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Mathematical Aspects of Consciousness
Note: Newest contributions are at the top!
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.
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.
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.
The chapter Does TGD Predict the Spectrum of Planck Constants? contains the detailed calculations.
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.
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.
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.
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.
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.
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
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?.
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?.
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)=αem/cos2(θW)≈ 1/105.3531 ,
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.
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.
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.
For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.
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= 2k ∏s 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
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= 2k ∏ns 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.
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.
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.
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.
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.
For details see the chapter Was von Neumann Right After All?.
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.
The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?
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 TGDThe 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 algebrasHyper-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.
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.
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.
5. Explicit general formula for S-matrix emerges also
This picture leads also to an explicit master formula for S-matrix.
The reader interested in details is recommended to look at the chapter Was von Neumann Right After All?