What's new inTopological Geometrodynamics: an OverviewNote: Newest contributions are at the top! 
Year 2007 
One element field, quantum measurement theory and its qvariant, and the Galois fields associated with infinite primesKea mentioned John Baez's This Week's Finds 259, where John talked about oneelement field  a notion inspired by the q=exp(i2π/n)→1 limit for quantum groups. This limit suggests that the notion of oneelement field for which 0=1  a kind of mathematical phantom for which multiplication and sum should be identical operations  could make sense. Physicist might not be attracted by this kind of identification. In the following I want to articulate some comments from the point of view of quantum measurement theory and its generalization to qmeasurement theory which I proposed for some years ago (see this). I also consider and alternative interpretation in terms of Galois fields assignable to infinite primes which form an infinite hierarchy. These Galois fields have infinite number of elements but the map to the real world effectively reduces the number of elements to 2: 0 and 1 remain different. 1. q→ 1 limit as transition from quantum physics to effectively classical physics? The q→limit of quantum groups at qintegers become ordinary integers and nD vector spaces reduce to nelement sets. For quantum logic the reduction would mean that 2^{N}D spinor space becomes 2^{N}element set. N qubits are replaced with N bits. This brings in mind what happens in the transition from wave mechanism to classical mechanics. This might relate in interesting manner to quantum measurement theory. Strictly speaking, q→1 limit corresponds to the limit q=exp(i2π/n), n→∞ since only roots of unity are considered. This also correspond to Jones inclusions at the limit when the discrete group Z_{n} or or its extensionboth subgroups of SO(3) to contain reflection has infinite elements. Therefore this limit where field with one element appears might have concrete physical meaning. Does the system at this limit behave very classically? In TGD framework this limit can correspond to either infinite or vanishing Planck constant depending on whether one consider orbifolds or coverings. For the vanishing Planck constant one should have classicality: at least naively! In perturbative gauge theory higher order corrections come as powers of g^{2}/4πhbar so that also these corrections vanish and one has same predictions as given by classical field theory. 2. Qmeasurement theory and q→ 1 limit. Qmeasurement theory differs from quantum measurement theory in that the coordinates of the state space, say spinor space, are noncommuting. Consider in the sequel qspinors for simplicity. Since the components of quantum spinor do not commute, one cannot perform state function reduction. One can however measure the modulus squared of both spinor components which indeed commute as operators and have interpretation as probabilities for spin up or down. They have a universal spectrum of eigen values. The interpretation would be in terms of fuzzy probabilities and finite measurement resolution but may be in different sense as in case of HFF:s. Probability would become the observable instead of spin for q not equal to 1. At q→ 1 limit quantum measurement becomes possible in the standard sense of the word and one obtains spin down or up. This in turn means that the projective ray representing quantum states is replaced with one of n possible projective rays defining the points of nelement set. For HFF:s of type II_{1} it would be Nrays which would become points, N the included algebra. One might also say that state function reduction is forced by this mapping to single object at q→ 1 limit. On might say that the set of orthogonal coordinate axis replaces the state space in quantum measurement. We do this replacement of space with coordinate axis all the time when at blackboard. Quantum consciousness theorist inside me adds that this means a creation of symbolic representations and that the function of quantum classical correspondences is to build symbolic representations for quantum reality at spacetime level. q→ 1 limit should have spacetime correlates by quantum classical correspondence. A TGD inspired geometrotopological interpretation for the projection postulate might be that quantum measurement at q→1 limit corresponds to a leakage of 3surface to a dark sector of imbedding space with q→ 1 (Planck constant near to 0 or ∞ depending on whether one has n→∞ covering or division of M^{4} or CP_{2} by a subgroup of SU(2) becoming infinite cyclic  very roughly!) and Hilbert space is indeed effectively replaced with n rays. For q not equal to 1 one would have only probabilities for different outcomes since things would be fuzzy. In this picture classical physics and classical logic would be the physical counterpart for the shadow world of mathematics and would result only as an asymptotic notion. 3. Could 1element fields actually correspond to Galois fields associated with infinite primes? Finite field G_{p} corresponds to integers modulo p and product and sum are taken only modulo p. An alternative representation is in terms of phases exp(ik2π/p), k=0,...,p1 with sum and product performed in the exponent. The question is whether could one define these fields also for infinite primes (see this) by identifying the elements of this field as phases exp(ik2π/Π) with k taken to be finite integer and Π an infinite prime (recall that they form infinite hierarchy). Formally this makes sense. 1element field would be replaced with infinite hierarchy of Galois fields with infinite number of elements! The probabilities defined by components of quantum spinor make sense only as real numbers and one can indeed map them to real numbers by interpreting q as an ordinary complex number. This would give same results as q→ 1 limit and one would have effectively 1element field but actually a Galois field with infinite number of elements. If one allows k to be also infinite integer but not larger than than Π in real sense, the phases exp(k2π/Π) would be well defined as real numbers and could differ from 1. All real numbers in the range [1,1] would be obtained as values of cos(k2π/Π) so that this limit would effectively give real numbers. This relates also interestingly to the question whether the notion of padic field makes sense for infinite primes. The padic norm of any infinitep padic number would be power of π either infinite, zero, or 1. Excluding infinite normed numbers one would have effectively only padic integers in the range 1,...Π1 and thus only the Galois field G_{Π} representable also as quantum phases. I conclude with a nice string of text from John'z page: What's a mathematical phantom? According to Wraith, it's an object that doesn't exist within a given mathematical framework, but nonetheless "obtrudes its effects so convincingly that one is forced to concede a broader notion of existence". and unashamedely propose that perhaps Galois fields associated with infinite primes might provide this broader notion of existence! In equally unashamed tone I ask whether there exists also hierarchy of conscious entities at q=1 levels in real sense and whether we might identify ourselves as this kind of entities? Note that if cognition corresponds to padic spacetime sheets, our cognitive bodies have literally infinite geometric size in real sense. For details see the chapter Was von Neumann Right After All?.

Connes tensor product and perturbative expansion in terms of generalized braid diagrams
Many steps of progress have occurred in TGD lately.
In the previous posting I explained how generalized braid diagrams emerge naturally as orbits of the minima of Higgs defined as a generalized eigenvalue of the modified Dirac operator. The association of generalized braid diagrams to incoming and outgoing 3D partonic legs and possibly also vertices of the generalized Feynman diagrams forces to ask whether the generalized braid diagrams could give rise to a counterpart of perturbation theoretical formalism via the functional integral over configuration space degrees of freedom. The question is how the functional integral over configuration space degrees of freedom relates to the generalized braid diagrams. The basic conjecture motivated also number theoretically is that radiative corrections in this sense sum up to zero for critical values of Kähler coupling strength and Kähler function codes radiative corrections to classical physics via the dependence of the scale of M^{4} metric on Planck constant. Cancellation occurs only for critical values of Kähler coupling strength α_{K}: for general values of α_{K} cancellation would require separate vanishing of each term in the sum and does not occur. The natural guess is that finite measurement resolution in the sense of Connes tensor product can be described as a cutoff to the number of generalized braid diagrams. Suppose that the cutoff due to the finite measurement resolution can be described in terms of inclusions and Mmatrix can be expressed as a Connes tensor product. Suppose that the improvement of the measurement resolution means the introduction of zero energy states and corresponding lightlike 3surfaces in shorter time scales bringing in increasingly complex 3topologies. This would mean following.
There are still some questions. Radiative corrections around given 3topology vanish. Could radiative corrections sum up to zero in an ideal measurement resolution also in 2D sense so that the initial and final partonic 2surfaces associated with a partonic 3surface of minimal complexity would determine the outcome completely? Could the 3surface of minimal complexity correspond to a trivial diagram so that free theory would result in accordance with asymptotic freedom as measurement resolution becomes ideal? The answer to these questions seems to be 'No'. In the padic sense the ideal limit would correspond to the limit p→ 0 and since only p→ 2 is possible in the discrete length scale evolution defined by primes, the limit is not a free theory. This conforms with the view that CP_{2} length scale defines the ultimate UV cutoff. For more details see the chapter Configuration Space Spinor Structure.

Number theoretic braids and global view about anticommutations of induced spinor fields
The anticommutations of induced spinor fields are reasonably well understood locally. The basic objects are 3dimensional lightlike 3surfaces. These surfaces can be however seen as random lightlike orbits of partonic 2surfaces taking which would thus seem to take the role of fundamental dynamical objects. Conformal invariance in turn seems to make the 2D partons 1D objects and number theoretical braids in turn discretizes strings. And it also seems that the strands of number theoretic braid can in turn be discretized by considering the minima of Higgs potential in 3D sense. Somehow these apparently contradictory views should be unifiable in a more global view about the situation allowing to understand the reduction of effective dimension of the system as one goes to short scales. The notions of measurement resolution and number theoretic braid indeed provide the needed insights in this respect. 1. Anticommutations of the induced spinor fields and number theoretical braids The understanding of the number theoretic braids in terms of Higgs minima and maxima allows to gain a global view about anticommutations. The coordinate patches inside which Higgs modulus is monotonically increasing function define a division of partonic 2surfaces X^{2}_{t}= X^{3}_{l}\intersection δ M^{4}_{+/,t} to 2D patches as a function of time coordinate of X^{3}_{l} as lightcone boundary is shifted in preferred time direction defined by the quantum critical submanifold M^{2}× CP_{2}. This induces similar division of the lightlike 3surfaces X^{3}_{l} to 3D patches and there is a close analogy with the dynamics of ordinary 2D landscape. In both 2D and 3D case one can ask what happens at the common boundaries of the patches. Do the induced spinor fields associated with different patches anticommute so that they would represent independent dynamical degrees of freedom? This seems to be a natural assumption both in 2D and 3D case and correspond to the idea that the basic objects are 2 resp. 3dimensional in the resolution considered but this in a discretized sense due to finite measurement resolution, which is coded by the patch structure of X^{3}_{l}. A dimensional hierarchy results with the effective dimension of the basic objects increasing as the resolution scale increases when one proceeds from braids to the level of X^{3}_{l}. If the induced spinor fields associated with different patches anticommute, patches indeed define independent fermionic degrees of freedom at braid points and one has effective 2dimensionality in discrete sense. In this picture the fundamental stringy curves for X^{2}_{t} correspond to the boundaries of 2D patches and anticommutation relations for the induced spinor fields can be formulated at these curves. Formally the conformal time evolution scaled down the boundaries of these patches. If anticommutativity holds true at the boundaries of patches for spinor fields of neighboring patches, the patches would indeed represent independent degrees of freedom at stringy level. The cutoff in transversal degrees of freedom for the induced spinor fields means cutoff n≤ n_{max} for the conformal weight assignable to the holomorphic dependence of the induced spinor field on the complex coordinate. The dropping of higher conformal weights should imply the loss of the anticommutativity of the induced spinor fields and its conjugate except at the points of the number theoretical braid. Thus the number theoretic braid should code for the value of n_{max}: the naive expectation is that for a given stringy curve the number of braid points equals to n_{max}. 2. The decomposition into 3D patches and QFT description of particle reactions at the level of number theoretic braids What is the physical meaning of the decomposition of 3D lightlike surface to patches? It would be very desirable to keep the picture in which number theoretic braid connects the incoming positive/negative energy state to the partonic 2surfaces defining reaction vertices. This is not obvious if X^{3}_{l} decomposes into causally independent patches. One can however argue that although each patch can define its own fermion state it has a vanishing net quantum numbers in zero energy ontology, and can be interpreted as an intermediate virtual state for the evolution of incoming/outgoing partonic state. Another problem  actually only apparent problem has been whether it is possible to have a generalization of the braid dynamics able to describe particle reactions in terms of the fusion and decay of braid strands. For some strange reason I had not realized that number theoretic braids naturally allow fusion and decay. Indeed, cusp catastrophe is a canonical representation for the fusion process: cusp region contains two minima (plus maximum between them) and the complement of cusp region single minimum. The crucial control parameter of cusp catastrophe corresponds to the time parameter of X^{3}_{l}. More concretely, two valleys with a mountain between them fuse to form a single valley as the two real roots of a polynomial become complex conjugate roots. The continuation of lightlike surface to slicing of X^{4} to lightlike 3surfaces would give the full cusp catastrophe. In the catastrophe theoretic setting the time parameter of X^{3}_{l} appears as a control variable on which the roots of the polynomial equation defining minimum of Higgs depend: the dependence would be given by a rational function with rational coefficients. This picture means that particle reactions occur at several levels which brings in mind a kind of universal mimicry inspired by Universe as a Universal Computer hypothesis. Particle reactions in QFT sense correspond to the reactions for the number theoretic braids inside partons. This level seems to be the simplest one to describe mathematically. At parton level particle reactions correspond to generalized Feynman diagrams obtained by gluing partonic 3surfaces along their ends at vertices. Particle reactions are realized also at the level of 4D spacetime surfaces. One might hope that this multiple realization could code the dynamics already at the simple level of single partonic 3surface. 3. About 3D minima of Higgs potential The dominating contribution to the modulus of the Higgs field comes from δ M^{4}_{+/} distance to the axis R_{+} defining quantization axis. Hence in scales much larger than CP_{2} size the geometric picture is quite simple. The orbit for the 2D minimum of Higgs corresponds to a particle moving in the vicinity of R_{+} and minimal distances from R_{+} would certainly give a contribution to the Dirac determinant. Of course also the motion in CP_{2} degrees of freedom can generate local minima and if this motion is very complex, one expects large number of minima with almost same modulus of eigenvalues coding a lot of information about X^{3}_{l}. It would seem that only the most essential information about surface is coded: the knowledge of minima and maxima of height function indeed provides the most important general coordinate invariant information about landscape. In the rational category where X^{3}_{l} can be characterized by a finite set of rational numbers, this might be enough to deduce the representation of the surface. What if the situation is stationary in the sense that the minimum value of Higgs remains constant for some time interval? Formally the Dirac determinant would become a continuous product having an infinite value. This can be avoided by assuming that the contribution of a continuous range with fixed value of Higgs minimum is given by the contribution of its initial point: this is natural if one thinks the situation information theoretically. Physical intuition suggests that the minima remain constant for the maxima of Kähler function so that the initial partonic 2surface would determine the entire contribution to the Dirac determinant. For more details see the chapter Configuration Space Spinor Structure.

Geometric view about Higgs mechanismThe improved understanding of the generalization of the imbedding space concept forced by the hierarchy of Planck constants led to a considerable progress in TGD. For instance, I understand now how fractional quantum Hall effect emerges in TGD framework. I have also a rather satisfactory understanding of the notion of number theoretic braid: in particular the question how the cutoff implying that the number of strands is finite, emerges from inherent geometry of the partonic 2surface. Also a beautiful geometric interpretation of the generalized eigenstates and eigenvalues of the modified Dirac operator and understanding of supercanonical conforma weights emerges. It became already earlier clear that the generalized eigenvalue of Dirac operator which are scalar fields correspond to Higgs expectation value physically. The problem was to deduce what this expectation value is and I have now very beautiful geometric construction of Higgs expectation value as a coder of rather simple but fundamental geometric information about partonic surface. This leads also to an expression for the zeta function associated with number theoretic braid and understanding of what geometric information it codes about partonic 2surface. Also the finiteness of the theory becomes manifest since the number of generalized eigenvalues is finite. In the following I describe the arguments related to the geometrization of Higgs expectation. I attach the text which can be also found from the chapter Construction of Quantum Theory Symmetries of "Towards Smatrix".
The identification of the generalized eigenvalues of the modified Dirac operator as Higgs field suggests the possibility of understanding the spectrum of D purely geometrically by combining physical and geometric constraints. The standard zeta function associated with the eigenvalues of the modified Dirac action is the best candidate concerning the interpretation of supercanonical conformal weights as zeros of ζ. This ζ should have very concrete geometric and physical interpretation related to the quantum criticality. This would be the case if these eigenvalues, eigenvalue actually, have geometric based on geometrization of Higgs field. Before continuing it its convenient to introduce some notations. Denote the complex coordinate of a point of X^{2} by w, its H=M^{4}× CP_{2} coordinates by h=(m,s), and the H coordinates of its R_{+}× S^{2}_{II} projection by h_{c}=(r_{+},s_{II}). 1. Interpretation of eigenvalues of D as Higgs field The eigenvalues of the modified Dirac operator have a natural interpretation as Higgs field which vanishes for unstable extrema of Higgs potential. These unstable extrema correspond naturally to quantum critical points resulting as intersection of M^{4} resp. CP_{2} projection of the partonic 2surface X^{2} with S^{2}_{r} resp. S^{2}_{II}. Quantum criticality suggests that the counterpart of Higgs potential could be identified as the modulus square of Higgs V(H(s))= H(s)^{2} . which indeed has the points s with V(H(s))=0 as extrema which would be unstable in accordance with quantum criticality. The fact that for ordinary Higgs mechanism minima of V are the important ones raises the question whether number theoretic braids might more naturally correspond to the minima of V rather than intersection points with S^{2}. This turns out to be the case. It will also turn out that the detailed form of Higgs potential does not matter: the only thing that matters is that V is monotonically decreasing function of the distance from the critical manifold.
2. Purely geometric interpretation of Higgs Geometric interpretation of Higgs field suggests that critical points with vanishing Higgs correspond to the maximally quantum critical manifold R_{+}× S^{2}_{II}. The value of H should be determined once h(w) and R_{+}× S^{2}_{II} projection h_{c}(w) are known. H should increase with the distance between these points. The question is whether one can assign to a given point pair (h(w),h_{c}(w)) naturally a value of H. The first guess is that the value of H is determined by the shortest geodesic line connecting the points (product of geodesics of δM^{4} and CP_{2}). The value should be in general complex and invariant under the isometries of δH affecting h and h_{c}(w). The minimal geodesic distance d(h,h_{c}) between the two points would define the first candidate for the modulus of H. This guess turns need not be quite correct. An alternative guess is that M^{4} projection is indeed geodesic but that M^{4} projection extremizes itse length subject to the constraint that the absolute value of the phase defined by onedimensional Käahler action ∫ A_{μ}dx^{μ} is minimized: this point will be discussed below. If this inclusion is allowed then internal consistency requires also the extremization of ∫ A_{μ}dx^{μ} so that geodesic lines are not allowed in CP_{2}. The value should be in general complex and invariant under the isometries of δ H affecting h and h_{c}. The minimal distance d(h,h_{c}) between the two points constrained by extremal property of phase would define the first candidate for the modulus of H. The phase factor should relate close to the Kähler structure of CP_{2} and one possibility would be the nonintegrable phase factor U(s,s_{II}) defined as the integral of the induced Kähler gauge potential along the geodesic line in question. Hence the first guess for the Higgs would be as H(w)= d(h,h_{c}(w))× U(s,s_{II}) , d(h,h_{c}(w))=∫_{h}^{hc}ds , U(s,s_{II}) = exp[i∫_{s}^{sII}A_{k}ds^{k}] . This gives rise to a holomorphic function is X^{2} the local complex coordinate of X^{2} is identified as w= d(h,h_{c})U(s,s_{II}) so that one would have H(w)=w locally. This view about H would be purely geometric. One can ask whether one should include to the phase factor also the phase obtained using the Kähler gauge potential associated with S^{2}_{r} having expression (A_{θ},A_{φ})=(k,cos(θ)) with k even integer from the requirement that the nonintegral phase factor at equator has the same value irrespective of whether it is calculated with respect to North or South pole. For k=0 the contribution would be vanishing. The value of k might correlate directly with the value of quantum phase. The objection against inclusion of this term is that Kähler action defining Kähler function should contain also M^{4} part if this term is included. In each coordinate patch Higgs potential would be simply the quadratic function V= ww*. Negative sign is required by quantum criticality. Potential could indeed have minima as minimal distance of X^{2}_{CP2} point from S^{2}_{II}. Earth's surface with zeros as tops of mountains and bottoms of valleys as minima would be a rather precise visualization of the situation for given value of r_{+}. Mountains would have a shape of inverted rotationally symmetry parabola in each local coordinate system.
3. Consistency with the vacuum degeneracy of Käahler action and explicit construction of preferred extremals An important constraint comes from the condition that the vacuum degeneracy of Käahler action should be understood from the properties of the Dirac determinant. In the case of vacuum extremals Dirac determinant should have unit modulus. Suppose that the spacetime sheet associated with the vacuum parton X^{2} is indeed vacuum extremal. This requires that also X^{3}_{l} is a vacuum extremal: in this case Dirac determinant must be real although it need not be equal to unity. The CP_{2} projection of the vacuum extremal belongs to some Lagrangian submanifold Y^{2} of CP_{2}. For this kind of vacuum partons the ratio of the product of minimal H distances to corresponding M^{4}_{+/} distances must be equal to unity, in other words minima of Higgs potential must belong to the intersection X^{2}\cap S^{2}_{II} or to the intersection X^{2}\cap R_{+} so that distance reduces to M^{4} or CP_{2} distance and Dirac determinant to a phase factor. Also this phase factor should be trivial. It seems however difficult to understand how to obtain nontrivial phase in the generic case for all points if the phase is evaluated along geodesic line in CP_{2} degrees of freedom. There is however no deep reason to do this and the way out of difficulty could be based on the requirement that the phase defined by the Kähler gauge potential is evaluated along a curve either minimizing the absolute value of the phase modulo 2π. One must add the condition that curve is not shorter than the geodesic line between points. For a given curve length s_{0} the action must contain as a Lagrange multiplier the curve length so that the action using curve length s as a coordinate reads as S= ∫ A_{s}ds +λ(∫ dss_{0}). This gives for the extremum the equation of motion for a charged particle with Kähler charge Q_{K}= 1/λ: D^{2}s^{k}/ds^{2} + (1/λ)× J^{k}_{l}ds^{l}/ds=0 , D^{2}m^{k}/ds^{2}=0 . The magnitude of the phase must be further minimized as a function of curve length s. If the extremum curve in CP_{2} consists of two parts, first belonging to X^{2}_{II} and second to Y^{2}, the condition is satisfied. Hence, if X^{2}_{CP2}× Y^{2} is not empty, the phases are trivial. In the generic case 2D submanifolds of CP_{2} have intersection consisting of discrete points (note again the fundamental role of 4dimensionality of CP_{2}). Since S^{2}_{II} itself is a Lagrangian submanifold, it has especially high probably to have intersection points with S^{2}_{II}. If this is not the case one can argue that X^{3}_{l} cannot be vacuum extremal anymore. The construction gives also a concrete idea about how the 4D spacetime sheet X^{4}(X^{3}_{l}) becomes assigned with X^{3}_{l}. The point is that the construction extends X^{2} to 3D surface by connecting points of X^{2} to points of S^{2}_{II} using the proposed curves. This process can be carried out in each intersection of X^{3}_{l} and M^{4}_{+} shifted to the direction of future. The natural conjecture is that the resulting spacetime sheet defines the 4D preferred extremum of Käahler action.
4. About the definition of the Dirac determinant and number theoretic braids The definition of Dirac determinant should be independent of the choice of complex coordinate for X^{2} and local complex coordinate implied by the definition of Higgs is a unique choice for this coordinate. The physical intuition based on Higgs mechanism suggests strongly that the Dirac determinant should be defined simply as products of the eigenvalues of D, that is those of Higgs field, associated with the number theoretic braid. If only single kind of braid is allowed then the minima of Higgs field define the points of the braid very naturally. The points in R_{+}× S^{2}_{II} cannot contribute to the Dirac determinant since Higgs vanishes at the critical manifold. Note that at S^{2}_{II} criticality Higgs values become real and the exponent of Kähler action should become equal to one. This is guaranteed if Dirac determinant is normalized by dividing it with the product of δM^{4}_{+/}distances of the extrema from R_{+}. The value of the determinant would equal to one also at the limit R_{+}× S^{2}_{II}. One would define the Dirac determinant as the product of the values of Higgs field over all minima of local Higgs potential det(D)= [∏_{k} H(w_{k})]/[∏_{k} H_{0}(w_{k})]= ∏_{k}[w_{k}/w^{0}_{k}]. Here w^{0}_{k} are M^{4} distances of extrema from R_{+}. Equivalently: one can identify the values of Higgs field as dimensionless numbers w_{k}/w^{0}_{k}. The modulus of Higgs field would be the ratio of H and M^{4}_{+/} distances from the critical submanifold. The modulus of the Dirac determinant would be the product of the ratios of H and M^{4} depths of the valleys. This definition would be general coordinate invariant and independent of the topology of X^{2}. It would also introduce a unique conformal structure in X^{2} which should be consistent with that defined by the induced metric. Since the construction used relies on the induced metric this looks natural. The number of eigen modes of D would be automatically finite and eigenvalues would have a purely geometric interpretation as ratios of distances on one hand and as masses on the other hand. The inverse of CP_{2} length defines the natural unit of mass. The determinant is invariant under the scalings of H metric as are also Kähler action and ChernSimons action. This excludes the possibility that Dirac determinant could also give rise to the exponent of the area of X^{2}. Number theoretical constraints require that the numbers w_{k} are algebraic numbers and this poses some conditions on the allowed partonic 2surfaces unless one drops from consideration the points which do not belong to the algebraic extension used.
5. Physical identification of zeta function The proposed picture supports the identification of the eigenvalues of D in terms of a Higgs fields having purely geometric meaning. The identification of Higgs as the inverse of ζ function is not favored. It also seems that number theoretic braids must be identified as minima of Higgs potential in X^{2}. Furthermore, the braiding operation could be defined for all intersections of X^{3}_{l} defined by shifts M^{4}_{+/} as orbits of minima of Higgs potential. Second option is braiding by Kähler magnetic flux lines. The question is then how to understand supercanonical conformal weights for which the identification as zeros of a zeta function of some kind is highly suggestive. The natural answer would be that the eigenvalues of D defines this zeta function as ζ(s)= ∑_{k} [H(w_{k})/H(w^{0}_{k})]^{s} . The number of eigenvalues contributing to this function would be finite and H(w_{k})/H(w^{0}_{k} should be rational or algebraic at most. ζ function would have a precise meaning consistent with the usual assignment of zeta function to Dirac determinant. The ζ function would directly code the basic geometric properties of X^{2} since the moduli of the eigenvalues characterize the depths of the valleys of the landscape defined by X^{2} and the associated nonintegrable phase factors. The degeneracies of eigenvalues would in turn code for the number of points with same distance from a given zero intersection point. The zeros of this ζ function would in turn define natural candidates for supercanonical conformal weights and their number would thus be finite in accordance with the idea about inherent cutoff also in configuration space degrees of freedom. Note that supercanonical conformal weights would be functionals of X^{2}. The scaling of λ by a constant depending on padic prime factors out from the zeta so that zeros are not affected: this is in accordance with the renormalization group invariance of both supercanonical conformal weights and Dirac determinant. The zeta function should exist also in padic sense. This requires that the numbers λ:s at the points s of S^{2}_{II} which corresponds to the number theoretic braid are algebraic numbers. The freedom to scale λ could help to achieve this. 6. The relationship between λ and Higgs field The generalized eigenvalue λ(w) is only proportional to the vacuum expectation value of Higgs, not equal to it. Indeed, Higgs and gauge bosons as elementary particles correspond to wormhole contacts carrying fermion and antifermion at the two wormhole throats and must be distinguished from the spacetime correlate of its vacuum expectation as something proportional to λ. In the fermionic case the vacuum expectation value of Higgs does not seem to be even possible since fermions do not correspond to wormhole contacts between two spacetime sheets but possess only single wormhole throat (padic mass calculations are consistent with this). Gauge bosons can have Higgs expectation proportional to λ. The proportionality must be of form <H> propto λ/p^{n/2} if gauge boson mass squared is of order 1/p^{n}. The pdependent scaling factor of λ is expected to be proportional to log(p) from padic coupling constant evolution. 7. Possible objections related to the interpretation of Dirac determinant Suppose that that Dirac determinant is defined as a product of determinants associated with various points z_{k} of number theoretical braids and that these determinants are defined as products of corresponding eigenvalues. Since Dirac determinant is not real and is not invariant under isometries of CP_{2} and of δ M^{4}_{+/}, it cannot give only the exponent of Kähler function which is real and SU(3)× SO(3,1) invariant. The natural guess is that Dirac determinant gives also the ChernSimons exponential. The objection is that ChernSimons action depends not only on X^{2} but its lightlike orbit X^{3}_{l}.
The fact that general Poincare transformations fail to be symmetries of Dirac determinant is not in conflict with Poincare invariance of Kähler action since preferred extremals of Kähler action are in question and must contain the fixed partonic 2surfaces at δ M^{4}_{+/} so that these symmetries are broken by boundary conditions which does not require that the variational principle selecting the preferred extremals breaks these symmetries. One can exclude the possibility that the exponent of the stringy action defined by the area of X^{2} emerges also from the Dirac determinant. The point is that Dirac determinant is invariant under the scalings of H metric whereas the area action is not. The condition that the number of eigenvalues is finite is most naturally satisfied if generalized ζ coding information about the properties of partonic 2surface and expressible as a rational function for which the inverse has a finite number of branches is in question.
8. How unique the construction of Higgs field really is? Is the construction of spacetime correlate of Higgs as λ really unique? The replacement of H with its power H^{r}, r>0, leaves the minima of H invariant as points of X^{2} so that number theoretic braid is not affected. As a matter fact, the group of monotonically increasing maps realanalytic maps applied to H leaves number theoretic braids invariant. Polynomials with positive rational coefficients suggest themselves. The map H→ H^{r} scales Kähler function to its rmultiple, which could be interpreted in terms of 1/rscaling of the Kähler coupling strength. Also supercanonical conformal weights identified as zeros of ζ are scaled as h→ h/r and ChernSimons charge k is replaced with k/r so that at least r=1/n might be allowed. One can therefore ask whether the powers of H could define a hierarchy of quantum phases labelled by different values of k and α_{K}. The interpretation as separate phases would conform with the idea that D in some sense has entire spectrum of generalized eigenvalues. Note however that this would imply fractional powers for H. For more details see the chapter Overall View about Quantum TGD. 
Fractional Quantum Hall effect in TGD frameworkThe generalization of the imbedding space discussed in previous posting allows to understand fractional quantum Hall effect (see this and this). The formula for the quantized Hall conductance is given by σ= ν× e^{2}/h,ν=m/n. Series of fractions in ν=1/3, 2/5 3/7, 4/9, 5/11, 6/13, 7/15..., 2/3, 3/5, 4/7 5/9, 6/11, 7/13..., 5/3, 8/5, 11/7, 14/9... 4/3 7/5, 10/7, 13/9... , 1/5, 2/9, 3/13..., 2/7 3/11..., 1/7.. with odd denominator have bee observed as are also ν=1/2 and ν=5/2 state with even denominator. The model of Laughlin [Laughlin] cannot explain all aspects of FQHE. The best existing model proposed originally by Jain [Jain] is based on composite fermions resulting as bound states of electron and even number of magnetic flux quanta. Electrons remain integer charged but due to the effective magnetic field electrons appear to have fractional charges. Composite fermion picture predicts all the observed fractions and also their relative intensities and the order in which they appear as the quality of sample improves. I have considered earlier a possible TGD based model of FQHE not involving hierarchy of Planck constants. The generalization of the notion of imbedding space suggests the interpretation of these states in terms of fractionized charge and electron number.
[Laughlin] R. B. Laughlin (1983), Phys. Rev. Lett. 50, 1395. For more details see the chapter Does TGD Predict the Spectrum of Planck Constants.

A further generalization of the notion of imbedding spaceThe hypothesis that Planck constant is quantized having in principle all possible rational values but with some preferred values implying algebraically simple quantum phases has been one of the main ideas of TGD during last years. The mathematical realization of this idea leads to a profound generalization of the notion of imbedding space obtained by gluing together infinite number of copies of imbedding space along common 4dimensional intersection. The hope was that this generalization could explain charge fractionization but this does not seem to be the case. This problem led to a futher generalization of the imbedding space and this is what I want to discussed below. 1. Original view about generalized imbedding space The original generalization of imbedding space was basically following. Take imbedding space H=M^{4}×CP_{2}. Choose submanifold M^{2}×S^{2}, where S^{2} is homologically nontrivial geodesic submanifold of CP_{2}. The motivation is that for a given choice of Cartan algebra of Poincare algebra (translations in time direction and spin quantization axis plus rotations in plane orthogonal to this plane plus color hypercharge and isospin) this submanifold remains invariant under the transformations leaving the quantization axes invariant.
Form spaces M^{4}= M^{4}\M^{2} and CP_{2} = CP_{2}\S^{2} and their Cartesian product. Both spaces have a hole of codimension 2 so that the first homotopy group is Z. From these spaces one can construct an infinite hierarchy of factor spaces M^{4}/G_{a} and CP The hypothesis is that Planck constant is given by the ratio hbar= n_{a}/n_{b}, where n_{i} is the order of maximal cyclic subgroups of G_{i}. The hypothesis states also that the covariant metric of the Minkowski factor is scaled by the factor (n_{a}/n_{b})^{2}. One must take care of this in the gluing procedure. One can assign to the field bodies describing both self interactions and interactions between physical systems definite sector of generalized imbedding space characterized partially by the Planck constant. The phase transitions changing Planck constant correspond to tunnelling between different sectors of the imbedding space. 2. Fractionization of quantum numbers is not possible if only factor spaces are allowed The original idea was that the modification of the imbedding space inspired by the hierarchy of Planck constants could explain naturally phenomena like quantum Hall effect involving fractionization of quantum numbers like spin and charge. This does not however seem to be the case. G_{a}× G_{b} implies just the opposite if these quantum numbers are assigned with the symmetries of the imbedding space. For instance, quantization unit for orbital angular momentum becomes n_{a} where Z_{na} is the maximal cyclic subgroup of G_{a}. One can however imagine obtaining fractionization at the level of imbedding space for spacetime sheets, which are analogous to multisheeted Riemann surfaces (say Riemann surfaces associated with z^{1/n} since the rotation by 2π understood as a homotopy of M^{4} lifted to the spacetime sheet is a nonclosed curve. Continuity requirement indeed allows fractionization of the orbital quantum numbers and color in this kind of situation. Lifting up this idea to the level of imbedding space leads to the generalization of the notion of imbedding space. 3. Both covering spaces and factor spaces are possible The observation above stimulates the question whether it might be possible in some sense to replace H or its factors by their multiple coverings.
What could be the interpretation of these two kinds of spaces?

Does the quantization of Kähler coupling strength reduce to the quantization of ChernSimons coupling at partonic level?Kähler coupling strength associated with Kähler action (Maxwell action for the induced Kähler form) is the only coupling constant parameter in quantum TGD, and its value (or values) is in principle fixed by the condition of quantum criticality since Kähler coupling strength is completely analogous to critical temperature. The quantum TGD at parton level reduces to almost topological QFT for lightlike 3surfaces. This almost TQFT involves Abelian ChernSimons action for the induced Kähler form. This raises the question whether the integer valued quantization of the ChernSimons coupling k could predict the values of the Kähler coupling strength. I considered this kind of possibility already for more than 15 years ago but only the reading of the introduction of the recent paper by Witten about his new approach to 3D quantum gravity led to the discovery of a childishly simple argument that the inverse of Kähler coupling strength could indeed be proportional to the integer valued ChernSimons coupling k: 1/α_{K}=4k if all factors are correct. k=26 is forced by the comparison with some physical input. Also padic temperature could be identified as T_{p}=1/k. 1. Quantization of ChernSimons coupling strength For ChernSimons action the quantization of the coupling constant guaranteing so called holomorphic factorization is implied by the integer valuedness of the ChernSimons coupling strength k. As Witten explains, this follows from the quantization of the first ChernSimons class for closed 4manifolds plus the requirement that the phase defined by ChernSimons action equals to 1 for a boundaryless 4manifold obtained by gluing together two 4manifolds along their boundaries. As explained by Witten in his paper, one can consider also "anyonic" situation in which k has spectrum Z/n^{2} for nfold covering of the gauge group and in dark matter sector one can consider this kind of quantization. 2. Formula for Kähler coupling strength The quantization argument for k seems to generalize to the case of TGD. What is clear that this quantization should closely relate to the quantization of the Kähler coupling strength appearing in the 4D Kähler action defining Kähler function for the world of classical worlds and conjectured to result as a Dirac determinant. The conjecture has been that g_{K}^{2} has only single value. With some physical input one can make educated guesses about this value. The connection with the quantization of ChernSimons coupling would however suggest a spectrum of values. This spectrum is easy to guess.
It is not too difficult to believe to the formula 1/α_{K} =qk, q some rational. q=4 however requires a justification for the Wick rotation bringing the imaginary unit to ChernSimons action exponential lacking from Kähler function exponential. In this kind of situation one might hope that an additional symmetry might come in rescue. The guess is that number theoretic vision could justify this symmetry.
The action of CP_{2} type extremal is given as S=π/8α_{K}= kπ/2. Therefore the exponent of Kähler action appearing in the vacuum functional would be exp(kπ) known to be a transcendental number (Gelfond's constant). Also its powers are transcendental. If one wants to padicize also in 4D sense, this raises a problem. Before considering this problem, consider first the 4D padicization more generally.
Kähler coupling strength would have the same spectrum as padic temperature T_{p} apart from a multiplicative factor. The identification T_{p}=1/k is indeed very natural since also g_{K}^{2} is a temperature like parameter. The simplest guess is T_{p}= 1/k. Also gauge couplings strengths are expected to be proportional to g_{K}^{2} and thus to 1/k apart from a factor characterizing padic coupling constant evolution. That all basic parameters of theory would have simple expressions in terms of k would be very nice from the point of view quantum classical correspondence. If U(1) coupling constant strength at electron length scales equals α_{K}=1/104, this would give 1/T_{p}≈ 1/26. This means that photon, graviton, and gluons would be massless in an excellent approximation for say p=M_{89}, which characterizes electroweak gauge bosons receiving their masses from their coupling to Higgs boson. For fermions one has T_{p}=1 so that fermionic lightlike wormhole throats would correspond to the strongest possible coupling strength α_{K}=1/4 whereas gauge bosons identified as pairs of lightlike wormhole throats associated with wormhole contacts would correspond to α_{K}=1/104. Perhaps T_{p}=1/26 is the highest padic temperature at which gauge boson wormhole contacts are stable against splitting to fermionantifermion pair. Fermions and possible exotic bosons created by bosonic generators of supercanonical algebra would correspond to single wormhole throat and could also naturally correspond to the maximal value of padic temperature since there is nothing to which they can decay. A fascinating problem is whether k=26 defines internally consistent conformal field theory and is there something very special in it. Also the thermal stability argument for gauge bosons should be checked. What could go wrong with this picture? The different value for the fermionic and bosonic α_{K} makes sense only if the 4D spacetime sheets associated with fermions and bosons can be regarded as disjoint spacetime regions. Gauge bosons correspond to wormhole contacts connecting (deformed pieces of CP_{2} type extremal) positive and negative energy spacetime sheets whereas fermions would correspond to deformed CP_{2} type extremal glued to single spacetime sheet having either positive or negative energy. These spacetime sheets should make contact only in interaction vertices of the generalized Feynman diagrams, where partonic 3surfaces are glued together along their ends. If this gluing together occurs only in these vertices, fermionic and bosonic spacetime sheets are disjoint. For stringy diagrams this picture would fail. To sum up, the resulting overall vision seems to be internally consistent and is consistent with generalized Feynman graphics, predicts exactly the spectrum of α_{K}, allows to identify the inverse of padic temperature with k, allows to understand the differences between fermionic and bosonic massivation, and reduces Wick rotation to a number theoretic symmetry. One might hope that the additional objections (to be found sooner or later!) could allow to develop a more detailed picture. For more details see the chapter An Overview About Quantum TGD.

Smatrix as a functor and the groupoid like structure formed by SmatricesIn zero energy ontology Smatrix can be seen as a functor from the category of Feynman cobordisms to the category of operators. Smatrix can be identified as a "complex square root" of the positive energy density matrix S= ρ^{1/2}_{+}S_{0}, where S_{0} is a unitary matrix and ρ_{+} is the density matrix for positive energy part of the zero energy state. Obviously one has SS^{*}=ρ_{+}. S^{*}S=ρ_{} gives the density matrix for negative energy part of zero energy state. Clearly, Smatrix can be seen as a matrix valued generalization of Schrödinger amplitude. Note that the "indices" of the Smatrices correspond to configuration space spinors (fermions and their bound states giving rise to gauge bosons and gravitons) and to configuration space degrees of freedom (world of classical worlds). For hyperfinite factor of II_{1} it is not strictly speaking possible to speak about indices since the matrix elements are traces of the Smatrix multiplied by projection operators to infinitedimensional subspaces from right and left. The functor property of Smatrices implies that they form a multiplicative structure analogous but not identical to groupoid. Groupoid has associative product and there exist always right and left inverses and identity in the sense that ff^{1} and f^{1}f are defined but not identical in general, and one has fgg^{1}=f and f^{1}fg= g. The reason for the groupoid like property is that Smatrix is a map between state spaces associated with initial and final sets of partonic surfaces and these state spaces are different so that inverse must be replaced with right and left inverse. The defining conditions for the groupoid are however replaced with more general ones. Associativity holds also now but the role of inverse is taken by hermitian conjugate. Thus one has the conditions fgg^{*}=fρ_{g,+} and f^{*}fg= ρ_{f,}g, and the conditions ff^{*}=ρ_{+} and f^{*}f=ρ_{} are satisfied. Here ρ_{f+/} is density matrix associated with positive/negative energy parts of zero energy state. If the inverses of the density matrices exist, groupoid axioms hold true since f^{1}_{L}=f^{*}ρ_{f,+}^{1} satisfies ff^{1}_{L}= Id_{+} and f_{R}^{1}=ρ_{f,}^{1}f^{*} satisfies f^{1}_{R}f= Id_{}. There are good reasons to believe that also tensor product of its appropriate generalization to the analog of coproduct makes sense with nontriviality characterizing the interaction between the systems of the tensor product. If so, the Smatrices would form very beautiful mathematical structure bringing in mind the corresponding structures for 2tangles and Ntangles. Knowing how incredibly powerful the group like structures have been in physics one has good reasons to hope that groupoid like structure might help to deduce a lot of information about the quantum dynamics of TGD. A word about nomenclature is in order. S has strong associations to unitarity and it might be appropriate to replace S with some other letter. The interpretation of Smatrix as a generalized Schrödinger amplitude would suggest Ψmatrix. Since the interaction with Kea's Mtheory blog (with M denoting Monad or Motif in this context) helped to realize the connection with density matrix, also Mmatrix might work. Smatrix as a functor from the category of Feynman cobordisms in turn suggests C or F. Or could just Matrix denoted by M in formulas be enough? Certainly it would inspire feeling of awe but create associations with Mtheory in the stringy sense of the word but wouldn't it be fair if stringy Mtheory could leave at least some trace to physics;)! For details see the chapter An Overview About Quantum TGD.

Dark matter hierarchy corresponds to a hierarchy of quantum critical systems in modular degrees of freedomDark matter hierarchy corresponds to a hierarchy of conformal symmetries Z_{n} of partonic 2surfaces with genus g≥ 1 such that factors of n define subgroups of conformal symmetries of Z_{n}. By the decomposition Z_{n}=∏_{pn} Z_{p}, where pn tells that p divides n, this hierarchy corresponds to an hierarchy of increasingly quantum critical systems in modular degrees of freedom. For a given prime p one has a subhierarchy Z_{p}, Z_{p2}=Z_{p}× Z_{p}, etc... such that the moduli at n+1:th level are contained by n:th level. In the similar manner the moduli of Z_{n} are submoduli for each prime factor of n. This mapping of integers to quantum critical systems conforms nicely with the general vision that biological evolution corresponds to the increase of quantum criticality as Planck constant increases. The group of conformal symmetries could be also noncommutative discrete group having Z_{n} as a subgroup. This inspires a very shortlived conjecture that only the discrete subgroups of SU(2) allowed by Jones inclusions are possible as conformal symmetries of Riemann surfaces having g≥ 1. Besides Z_{n} one could have tedrahedral and icosahedral groups plus cyclic group Z_{2n} with reflection added but not Z_{2n+1} nor the symmetry group of cube. The conjecture is wrong. Consider the orbit of the subgroup of rotational group on standard sphere of E^{3}, put a handle at one of the orbits such that it is invariant under rotations around the axis going through the point, and apply the elements of subgroup. You obtain Riemann surface having the subgroup as its isometries. Hence all subgroups of SU(2) can act as conformal symmetries. The number theoretically simple rulerandcompass integers having as factors only first powers of Fermat primes and power of 2 would define a physically preferred subhierarchy of quantum criticality for which subsequent levels would correspond to powers of 2: a connection with padic length scale hypothesis suggests itself. Spherical topology is exceptional since in this case the space of conformal moduli is trivial and conformal symmetries correspond to the entire SL(2,C). This would suggest that only the fermions of lowest generation corresponding to the spherical topology are maximally quantum critical. This brings in mind Jones inclusions for which the defining subgroup equals to SU(2) and Jones index equals to M/N =4. In this case all discrete subgroups of SU(2) label the inclusions. These inclusions would correspond to fiber space CP_{2}→ CP_{2}/U(2) consisting of geodesic spheres of CP_{2}. In this case the discrete subgroup might correspond to a selection of a subgroup of SU(2)subset SU(3) acting nontrivially on the geodesic sphere. Cosmic strings X^{2}× Y^{2} subset M^{4}×CP_{2} having geodesic spheres of CP_{2} as their ends could correspond to this phase dominating the very early cosmology.
For more details see the chapter Construction of Elementary Particle Vacuum Functionals.

Elementary particle vacuum functionals for dark matter and why fermions can have only three familiesOne of the open questions is how dark matter hierarchy reflects itself in the properties of the elementary particles. The basic questions are how the quantum phase q=ep(2iπ/n) makes itself visible in the solution spectrum of the modified Dirac operator D and how elementary particle vacuum functionals depend on q. Considerable understanding of these questions emerged recently. One can generalize modular invariance to fractional modular invariance for Riemann surfaces possessing Z_{n} symmetry and perform a similar generalization for theta functions and elementary particle vacuum functionals. In particular, without any further assumptions n=2 dark fermions have only three families. The existence of spacetime correlate for fermionic 2valuedness suggests that fermions quite generally correspond to even values of n, so that this result would hold quite generally. Elementary bosons (actually exotic particles) would correspond to n=1, and more generally odd values of n, and could have also higher families. For more details see the chapter Construction of Elementary Particle Vacuum Functionals . 
Sierpinski topology and quantum measurement theory with finite measurement resolutionI have been trying to understand whether category theory might provide some deeper understanding about quantum TGD, not just as a powerful organizer of fuzzy thoughts but also as a tool providing genuine physical insights. Kea is also interested in categories but in much more technical sense. Her dream is to find a category theoretical formulation of Mtheory as something, which is not the 11D something making me rather unhappy as a physicist with second foot still deep in the muds of low energy phenomenology. Kea talks about topos, nlogos,... and their possibly existing quantum variants. I have used to visit Kea's blog in the hope of stealing some category theoretic intuition. It is also nice to represent comments knowing that they are not censored out immediately if their have the smell of original thought: this is quite too often the case in alpha male dominated blogs. It might be that I had luck this morning! 1. Locales, frames, Sierpinski topologies and Sierpinski space Kea mentioned the notions of locale and frame . In Wikipedia I learned that complete Heyting algebras, which are fundamental to category theory, are objects of three categories with differing arrows. CHey, Loc and its opposite category Frm (arrows reversed). Complete Heyting algebras are partially ordered sets which are complete lattices. Besides the basic logical operations there is also algebra multiplication. From Wikipedia I learned also that locales and the dual notion of frames form the foundation of pointless topology. These topologies are important in topos theory which does not assume the axiom of choice. So called particular point topology assumes a selection of single point but I have the physicist's feeling that it is otherwise rather near to pointless topology. Sierpinski topology is this kind of topology. Sierpinski topology is defined in a simple manner: set is open only if it contains a given point p. The dual of this topology defined in the obvious sense exists also. Sierpinski space consisting of just two points 0 and 1 is the universal building block of these topologies in the sense that a map of an arbitrary space to Sierpinski space provides it with Sierpinski topology as the induced topology. In category theoretical terms Sierpinski space is the initial object in the category of frames and terminal object in the dual category of locales. This category theoretic reductionism looks highly attractive to me. 2. Particular point topologies, their generalization, and finite measurement resolution Pointless, or rather particular point topologies might be very interesting from physicist's point of view. After all, every classical physical measurement has a finite spacetime resolution. In TGD framework discretization by number theoretic braids replaces partonic 2surface with a discrete set consisting of algebraic points in some extension of rationals: this brings in mind something which might be called a topology with a set of particular algebraic points. Perhaps the physical variant for the axiom of choice could be restricted so that only sets of algebraic points in some extension of rationals can be chosen freely. The extension would depend on the position of the physical system in the algebraic evolutionary hierarchy defining also a cognitive hierarchy. Certainly this would fit very nicely to the formulation of quantum TGD unifying real and padic physics by gluing real and padic number fields to single superstructure via common algebraic points. There is also a finite measurement resolution in Hilbert space sense not taken into account in the standard quantum measurement theory based on factors of type I. In TGD framework one indeed introduces quantum measurement theory with a finite measurement resolution so that complex rays becomes included hyperfinite factors of type II_{1} (HFF, see this).
This program, which I formulated only after this section had been written, might indeed make sense (ideas never learn to emerge in the logical order of things;)). The lucky association was to the ideas about fuzzy quantum logic realized in terms of quantum 2spinor that I had developed a couple of years ago. Fuzzy quantum logic would reflect the finite measurement resolution. I just list the pieces of the argument. Spinors and qbits: Spinors define a quantal variant of Boolean statements, qbits. One can however go further and define the notion of quantum qbit, qqbit. I indeed did this for couple of years ago (the last section in Was von Neumann Right After All?). Qspinors and qqbits: For qspinors the two components a and b are not commuting numbers but nonHermitian operators. ab= qba, q a root of unity. This means that one cannot measure both a and b simultaneously, only either of them. aa^{+} and bb^{+} however commute so that probabilities for bits 1 and 0 can be measured simultaneously. State function reduction is not possible to a state in which a or b gives zero! The interpretation is that one has qlogic is inherently fuzzy: there are no absolute truths or falsehoods. One can actually predict the spectrum of eigenvalues of probabilities for say 1. qSpinors bring in mind strongly the Hilbert space counterpart of Sierpinski space. One would however expect that fuzzy quantum logic replaces the logic defined by Heyting algebra. Qlocale: Could one think of generalizing the notion of locale to quantum locale by using the idea that sets are replaced by subspaces of Hilbert space in the conventional quantum logic. Qopenness would be defined by identifying quantum spinors as the initial object, qSierpinski space. a (resp. b for dual category) would define qopen set in this space. Qopen sets for other quantum spaces would be defined as inverse images of a (resp. b) for morphisms to this space. Only for q=1 one could have the qcounterpart of rather uninteresting topology in which all sets are open and every map is continuous. Qlocale and HFFs: The qSierpinski character of qspinors would conform with the very special role of Clifford algebra in the theory of HFFs, in particular, the special role of Jones inclusions to which one can assign spinor representations of SU(2). The Clifford algebra and spinors of the world of classical worlds identifiable as Fock space of quark and lepton spinors is the fundamental example in which 2spinors and corresponding Clifford algebra serves as basic building brick although tensor powers of any matrix algebra provides a representation of HFF. Qmeasurement theory: Finite measurement resolution (qquantum measurement theory) means that complex rays are replaced by subalgebra rays. This would force the Jones inclusions associated with SU(2) spinor representation and would be characterized by quantum phase q and bring in the qtopology and qspinors. Fuzzyness of qqbits of course correlates with the finite measurement resolution. Qnlogos: For other qrepresentations of SU(2) and for representations of compact groups (see appendix of this) one would obtain something which might have something to do with quantum nlogos, quantum generalization of nvalued logic. All of these would be however less fundamental and induced by qmorphisms to the fundamental representation in terms of spinors of the world of classical worlds. What would be however very nice that if these qmorphisms are constructible explicitly it would become possible to build up qrepresentations of various groups using the fundamental physical realization  and as I have conjectured (see this)  McKay correspondence and huge variety of its generalizations would emerge in this manner. The analogs of Sierpinski spaces: The discrete subgroups of SU(2), and quite generally, the groups Z_{n} associated with Jones inclusions and leaving the choice of quantization axes invariant, bring in mind the npoint analogs of Sierpinski space with unit element defining the particular point. Note however that n≥3 holds true always so that one does not obtain Sierpinski space itself. Could it be that all of these n preferred points belong to any open set? Number theoretical braids identified as subsets of the intersection of real and padic variants of algebraic partonic 2surface define second candidate for the generalized Sierpinski space with set of preferred points. Recall that the generalized imbedding space related to the quantization of Planck constant is obtained by gluing together coverings of M^{4}×CP_{2}→ M^{4}×CP_{2}/G_{a}×G_{b} along their common points. The topology in question would mean that if some point in the covering belongs to an open set, all of them do so. The interpretation could be that the points of fiber form a single inseparable quantal unit. For more details see the chapter Was von Neumann Right After All?. 
About microscopic description of dark matterEvery step of progress induces a handful of worried questions about consistency with the existing network of beliefs and almost as a rule the rules must be modified slightly or be made more precise. The construction of a model for the detection of gravitational radiation assuming that gravitons correspond to a gigantic gravitational constant was the last step of progress. It was carried out in TGD and Astrophysics, see also the earlier posting . One can say that dark gravitons are BoseEinstein condensates of ordinary gravitons. This suggests that BoseEinstein condensates of some kind could accompany and perhaps even characterize also the dark variants of ordinary elementary particles. The question is whether the new picture is consistent with the earlier dark rules. 1. Higgs boson BoseEinstein condensate as characterized of Planck constant The following picture is the simplest I have been able to imagine hitherto.
This scenario would allow to add some details to the general picture about particle massivation reducing to padic thermodynamics plus Higgs mechanism, both of them having description in terms of conformal weight.
This leaves still some questions.

Could also gauge bosons correspond to wormhole contacts?The developments in the formulation of quantum TGD which have taken place during the period 20052007 (see this, this, and this) suggest dramatic simplifications of the general picture about elementary particle spectrum. pAdic mass calculations (see this, this, this, and this) leave a lot of freedom concerning the detailed identification of elementary particles. The basic open question is whether the theory is free at parton level as suggested by the recent view about the construction of Smatrix and by the almost topological QFT property of quantum TGD at parton level (see this and this). Or more concretely: do partonic 2surfaces carry only free manyfermion states or can they carry also bound states of fermions and antifermions identifiable as bosons? What is known that Higgs boson corresponds naturally to a wormhole contact (see this). The wormhole contact connects two spacetime sheets with induced metric having Minkowski signature. Wormhole contact itself has an Euclidian metric signature so that there are two wormhole throats which are lightlike 3surfaces and would carry fermion and antifermion number in the case of Higgs. Irrespective of the identification of the remaining elementary particles MEs (massless extremals, topological light rays) would serve as spacetime correlates for elementary bosons. Higgs type wormhole contacts would connect MEs to the larger spacetime sheet and the coherent state of neutral Higgs would generate gauge boson mass and could contribute also to fermion mass. The basic question is whether this identification applies also to gauge bosons (certainly not to graviton). This identification would imply quite a dramatic simplification since the theory would be free at single parton level and the only stable parton states would be fermions and antifermions. As will be found this identification allows to understand the dramatic difference between graviton and other gauge bosons and the weakness of gravitational coupling, gives a connection with the string picture of gravitons, and predicts that stringy states are directly relevant for nuclear and condensed matter physics as has been proposed already earlier (see this, this, and this). 1. Option I: Only Higgs as a wormhole contact The only possibility considered hitherto has been that elementary bosons correspond to partonic 2surfaces carrying fermionantifermion pair such that either fermion or antifermion has a nonphysical polarization. For this option CP_{2} type extremals condensed on MEs and travelling with light velocity would serve as a model for both fermions and bosons. MEs are not absolutely necessary for this option. The couplings of fermions and gauge bosons to Higgs would be very similar topologically. Consider now the counter arguments.
2. Option II: All elementary bosons as wormhole contacts The hypothesis that quantum TGD reduces to a free field theory at parton level is consistent with the almost topological QFT character of the theory at this level. Hence there are good motivations for studying explicitly the consequences of this hypothesis. 2.1 Elementary bosons must correspond to wormhole contacts if the theory is free at parton level Also gauge bosons could correspond to wormhole contacts connecting MEs (see this) to larger spacetime sheet and propagating with light velocity. For this option there would be no need to assume the presence of nonphysical fermion or antifermion polarization since fermion and antifermion would reside at different wormhole throats. Only the definition of what it is to be nonphysical would be different on the lightlike 3surfaces defining the throats. The difference would naturally relate to the different time orientations of wormhole throats and make itself manifest via the definition of lightlike operator o=x^{k}γ_{k} appearing in the generalized eigenvalue equation for the modified Dirac operator (see this and this). For the first throat o^{k} would correspond to a lightlike tangent vector t^{k}of the partonic 3surface and for the second throat to its M^{4} dual t_{d}^{k} in a preferred rest system in M^{4} (implied by the basic construction of quantum TGD). What is nice that this picture nonasks the question whether t^{k}or t_{d}^{k}should appear in the modified Dirac operator. Rather satisfactorily, MEs (massless extremals, topological light rays) would be necessary for the propagation of wormhole contacts so that they would naturally emerge as classical correlates of bosons. The simplest model for fermions would be as CP_{2} type extremals topologically condensed on MEs and for bosons as pieces of CP_{2} type extremals connecting ME to the larger spacetime sheet. For fermions topological condensation is possible to either spacetime sheet. 2.2 Phase conjugate states and matterantimatter asymmetry By fermion number conservation fermionboson and bosonboson couplings must involve the fusion of partonic 3surfaces along their ends identified as wormhole throats. Bosonic couplings would differ from fermionic couplings only in that the process would be 2→ 4 rather than 1→ 3 at the level of throats. The decay of boson to an ordinary fermion pair with fermion and antifermion at the same spacetime sheet would take place via the basic vertex at which the 2dimensional ends of lightlike 3surfaces are identified. The sign of the boson energy would tell whether boson is ordinary boson or its phase conjugate (say phase conjugate photon of laser light) and also dictate the sign of the time orientation of fermion and antifermion resulting in the decay. Also a candidate for a new kind interaction vertex emerges. The splitting of bosonic wormhole contact would generate fermion and timereversed antifermion having interpretation as a phase conjugate fermion. This process cannot correspond to a decay of boson to ordinary fermion pair. The splitting process could generate matterantimatter asymmetry in the sense that fermionic antimatter would consist dominantly of negative energy antifermions at spacetime sheets having negative time orientation (see this and this). This vertex would define the fundamental interaction between matter and phase conjugate matter. Phase conjugate photons are in a key role in TGD based quantum model of living matter. This involves a model for memory as communications in time reversed direction, mechanism of intentional action involving signalling to geometric past, and mechanism of remote metabolism involving sending of negative energy photons to the energy reservoir (see this). The splitting of wormhole contacts has been considered as a candidate for a mechanism realizing Boolean cognition in terms of "cognitive neutrino pairs" resulting in the splitting of wormhole contacts with net quantum numbers of Z^{0} boson (see this). 3. Graviton and other stringy states Fermion and antifermion can give rise to only single unit of spin since it is impossible to assign angular momentum with the relative motion of wormhole throats. Hence the identification of graviton as single wormhole contact is not possible. The only conclusion is that graviton must be a superposition of fermionantifermion pairs and bosonantiboson pairs with coefficients determined by the coupling of the parton to graviton. Gravitongraviton pairs might emerge in higher orders. Fermion and antifermion would reside at the same spacetime sheet and would have a nonvanishing relative angular momentum. Also bosons could have nonvanishing relative angular momentum and Higgs bosons must indeed possess it. Gravitons are stable if the throats of wormhole contacts carry nonvanishing gauge fluxes so that the throats of wormhole contacts are connected by flux tubes carrying the gauge flux. The mechanism producing gravitons would the splitting of partonic 2surfaces via the basic vertex. A connection with string picture emerges with the counterpart of string identified as the flux tube connecting the wormhole throats. Gravitational constant would relate directly to the value of the string tension. The TGD view about coupling constant evolution (see this) predicts G propto L_{p}^{2}, where L_{p} is padic length scale, and that physical graviton corresponds to p=M_{127}=2^{127}1. Thus graviton would have geometric size of order Compton length of electron which is something totally new from the point of view of usual Planck length scale dogmatism. In principle an entire padic hierarchy of gravitational forces is possible with increasing value of G. The explanation for the small value of the gravitational coupling strength serves as a test for the proposed picture. The exchange of ordinary gauge boson involves the exchange of single CP_{2} type extremal giving the exponent of Kähler action compensated by state normalization. In the case of graviton exchange two wormhole contacts are exchanged and this gives second power for the exponent of Kähler action which is not compensated. It would be this additional exponent that would give rise to the huge reduction of gravitational coupling strength from the naive estimate G ≈ L_{p}^{2}. Gravitons are obviously not the only stringy states. For instance, one obtains spin 1 states when the ends of string correspond to gauge boson and Higgs. Also nonvanishing electroweak and color quantum numbers are possible and stringy states couple to elementary partons via standard couplings in this case. TGD based model for nuclei as nuclear strings having length of order L(127) (see this) suggests that the strings with light M_{127}quark and antiquark at their ends identifiable as companions of the ordinary graviton are responsible for the strong nuclear force instead of exchanges of ordinary mesons or color van der Waals forces. Also the TGD based model of high T_{c} superconductivity involves stringy states connecting the spacetime sheets associated with the electrons of the exotic Cooper pair (see this and this). Thus stringy states would play a key role in nuclear and condensed matter physics, which means a profound departure from stringy wisdom, and breakdown of the standard reductionistic picture. 4. Spectrum of nonstringy states The 1throat character of fermions is consistent with the generationgenus correspondence. The 2throat character of bosons predicts that bosons are characterized by the genera (g_{1},g_{2}) of the wormhole throats. Note that the interpretation of fundamental fermions as wormhole contacts with second throat identified as a Fock vacuum is excluded. The general bosonic wavefunction would be expressible as a matrix M_{g1,g2} and ordinary gauge bosons would correspond to a diagonal matrix M_{g1,g2}=δ_{g1,g2} as required by the absence of neutral flavor changing currents (say gluons transforming quark genera to each other). 8 new gauge bosons are predicted if one allows all 3× 3 matrices with complex entries orthonormalized with respect to trace meaning additional dynamical SU(3) symmetry. Ordinary gauge bosons would be SU(3) singlets in this sense. The existing bounds on flavor changing neutral currents give bounds on the masses of the boson octet. The 2throat character of bosons should relate to the low value T=1/n<< 1 for the padic temperature of gauge bosons as contrasted to T=1 for fermions. If one forgets the complications due to the stringy states (including graviton), the spectrum of elementary fermions and bosons is amazingly simple and almost reduces to the spectrum of standard model. In the fermionic sector one would have fermions of standard model. By simple counting leptonic wormhole throat could carry 2^{3}=8 states corresponding to 2 polarization states, 2 charge states, and sign of lepton number giving 8+8=16 states altogether. Taking into account phase conjugates gives 16+16=32 states. In the nonstringy boson sector one would have bound states of fermions and phase conjugate fermions. Since only two polarization states are allowed for massless states, one obtains (2+1)× (3+1)=12 states plus phase conjugates giving 12+12=24 states. The addition of color singlet states for quarks gives 48 gauge bosons with vanishing fermion number and color quantum numbers. Besides 12 electroweak bosons and their 12 phase conjugates there are 12 exotic bosons and their 12 phase conjugates. For the exotic bosons the couplings to quarks and leptons are determined by the orthogonality of the coupling matrices of ordinary and boson states. For exotic counterparts of Wbosons and Higgs the sign of the coupling to quarks is opposite. For photon and Z^{0} also the relative magnitudes of the couplings to quarks must change. Altogether this makes 48+16+16=80 states. Gluons would result as color octet states. Family replication would extend each elementary boson state into SU(3)octet and singlet and elementary fermion states into SU(3)triplets. 5. Higgs mechanism Consider next the generation of mass as a vacuum expectation value of Higgs when also gauge bosons correspond to wormhole contacts. The presence of Higgs condensate should make the simple rectilinear ME curved so that the average propagation of fields would occur with a velocity less than light velocity. Field equations allow MEs of this kind as solutions (see this). The finite range of interaction characterized by the gauge boson mass should correlate with the finite range for the free propagation of wormhole contacts representing bosons along corresponding ME. The finite range would result from the emission of Higgs like wormhole contacts from gauge boson like wormhole contact leading to the generation of coherent states of neutral Higgs particles. The emission would also induce nonrectilinearity of ME as a correlate for the recoil in the emission of Higgs. For more details see the chapter Massless states and Particle Massivation.

Jones inclusions and construction of Smatrix and U matrixTGD leads naturally to zero energy ontology which reduces to the positive energy ontology of the standard model only as a limiting case. In this framework one must distinguish between the Umatrix characterizing the unitary process associated with the quantum jump (and followed by state function reduction and state preparation) and the Smatrix defining timelike entanglement between positive and negative energy parts of the zero energy state and coding the rates for particle reactions which in TGD framework correspond to quantum measurements reducing timelike entanglement. 1. Smatrix In zero energy ontology Smatrix characterizes time like entanglement of zero energy states (this is possible only for HFFs for which Tr(SS^{+})=Tr(Id)=1 holds true). Smatrix would code for transition rates measured in particle physics experiments with particle reactions interpreted as quantum measurements reducing time like entanglement. In TGD inspired quantum measurement theory measurement resolution is characterized by Jones inclusion (the group G defines the measured quantum numbers), N subset M takes the role of complex numbers, and state function reduction leads to N ray in the space M/N regarded as N module and thus from a factor to a subfactor. The finite number theoretic braid having Galois group G as its symmetries is the spacetime correlate for both the finite measurement resolution and the effective reduction of HFF to that associated with a finitedimensional quantum Clifford algebra M/N. SU(2) inclusions would allow angular momentum and color quantum numbers in bosonic degrees of freedom and spin and electroweak quantum numbers in spinorial degrees of freedom. McKay correspondence would allow to assign to G also compact ADE type Lie group so that also Lie group type quantum numbers could be included in the repertoire. Galois group G would characterize subspaces of the configuration space ("world of classical worlds") number theoretically in a manner analogous to the rough characterization of physical states by using topological quantum numbers. Each braid associated with a given partonic 2surface would correspond to a particular G that the state would be characterized by a collection of groups G. G would act as symmetries of zero energy states and thus of Smatrix. Smatrix would reduce to a direct integral of Smatrices associated with various collections of Galois groups characterizing the number theoretical properties of partonic 2surfaces. It is not difficult to criticize this picture.
2. Umatrix In a welldefined sense U process seems to be the reversal of state function reduction. Hence the natural guess is that Umatrix means a quantum transition in which a factor becomes a subfactor whereas state function reduction would lead from a factor to a subfactor. Various arguments suggest that U matrix could be almost trivial and has as a basic building block the so called factorizing Smatrices for integrable quantum field theories in 2dimensional Minkowski space. For these Smatrices particle scattering would mean only a permutation of momenta in momentum space. If Smatrix is invariant under inclusion then U matrix should be in a welldefined sense almost trivial apart from a dispersion in zero modes leading to a superpositions of states characterized by different collections of Galois groups. 3. Relation to TGD inspired theory of consciousness Umatrix could be almost trivial with respect to the transitions which are diagonal with respect to the number field. What would however make U highly interesting is that it would predict the rates for the transitions representing a transformation of intention to action identified as a padictoreal transition. In this context almost triviality would translate to a precise correlation between intention and action. The general vision about the dynamics of quantum jumps suggests that the extension of a subfactor to a factor is followed by a reduction to a subfactor which is not necessarily the same. Breathing would be an excellent metaphor for the process. Breathing is also a metaphor for consciousness and life. Perhaps the essence of living systems distinguishing them from subsystems with a fixed state space could be cyclic breathing like process N→ M supset N → N_{1} subset M→ .. extending and reducing the state space of the subsystem by entanglement followed by deentanglement. One could even ask whether the unique role of breathing exercise in meditation practices relates directly to this basic dynamics of living systems and whether the effect of these practices is to increase the value of M:N and thus the order of Galois group G describing the algebraic complexity of "partonic" 2surfaces involved (they can have arbitrarily large sizes). The basic hypothesis of TGD inspired theory of cognition indeed is that cognitive evolution corresponds to the growth of the dimension of the algebraic extension of padic numbers involved. If one is willing to consider generalizations of the existing picture about quantum jump, one can imagine that unitary process can occur arbitrary number of times before it is followed by state function reduction. Unitary process and state function reduction could compete in this kind of situation. 4. Fractality of Smatrix and translational invariance in the lattice defined by subfactors Fractality realized as the invariance of the Smatrix in Jones inclusion means that the Smatrices of N and M relate by the projection P: M→N as S_{N}=PS_{M}P. S_{N} should be equivalent with S_{M} with a trivial relabelling of strands of infinite braid. Inclusion invariance would mean translational invariance of the Smatrix with respect to the index n labelling strands of braid defined by the projectors e_{i}. Translations would act only as a semigroup and Smatrix elements would depend on the difference mn only. Transitions can occur only for mn≥ 0, that is to the direction of increasing label of strand. The group G leaving N elementwise invariant would define the analog of a unit cell in lattice like condensed matter systems so that translational invariance would be obtained only for translations m→ m+ nk, where one has n≥ 0 and k is the number of M(2,C) factors defining the unit cell. As a matter fact, this picture might apply also to ordinary condensed matter systems. For more details see the chapter Was von Neumann Right After All?.
