subgroups G subset of SU(2) subset of SU(3) given by ADE diagrams would emerge (An, D2n, E6 and E8 are possible). What would be also nice that CP2 would make itself directly manifest at the level of condensed matter physics.

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

Large values of Planck constant and coupling constant evolution

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

1. hbargr as CP2 Planck constant

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

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

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

The expression of the basic parameter v0=2-11 appearing in the formula of hbargr=GMm/v0 in terms of basic parameters of TGD leads to the unexpected conclusion that αK in electron length scale can be identified as electro-weak U(1) coupling strength αU(1). This identification, or actually something slightly complex (see below), is what group theory suggests but I had given it up since the resulting evolution for gravitational coupling predicted G to be proportional to Lp2 and thus completely un-physical. However, if gravitational interactions are mediated by space-time sheets characterized by Mersenne prime, the situation changes completely since M127 is the largest non-super-astrophysical p-adic length scale.

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

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

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.

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

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

αK(p)= a/log(pK) , K= R2/G .

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

If one assumes that αK is of order fine structure constant in electron length scale, the value of the parameter a cannot be far from unity. A more precise condition would result by identifying αK with weak U(1) coupling strength αK = αU(1)em/cos2W)≈ 1/105.3531 ,

sin2W)≈ .23120(15),

αem= 0.00729735253327 .

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

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

1. Are the ratios π/log(q) rational?

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

v02= qlog(pK)/π, q rational.

  1. Since v02 should be rational, the minimal conclusion would be that the number log(pK)/π should be rational for some preferred prime p=p0. If this miracle occurs, the p-adic coupling constant evolution of Kähler coupling strength, the only coupling constant in TGD, would be completely fixed. Same would also hold true for the ratio of CP2 to length characterized by K1/2.

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

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

a= (log(pK)/π) × (22m/K).

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

The value of αK in the length scale Lp0 in which condition of the first equation holds true is given by

1/αK= 221/K≈ 106.379 .

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

The condition for v0 can hold only for a single p-adic length scale Lp0. This correspondence would presumably mean that gravitational interaction is mediated along the space-time sheets characterized by p0, or even that gravitons are characterized by p0.

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

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

    It took some time to really understand what the result means and I leave the explanation to a later posting.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.

New Results in Planetary Bohr Orbitology

The understanding of how the quantum octonionic local version of infinite-dimensional Clifford algebra of 8-dimensional space (the only possible local variant of this algebra) implies entire quantum and classical TGD led also to the understanding of the quantization of Planck constant. In the model for planetary orbits based on gigantic gravitational Planck constant this means powerful constraints on the number theoretic anatomy of gravitational Planck constants and therefore of planetary mass ratios. These very stringent predictions are immediately testable.

1. Preferred values of Planck constants and ruler and compass polygons

The starting point is that the scaling factor of M4 Planck constant is given by the integer n characterizing the quantum phase q= exp(iπ/n). The evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible p-adically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant.

One expects that quantum phases q=exp(iπ/n) which are expressible using only square roots of rationals are number theoretically very special since they correspond to algebraic extensions of p-adic numbers involving only square roots which should emerge first and therefore systems involving these values of q should be especially abundant in Nature.

These polygons are obtained by ruler and compass construction and Gauss showed that these polygons, which could be called Fermat polygons, have

nF= 2ks Fns

sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes Fn=22n+1 correspond to n=0,1,2,3,4 with F0=3, F1=5, F2=17, F3=257, F4=65537. It is not known whether there are higher Fermat primes. n=3,5,15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter.

2. Application to planetary Bohr orbitology

The understanding of the quantization of Planck constants in M4 and CP2 degrees of freedom led to a considerable progress in the understanding of the Bohr orbit model of planetary orbits proposed by Nottale, whose TGD version initiated "the dark matter as macroscopic quantum phase with large Planck constant" program.

Gravitational Planck constant is given by

hbargr/hbar0= GMm/v0

where an estimate for the value of v0 can be deduced from known masses of Sun and planets. This gives v0≈ 4.6× 10-4.

Combining this expression with the above derived expression one obtains

GMm/v0= nF= 2kns Fns

In practice only the Fermat primes 3,5,17 appearing in this formula can be distinguished from a power of 2 so that the resulting formula is extremely predictive. Consider now tests for this prediction.

  1. The first step is to look whether planetary mass ratios can be reproduced as ratios of Fermat primes of this kind. This turns out to be the case if Nottale's proposal for quantization in which outer planets correspond to v0/5: TGD provides a mechanism explaining this modification of v0. The accuracy is better than 10 per cent.

  2. Second step is to look whether GMm/v0 for say Earth allows the expression above. It turns out that there is discrepancy: allowing second power of 17 in the formula one obtains an excellent fit. Only first power is allowed. Something goes wrong! 16 is the nearest power of two available and gives for v0 the value 2-11 deduced from biological applications and consistent with p-adic length scale hypothesis. Amusingly, v0(exp)= 4.6 × 10-4 equals with 1/(27× F2)= 4.5956× 10-4 within the experimental accuracy.

    A possible solution of the discrepancy is that the empirical estimate for the factor GMm/v0 is too large since m contains also the the visible mass not actually contributing to the gravitational force between dark matter objects. M is known correctly from the knowledge of gravitational field of Sun. The assumption that the dark mass is a fraction 1/(1+ε) of the total mass for Earth gives 1+ε= 17/16 in an excellent approximation. This gives for the fraction of the visible matter the estimate ε=1/16≈ 6 per cent. The estimate for the fraction of visible matter in cosmos is about 4 per cent so that estimate is reasonable and would mean that most of planetary and solar mass would be also dark as TGD indeed predicts and for which there are already now several experimental evidence (consider only the evidence that photosphere has solid surface discussed earlier in this blog ).

To sum up, it seems that everything is now ready for the great revolution. I would be happy to share this flood of discoveries with colleagues but all depends on what establishment decides. To my humble opinion twenty one years in a theoretical desert should be enough for even the most arrogant theorist. There is now a book of 800 A4 pages about TGD at Amazon: Topological Geometrodynamics so that it is much easier to learn what TGD is about.

The reader interested in details is recommended to look at the chapter Does TGD Predict the Spectrum of Planck Constants? of this book and the chapter of "Quantum Hardware of Living Systems".

Connes tensor product as universal interaction, quantization of Planck constant, McKay correspondence, etc...

It seems that discussion both in Peter Woit's blog, John Baez's This Week's Findings, and in h Lubos Motl's blog happen to tangent very closely what I have worked with during last weeks: ADE and Jones inclusions.

1. Some background.

  1. It has been for few years clear that TGD could emerge from the mere infinite-dimensionality of the Clifford algebra of infinite-dimensional "world of classical worlds" and from number theoretical vision in which classical number fields play a key role and determine imbedding space and space-time dimensions. This would fix completely the "world of classical worlds".

  2. Infinite-D Clifford algebra is a standard representation for von Neumann algebra known as a hyper-finite factor of type II1. In TGD framework the infinite tensor power of C(8), Clifford algebra of 8-D space would be the natural representation of this algebra.

2. How to localize infinite-dimensional Clifford algebra?

The basic new idea is to make this algebra local: local Clifford algebra as a generalization of gamma field of string models.

  1. Represent Minkowski coordinate of Md as linear combination of gamma matrices of D-dimensional space. This is the first guess. One fascinating finding is that this notion can be quantized and classical Md is genuine quantum Md with coordinate values eigenvalues of quantal commuting Hermitian operators built from matrix elements. Euclidian space is not obtained in this manner! Minkowski signature is something quantal! Standard quantum group Gl(2,q)(C) gives M4.

  2. Form power series of the Md coordinate represented as linear combination of gamma matrices with coefficients in corresponding infinite-D Clifford algebra. You would get tensor product of two algebra.

  3. There is however a problem: one cannot distinguish the tensor product from the original infinite-D Clifford algebra. D=8 is however an exception! You can replace gammas in the expansion of M8 coordinate by hyper-octonionic units which are non-associative (or octonionic units in quantum complexified-octonionic case). Now you cannot anymore absorb the tensor factor to the Clifford algebra and you get genuine M8-localized factor of type II1. Everything is determined by infinite-dimensional gamma matrix fields analogous to conformal super fields with z replaced by hyperoctonion.

  4. Octonionic non-associativity actually reproduces whole classical and quantum TGD: space-time surface must be associative sub-manifolds hence hyper-quaternionic surfaces of M8. Representability as surfaces in M4xCP2 follows naturally, the notion of configuration space of 3-surfaces, etc..

3. Connes tensor product for free fields as a universal definition of interaction quantum field theory

This picture has profound implications. Consider first the construction of S-matrix.

  1. A non-perturbative construction of S-matrix emerges. The deep principle is simple. The canonical outer automorphism for von Neumann algebras defines a natural candidate unitary transformation giving rise to propagator. This outer automorphism is trivial for II1 factors meaning that all lines appearing in Feynman diagrams must be on mass shell states satisfying Virasoro conditions. You can allow all possible diagrams: all on mass shell loop corrections vanish by unitarity and what remains are diagrams with single N-vertex!

  2. At 2-surface representing N-vertex space-time sheets representing generalized Bohr orbits of incoming and outgoing p"../articles/ meet. This vertex involves von Neumann trace (finite!) of localized gamma matrices expressible in terms of fermionic oscillator operators and defining free fields satisfying Super Virasoro conditions.

  3. For free fields ordinary tensor product would not give interacting theory. What makes S-matrix non-trivial is that *Connes tensor product* is used instead of the ordinary one. This tensor product is a universal description for interactions and we can forget perturbation theory! Interactions result as a deformation of tensor product. Unitarity of resulting S-matrix is unproven but I dare believe that it holds true.

  4. The subfactor N defining the Connes tensor product has interpretation in terms of the interaction between experimenter and measured system and each interaction type defines its own Connes tensor product. Basically N represents the limitations of the experimenter. For instance, IR and UV cutoffs could be seen as primitive manners to describe what N describes much more elegantily. At the limit when N contains only single element, theory would become free field theory but this is ideal situation never achievable.

4. The quantization of Planck constant and ADE hierarchies

The quantization of Planck constant has been the basic them of TGD for more than one and half years and leads also the understanding of ADE correspondences (../index ≤ 4 and index=4) from the point of view of Jones inclusions.

  1. The new view allows to understand how and why Planck constant is quantized and gives an amazingly simple formula for the separate Planck constants assignable to M4 and CP2 and appearing as scaling constants of their metrics. This in terms of a mild generalizations of standard Jones inclusions. The emergence of imbedding space means only that the scaling of these metrics have spectrum: no landscape.

  2. In ordinary phase Planck constants of M4 and CP2 are same and have their standard values. Large Planck constant phases correspond to situations in which a transition to a phase in which quantum groups occurs. These situations correspond to standard Jones inclusions in which Clifford algebra is replaced with a sub-algebra of its G-invariant elements. G is product Ga×Gb of subgroups of SL(2,C) and SU(2)Lx×U(1) which also acts as a subgroup of SU(3). Space-time sheets are n(Gb) fold coverings of M4 and n(Ga) fold coverings of CP2 generalizing the picture which has emerged already. An elementary study of these coverings fixes the values of scaling factors of M4 and CP2 Planck constants to orders of the maximal cyclic sub-groups. Mass spectrum is invariant under these scalings.

  3. This predicts automatically arbitrarily large values of Planck constant and assigns the preferred values of Planck constant to quantum phases q=exp(iπ/n) expressible in terms of square roots of rationals: these correspond to polygons obtainable by compass and ruler construction. In particular, experimentally favored values of hbar in living matter correspond to these special values of Planck constant. This model reproduces also the other aspects of the general vision. The subgroups of SL(2,C) in turn can give rise to re-scaling of SU(3) Planck constant. The most general situation can be described in terms of Jones inclusions for fixed point subalgebras of number theoretic Clifford algebras defined by Ga× Gb in SL(2,C)× SU(2).

  4. These inclusions (apart from those for which Ga contains infinite number of elements) are represented by ADE or extended ADE diagrams depending on the value of index. The group algebras of these groups give rise to additional degrees of freedom which make possible to construct the multiplets of the corresponding gauge groups. For index&le4 all gauge groups allowed by the ADE correspondence (An,D2n, E6,E8) are possible so that TGD seems to be able to mimick these gauge theories. For index=4 all ADE Kac Moody groups are possible and again mimicry becomes possible: TGD would be kind of universal physics emulator but it would be anyonic dark matter which would perform this emulation.

  5. Large hbar phases provide good hopes of realizing topological quantum computation. There is an additional new element. For quantum spinors state function reduction cannot be performed unless quantum deformation parameter equals to q=1. The reason is that the components of quantum spinor do not commute: it is however possible to measure the commuting operators representing moduli squared of the components giving the probabilities associated with 'true' and 'false'. The universal eigenvalue spectrum for probabilities does not in general contain (1,0) so that quantum qbits are inherently fuzzy. State function reduction would occur only after a transition to q=1 phase and decoherence is not a problem as long as it does not induce this transition.

For details see the chapter Was von Neumann Right After All?.

Von Neumann inclusions, quantum group, and quantum model for beliefs

Configuration space spinor fields live in "the world of classical worlds", whose points correspond to 3-surfaces in H=M4×CP2. These fields represent the quantum states of the universe. Configuration space spinors (to be distinguished from spinor fields) have a natural interpretation in terms of a quantum version of Boolean algebra obtained by applying fermionic operators to the vacuum state. Both fermion number and various spinlike quantum numbers can be interpreted as representations of bits. Once you have true and false you have also beliefs and the question is whether it is possible to construct a quantum model for beliefs.

1. Some background about number theoretic Clifford algebras

Configuration space spinors are associated with an infinite-dimensional Clifford algebra spanned by configuration space gamma matrices: spinors are created from vacuum state by complexified gamma matrices acting like fermionic oscillator operators carrying quark and lepton numbers. In a rough sense this algebra could be regarded as an infinite tensor power of M2(F), where F would correspond to complex numbers. In fact, also F=H (quaternions) and even F=O (octonions) can and must(!) be considered although the definitions involve some delicacies in this case. In particular, the non-associativy of octonions poses an interpretational problem whose solution actually dictates the physics of TGD Universe.

These Clifford algebras can be extended local algebras representable as powers series of hyper-F coordinate (hyper-F is obtained by multiplying imaginary part of F number with a commuting additional imaginary unit) so that a generalization of conformal field concept results with powers of complex coordinate replaced with those of hyper-complex numerg, hyper-quaternion or octonion. TGD could be seen as a generalization of superstring models by adding H and O layers besides C so that space-time and imbedding space emerge without ad hoc tricks of spontaneous compactification and adding of branes non-perturbatively.

The inclusion sequence C in H in O induces generalization of Jones inclusion sequence for the local versions of the number theoretic Clifford algebras allowing to reduce quantum TGD to a generalized number theory. That is, classical and quantum TGD emerge from the natural number theoretic Jones inclusion sequence. Even more, an explicit master formula for S-matrix emerges consistent with the earlier general ideas. It seems safe to say that one chapter in the evolution of TGD is now closed and everything is ready for the technical staff to start their work.

2. Brahman=Atman property of hyper-finite type II1 factors makes them ideal for realizing symbolic and cognitive representations

Infinite-dimensional Clifford algebras provide a canonical example of von Neumann algebras known as hyper-finite factors of type II1 having rather marvellous properties. In particular, they possess Brahman= Atman property making it possible to imbed this kind of algebra within itself unitarily as a genuine sub-algebra. One obtains what infinite Jones inclusion sequences yielding as a by-product structures like quantum groups.

Jones inclusions are ideal for cognitive and symbolic representations since they map the fermionic state space of one system to a sub-space of the fermionic statespace of another system. Hence there are good reasons to believe that TGD universe is busily mimicking itself using Jones inclusions and one can identify the space-time correlates (braids connecting two subsystems consisting of magnetic flux tubes). p-Adic and real spinors do not differ in any manner and real-to-p-adic inclusions would give cognitive representations, real-to-real inclusions symbolic representations.

3. Jones inclusions and cognitive and symbolic representations

As already noticed, configuration space spinors provide a natural quantum model for the Boolean logic. When you have logic you have the notions of truth and false, and you have soon also the notion of belief. Beliefs of various kinds (knowledge, misbelief, delusion,...) are the basic element of cognition and obviously involve a representation of the external world or part of it as states of the system defining the believer. Jones inclusions for the mediating unitary mappings between the spaces of configuration spaces spinors of two systems are excellent candidates for these maps, and it is interesting to find what one kind of model for beliefs this picture leads to.

The resulting quantum model for beliefs provides a cognitive interpretation for quantum groups and predicts a universal spectrum for the probabilities that a given belief is true following solely from the commutation relations for the coordinates of complex quantum plane interpreted now as complex spinor components. This spectrum of probabilities depends only on the integer n characterizing the quantum phase q=exp(iπ/n) characterizing the Jones inclusion. For n < ∞ the logic is inherently fuzzy so that absolute knowledge is impossible. q=1 gives ordinary quantum logic with qbits having precise truth values after state function reduction.

One can make two conclusions.

  1. Quantum logics might have most interesting applications in the realm of consciousness theory and quantum spinors rather than quantum space-times seem to be more natural for the inclusions of factors of type II1.

  2. For n< ∞ inclusions quantum physical constraints pose fundamental restriction on how precisely it is possible to know and are reflected by the quantum dimension d<2 of quantum spinors telling the effective number of truth values smaller than one by correlations between non-commuting spinor components representing truth values. One could speak about Uncertainty Principle of Cognition for these inclusions.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?

Does TGD reduce to inclusion sequence of number theoretic von Neumann algebras?

The idea that the notion of space-time somehow from quantum theory is rather attractive. In TGD framework this would basically mean that the identification of space-time as a surface of 8-D imbedding space H=M4× CP2 emerges from some deeper mathematical structure. It seems that the series of inclusions for infinite-dimensional Clifford algebras associated with classical number fields F=R,C,H,O defining von Neumann algebras known as hyper-finite factors of type II1, could be this deeper mathematical structure.

1. Quaternions, octonions, and TGD

The dimensions of quaternions and octonions are 4 and 8 and same as the dimensions of space-time surface and imbedding space in TGD. It is difficult to avoid the feeling that TGD physics could somehow reduce to the structures assignable to the classical number fields. This vision is already now rather detailed. For instance, a proposal for a general solution of classical field equations is one outcome of this vision.

TGD suggests also what I call HO-H duality. Space-time can be regarded either as surface in H or as hyper-quaternionic sub-manifold of the space HO of hyper-octonions obtained by multiplying imaginary parts of octonions with a commuting additional imaginary unit.

The 2-dimensional partonic surfaces X2 are of central importance in TGD and it seems that the inclusion sequence C in H in O (complex numbers, quaternions, octonions) somehow corresponds to the inclusion sequence X2 in X4 in H. This inspires the that that whole TGD emerges from a generalized number theory and I have already proposed arguments for how this might happen.

2. Number theoretic Clifford algebras

Hyper-finite factors of type II1 defined by infinite-dimensional Clifford algebras is one thread in the multiple strand of number-theoretic ideas involving p-adic numbers fields and their fusion with reals along common rationals to form a generalized number system, classical number fields, hierarchy of infinite primes and integers, and von Neumann algebras and quantum groups. The new ideas allow to fuse von Neumans strand with the classical number field strand.
  1. The mere assumption that physical states are represented by spinor fields in the infinite-dimensional "world of classical worlds" implies the notion of infinite-dimensional Clifford algebra identifiable as generated by gamma matrices of infinite-dimensional separable Hilbert space. This algebra provides a standard representation for hyperfinite factors of type II1.

  2. Von Neumann algebras known as hyperfinite factors of type II1 are rather miraculous objects. The almost defining property is that the trace of unit operator is unity instead of infinity. This justifies the attribute hyperfinite and gives excellent hopes that the resulting quantum theory is free of infinities. These algebras are strange fractal like creatures in the sense that they can be imbedded unitarily within itself endlessly and one obtains infinite hierarchies of Jones inclusions. This means what might be called Brahman=Atman property: subsystem can represent in its state the state of the entire universe and this indeed leads to the idea that symbolic and cognitive representations are realized as Jones inclusions and that Universe is busily mimicking itself in this manner.

  3. Classical number fields F=R,C,H,O define four Clifford algebras using infinite tensor power of 2x2 Clifford algebra M2(F) associated with 2-spinors. The tensor powers associated with R and C are straightforward to define. The non-commutativity of H with C requires Connes tensor product which by definition guarantees that left and right multiplications of tensor product M2(H)×M2(H) by complex numbers are equivalent. For F=O the matrix algebra is not anymore associative but this implies only interpretational problems and means a slight generalization of von Neumann algebras which as far as I know are usually assumed to be associative. Denote by Cl(F) the infinite-dimensional Clifford algebras obtained in this manner. Perhaps I should not have said "only interpretational" since the solution of these problems dictates the classical and quantum dynamics.

3. TGD does not quite emerge from Jones inclusions for number theoretic Clifford algebras

Physics as a generalized number theory vision suggests that TGD physics is contained by the Jones inclusion sequence Cl(C) in Cl(H) in Cl(O) induced by C in H in O. This sequence could alone explain partonic, space-time, and imbedding space dimensions as dimensions of classical number fields. The dream is that also imbedding space H=M4× CP2 would emerge as a unique choice allowed by mathematical existence.

  1. CP2 indeed emerges naturally: it labels the possible H-planes of O and this observation stimulated the emergence idea for few years ago.

  2. Also Minkowski space M4 is wanted. In particular, future lightcones are needed since the super-canonical algebra defining the second super-conformal invariance of TGD is associated with the canonical algebra of δM4× CP2. The generalized conformal and symplectic structures of 4-D(!) lightcone boundary are crucial element here. Ordinary Super Kac-Moody algebra assignable with lightlike 3-D causal determinants is associated with the inclusion of partonic 2-surface X2 to X4 corresponding to C in H. Imbedding space cannot be dynamical anymore since no 16-D number field exists.

  3. The representation of space-times as surfaces of H should emerge as well as the space of configuration space spinor fields (not only spinors) defined in the space of 3-surfaces (or equivalently 4-surfaces which are generalizations of Bohr orbits).

  4. These surfaces should also have interpretation as hyper-quaternionic sub-manifold of hyper-octonionic 8-space HO (this would dictate the classical dynamics).

This has been the picture before the lacking string of ideas emerged.

4. Number-theoretic localization of infinite-dimensional number theoretic Clifford algebras as a lacking piece of puzzle The lacking piece of the big argument is below.

  1. Sequences of inclusions C in H in F allow to interpret infinite-D spinors in Cl(O) as a module having quaternionic spinors Cl(H) as coefficients multiplying quantum spinors with finite quantum dimension not larger than 16: this conforms with the fact that OH spinors indeed are complex 8+8 spinors (quarks, leptons). Configuration space spinors can be seen as quantized imbedding space spinors. Infinite-dimensional Cl(H) spinors in turn can be seen as 4-D quantum spinors having CL(C) spinors as coefficients. Quantum groups emerge naturally and relate to inclusions as does also Kac-Moody algebra.

  2. The key idea is to extend infinite-dimensional Clifford algebras to local algebras by allowing power series in hyper-F numbers with coefficients in Cl(F). Using algebraic terminology this means a direct integral of the factors. The resulting objects are generalizations of conformal fields (or quantum fields) defined in the space of hyper-complex numbers (string orbits), hyper-quaternions (space-time surface), hyper-octonions (HO). Their argument is hyper-F number instead of z. Very natural number theoretic generalization of gamma matrix fields (generators of local Clifford algebra!) of super string model is thus in question.

  3. Associativity at the space-time level becomes the fundamental physical law. This requires that physical Clifford algebra is associative. For Cl(O) this means that a quaternionic plane in O parametrized by a point of CP2 is selected at each point hyper-quaternionic point. For the local version of Cl(O) this means that powers of hyper-octonions in powers series are restricted to be hyperquaternions assignable to some hyper-quaternionic sub-manifold of HO (classical dynamics!). But since ordinary inclusion assigns CP2 point to given point of M4 represented by a hyper-quaternion one can regard space-time surface also as a surface of H! This means HO-H duality. Parton level emerges from the requirement of commutativity implying that partonic 2-surface correspond to commutative sub-manifolds of HO and thus also of H.

  4. Also the super-canonical invariance comes out naturally. The point is that light like hyper-quaternions do not possess inverse so that Laurent series for local Cl(F) elements does not exist at the boundaries lightcones of M4 which are thus causal determinants (note the analogy with pole of analytic function). Super-canonical algebra emerges at their boundaries and the intersections of space-time surfaces with the boundaries define a natural gauge fixing for the general coordinate invariance. Configuration space spinor fields are obtained by allowing quantum superpositions of these 3-surfaces (equivalently corresponding 4-surfaces).

Here is the entire quantum TGD believe it or not! I cannot tell whom I admire more: von Neumann or Chopin!

5. Explicit general formula for S-matrix emerges also

This picture leads also to an explicit master formula for S-matrix.

  1. The resulting S-matrix is consistent with the generalized duality symmetry implying that S-matrix element can be always expressed using a single diagram having single vertex from which lines identified as space-time surfaces emanate. There is analogy with effective action formalism in the sense that one proceeds in a direction reverse to that in the ordinary perturbative construction of S-matrix: from the vertex to the points defining tips of the boundaries of lightcones assignable to the incoming and outgoing p"../articles/ appearing in n-point function along the "lines". It remains to be shown that the generalized duality indeed holds true: now its basic implication is used to write the master formula for S-matrix.

  2. Configuration space integral over the 3-surfaces appearing as vertex is involved and corresponds to bosonic degrees of freedom in super string models. It is free of divergences since the exponent of Kähler function is a nonlocal functional of 3-surface, since ill-defined metric determinant is cancelled by ill-defined Gauss determinant, and since Ricci tensor for the configuration space vanishes implying the vanishing of further divergences coming from the metric determinant. Hyper-finiteness of type II1 factors (infinite-dimensional unit matrix has unit trace) is expected to imply the cancellation of the infinities in fermionic sector.

  3. Diagrams obtained by gluing of space-time sheets along their ends at the vertex rather than stringy diagrams turn indeed be the Feynman diagrams in TGD framework as previously concluded on basis of physical and algebraic arguments. These singular four-manifolds are not real solutions of field equation but only a construct emerging naturally in the definition of S-matrix based on general coordinate invariance implying that configuration space spinor fields have same value for all Diff4 related 3-surfaces along the space-time surface. S-matrix is automatically non-trivial.

The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?

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