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Topological Geometrodynamics: an Overview

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Year 2007

One element field, quantum measurement theory and its q-variant, and the Galois fields associated with infinite primes

Kea mentioned John Baez's This Week's Finds 259, where John talked about one-element field - a notion inspired by the q=exp(i2π/n)→1 limit for quantum groups. This limit suggests that the notion of one-element 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 q-measurement 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 q-integers become ordinary integers and n-D vector spaces reduce to n-element sets. For quantum logic the reduction would mean that 2N-D spinor space becomes 2N-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 Zn or or its extension-both 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 g2/4πhbar so that also these corrections vanish and one has same predictions as given by classical field theory.

2. Q-measurement theory and q→ 1 limit.

Q-measurement theory differs from quantum measurement theory in that the coordinates of the state space, say spinor space, are non-commuting. Consider in the sequel q-spinors 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 n-element set. For HFF:s of type II1 it would be N-rays 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 space-time level.

q→ 1 limit should have space-time correlates by quantum classical correspondence. A TGD inspired geometro-topological interpretation for the projection postulate might be that quantum measurement at q→1 limit corresponds to a leakage of 3-surface 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 M4 or CP2 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 1-element fields actually correspond to Galois fields associated with infinite primes?

Finite field Gp 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,...,p-1 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. 1-element 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 1-element 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 p-adic field makes sense for infinite primes. The p-adic norm of any infinite-p p-adic number would be power of π either infinite, zero, or 1. Excluding infinite normed numbers one would have effectively only p-adic integers in the range 1,...Π-1 and thus only the Galois field GΠ 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 p-adic space-time 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.

  1. In a given measurement resolution characterized by the inclusion of HFFs of type II1 Connes tensor product defines an almost universal M-matrix apart from the non-uniqueness due to the facts that one has a direct sum of hyper-finite factors of type II1 (sum over conformal weights at least) and the fact that the included algebra defining the measurement resolution can be represented in a reducible manner. The S-matrices associated with irreducible factors would be unique in a given measurement resolution and the non-uniqueness would make possible non-trivial density matrices and thermodynamics.

  2. Higgs vacuum expectation is proportional to the generalized position dependent eigenvalue of the modified Dirac operator and its minima define naturally number theoretical braids as orbits for the minima of the universal Higgs potential: fusion and decay of braid strands emerge naturally. Thus the old speculation about a generalization of braid diagrams to Feynman diagram likes objects, which I already began to think to be too crazy to be true, finds a very natural realization.

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 3-D 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 M4 metric on Planck constant. Cancellation occurs only for critical values of Kähler coupling strength αK: for general values of αK 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 M-matrix 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 light-like 3-surfaces in shorter time scales bringing in increasingly complex 3-topologies.

This would mean following.

  1. One would not have perturbation theory around a given maximum of Kähler function but as a sum over increasingly complex maxima of Kähler function. Radiative corrections in the sense of perturbative functional integral around a given maximum would vanish (so that the expansion in terms of braid topologies would not make sense around single maximum). Radiative corrections would not vanish in the sense of a sum over 3-topologies obtained by adding radiative corrections as zero energy states in shorter time scale.

  2. Connes tensor product with a given measurement resolution would correspond to a restriction on the number of maxima of Kähler function labelled by the braid diagrams. For zero energy states in a given time scale the maxima of Kähler function could be assigned to braids of minimal complexity with braid vertices interpreted in terms of an addition of radiative corrections. Hence a connection with QFT type Feyman diagram expansion would be obtained and the Connes tensor product would have a practical computational realization.

  3. The cutoff in the number of topologies (maxima of Kähler function contributing in a given resolution defining Connes tensor product) would be always finite in accordance with the algebraic universality.

  4. The time scale resolution defined by the temporal distance between the tips of the causal diamond defined by the future and past light-cones applies to the addition of zero energy sub-states and one obtains a direct connection with p-adic length scale evolution of coupling constants since the time scales in question naturally come as negative powers of two. More precisely, p-adic p-adic primes near power of two are very natural since the coupling constant evolution comes in powers of two of fundamental 2-adic length scale.

There are still some questions. Radiative corrections around given 3-topology vanish. Could radiative corrections sum up to zero in an ideal measurement resolution also in 2-D sense so that the initial and final partonic 2-surfaces associated with a partonic 3-surface of minimal complexity would determine the outcome completely? Could the 3-surface 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 p-adic 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 CP2 length scale defines the ultimate UV cutoff.

For more details see the chapter Configuration Space Spinor Structure.

Number theoretic braids and global view about anti-commutations of induced spinor fields

The anti-commutations of induced spinor fields are reasonably well understood locally. The basic objects are 3-dimensional light-like 3-surfaces. These surfaces can be however seen as random light-like orbits of partonic 2-surfaces taking which would thus seem to take the role of fundamental dynamical objects. Conformal invariance in turn seems to make the 2-D partons 1-D 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 3-D 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. Anti-commutations 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 anti-commutations. The coordinate patches inside which Higgs modulus is monotonically increasing function define a division of partonic 2-surfaces X2t= X3l\intersection δ M4+/-,t to 2-D patches as a function of time coordinate of X3l as light-cone boundary is shifted in preferred time direction defined by the quantum critical sub-manifold M2× CP2. This induces similar division of the light-like 3-surfaces X3l to 3-D patches and there is a close analogy with the dynamics of ordinary 2-D landscape.

In both 2-D and 3-D case one can ask what happens at the common boundaries of the patches. Do the induced spinor fields associated with different patches anti-commute so that they would represent independent dynamical degrees of freedom? This seems to be a natural assumption both in 2-D and 3-D case and correspond to the idea that the basic objects are 2- resp. 3-dimensional in the resolution considered but this in a discretized sense due to finite measurement resolution, which is coded by the patch structure of X3l. 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 X3l.

If the induced spinor fields associated with different patches anti-commute, patches indeed define independent fermionic degrees of freedom at braid points and one has effective 2-dimensionality in discrete sense. In this picture the fundamental stringy curves for X2t correspond to the boundaries of 2-D patches and anti-commutation 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 anti-commutativity 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≤ nmax 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 anti-commutativity 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 nmax: the naive expectation is that for a given stringy curve the number of braid points equals to nmax.

2. The decomposition into 3-D patches and QFT description of particle reactions at the level of number theoretic braids

What is the physical meaning of the decomposition of 3-D light-like 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 2-surfaces defining reaction vertices. This is not obvious if X3l 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 X3l. 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 light-like surface to slicing of X4 to light-like 3-surfaces would give the full cusp catastrophe.

In the catastrophe theoretic setting the time parameter of X3l 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 3-surfaces along their ends at vertices. Particle reactions are realized also at the level of 4-D space-time surfaces. One might hope that this multiple realization could code the dynamics already at the simple level of single partonic 3-surface.

3. About 3-D minima of Higgs potential

The dominating contribution to the modulus of the Higgs field comes from δ M4+/- distance to the axis R+ defining quantization axis. Hence in scales much larger than CP2 size the geometric picture is quite simple. The orbit for the 2-D 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 CP2 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 X3l.

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 X3l 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 2-surface would determine the entire contribution to the Dirac determinant.

For more details see the chapter Configuration Space Spinor Structure.

Geometric view about Higgs mechanism

The 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 2-surface. Also a beautiful geometric interpretation of the generalized eigenstates and eigenvalues of the modified Dirac operator and understanding of super-canonical 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 2-surface. 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 S-matrix".

Geometrization of Higgs mechanism in TGD framework

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 super-canonical 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 X2 by w, its H=M4× CP2 coordinates by h=(m,s), and the H coordinates of its R+× S2II projection by hc=(r+,sII).

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 M4 resp. CP2 projection of the partonic 2-surface X2 with S2r resp. S2II.

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 S2. 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+× S2II. The value of H should be determined once h(w) and R+× S2II projection hc(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),hc(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 δM4 and CP2). The value should be in general complex and invariant under the isometries of δH affecting h and hc(w). The minimal geodesic distance d(h,hc) 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 M4 projection is indeed geodesic but that M4 projection extremizes itse length subject to the constraint that the absolute value of the phase defined by one-dimensional Käahler action ∫ Aμdxμ is minimized: this point will be discussed below. If this inclusion is allowed then internal consistency requires also the extremization of ∫ Aμdxμ so that geodesic lines are not allowed in CP2.

The value should be in general complex and invariant under the isometries of δ H affecting h and hc. The minimal distance d(h,hc) 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 CP2 and one possibility would be the non-integrable phase factor U(s,sII) 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,hc(w))× U(s,sII) ,

d(h,hc(w))=∫hhcds ,

U(s,sII) = exp[i∫ssIIAkdsk] .

This gives rise to a holomorphic function is X2 the local complex coordinate of X2 is identified as w= d(h,hc)U(s,sII) 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 S2r having expression (Aθ,Aφ)=(k,cos(θ)) with k even integer from the requirement that the non-integral 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 M4 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 X2CP2 point from S2II. 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 space-time sheet associated with the vacuum parton X2 is indeed vacuum extremal. This requires that also X3l is a vacuum extremal: in this case Dirac determinant must be real although it need not be equal to unity. The CP2 projection of the vacuum extremal belongs to some Lagrangian sub-manifold Y2 of CP2. For this kind of vacuum partons the ratio of the product of minimal H distances to corresponding M4+/- distances must be equal to unity, in other words minima of Higgs potential must belong to the intersection X2\cap S2II or to the intersection X2\cap R+ so that distance reduces to M4 or CP2 distance and Dirac determinant to a phase factor. Also this phase factor should be trivial.

It seems however difficult to understand how to obtain non-trivial phase in the generic case for all points if the phase is evaluated along geodesic line in CP2 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 s0 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= ∫ Asds +λ(∫ ds-s0).

This gives for the extremum the equation of motion for a charged particle with Kähler charge QK= 1/λ:

D2sk/ds2 + (1/λ)× Jkldsl/ds=0 ,

D2mk/ds2=0 .

The magnitude of the phase must be further minimized as a function of curve length s.

If the extremum curve in CP2 consists of two parts, first belonging to X2II and second to Y2, the condition is satisfied. Hence, if X2CP2× Y2 is not empty, the phases are trivial. In the generic case 2-D sub-manifolds of CP2 have intersection consisting of discrete points (note again the fundamental role of 4-dimensionality of CP2). Since S2II itself is a Lagrangian sub-manifold, it has especially high probably to have intersection points with S2II. If this is not the case one can argue that X3l cannot be vacuum extremal anymore.

The construction gives also a concrete idea about how the 4-D space-time sheet X4(X3l) becomes assigned with X3l. The point is that the construction extends X2 to 3-D surface by connecting points of X2 to points of S2II using the proposed curves. This process can be carried out in each intersection of X3l and M4+ shifted to the direction of future. The natural conjecture is that the resulting space-time sheet defines the 4-D 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 X2 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+× S2II cannot contribute to the Dirac determinant since Higgs vanishes at the critical manifold. Note that at S2II 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 δM4+/-distances of the extrema from R+. The value of the determinant would equal to one also at the limit R+× S2II.

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(wk)]/[∏k H0(wk)]= ∏k[wk/w0k].

Here w0k are M4 distances of extrema from R+. Equivalently: one can identify the values of Higgs field as dimensionless numbers wk/w0k. The modulus of Higgs field would be the ratio of H and M4+/- distances from the critical sub-manifold. The modulus of the Dirac determinant would be the product of the ratios of H and M4 depths of the valleys.

This definition would be general coordinate invariant and independent of the topology of X2. It would also introduce a unique conformal structure in X2 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 CP2 length defines the natural unit of mass. The determinant is invariant under the scalings of H metric as are also Kähler action and Chern-Simons action. This excludes the possibility that Dirac determinant could also give rise to the exponent of the area of X2.

Number theoretical constraints require that the numbers wk are algebraic numbers and this poses some conditions on the allowed partonic 2-surfaces 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 X2. Furthermore, the braiding operation could be defined for all intersections of X3l defined by shifts M4+/- as orbits of minima of Higgs potential. Second option is braiding by Kähler magnetic flux lines.

The question is then how to understand super-canonical 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(wk)/H(w0k)]-s .

The number of eigenvalues contributing to this function would be finite and H(wk)/H(w0k 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 X2 since the moduli of the eigenvalues characterize the depths of the valleys of the landscape defined by X2 and the associated non-integrable 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 super-canonical 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 super-canonical conformal weights would be functionals of X2. The scaling of λ by a constant depending on p-adic prime factors out from the zeta so that zeros are not affected: this is in accordance with the renormalization group invariance of both super-canonical conformal weights and Dirac determinant.

The zeta function should exist also in p-adic sense. This requires that the numbers λ:s at the points s of S2II 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 space-time 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 space-time sheets but possess only single wormhole throat (p-adic mass calculations are consistent with this). Gauge bosons can have Higgs expectation proportional to λ. The proportionality must be of form <H> propto λ/pn/2 if gauge boson mass squared is of order 1/pn. The p-dependent scaling factor of λ is expected to be proportional to log(p) from p-adic 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 zk 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 CP2 and of δ M4+/-, 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 Chern-Simons exponential.

The objection is that Chern-Simons action depends not only on X2 but its light-like orbit X3l.

  1. The first manner to circumvent this objection is to restrict the consideration to maxima of Kähler function which select preferred light-like 3-surfaces X3l. The basic conjecture forced by the number theoretic universality and allowed by TGD based view about coupling constant evolution indeed is that perturbation theory at the level of configuration space can be restricted to the maxima of Kähler function and even more: the radiative corrections given by this perturbative series vanish being already coded by Kähler function having interpretation as analog of effective action.

  2. There is also an alternative way out of the difficulty: define the Dirac determinant and zeta function using the minima of the modulus of the generalized Higgs as a function of coordinates of X3l so that continuous strands of braids are replaced by a discrete set of points in the generic case.

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 2-surfaces at δ M4+/- 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 X2 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 2-surface 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 space-time correlate of Higgs as λ really unique? The replacement of H with its power Hr, r>0, leaves the minima of H invariant as points of X2 so that number theoretic braid is not affected. As a matter fact, the group of monotonically increasing maps real-analytic maps applied to H leaves number theoretic braids invariant. Polynomials with positive rational coefficients suggest themselves.

The map H→ Hr scales Kähler function to its r-multiple, which could be interpreted in terms of 1/r-scaling of the Kähler coupling strength. Also super-canonical conformal weights identified as zeros of ζ are scaled as h→ h/r and Chern-Simons 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 αK. 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 framework

The 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× e2/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.

  1. The easiest manner to understand the observed fractions is by assuming that both M4 an CP2 correspond to covering spaces so that both spin and electric charge and fermion number are quantized. With this assumption the expression for the Planck constant becomes hbar/hbar0 =nb/na and charge and spin units are equal to 1/nb and 1/na respectively. This gives ν =nna/nb2. The values n=2,3,5,7,.. are observed. Planck constant can have arbitrarily large values. There are general arguments stating that also spin is fractionized in FQHE and for na=knb required by the observed values of ν charge fractionization occurs in units of k/nb and forces also spin fractionization. For factor space option in M4 degrees of freedom one would have ν= n/nanb2.

  2. The appearance of nb=2 would suggest that also Z2 appears as the homotopy group of the covering space: filling fraction 1/2 corresponds in the composite fermion model and also experimentally to the limit of zero magnetic fiel [Jain]. Also ν=5/2 has been observed.

  3. A possible problematic aspect of the TGD based model is the experimental absence of even values of nb except nb=2. A possible explanation is that by some symmetry condition possibly related to fermionic statistics kn/nb must reduce to a rational with an odd denominator for nb>2. In other words, one has k propto 2r, where 2r the largest power of 2 divisor of nb smaller than nb.

  4. Large values of nb emerge as B increases. This can be understood from flux quantization. One has eBS= nhbar= n(nb/na)hbar0. The interpretation is that each of the nb sheets contributes n/na units to the flux. As nb increases also the flux increases for a fixed value of na and area S: note that magnetic field strength remains more or less constant so that kind of saturation effect for magnetic field strength would be in question. For na=knb one obtains eBS/hbar0= n/k so that a fractionization of magnetic flux results and each sheet contributes 1/knb units to the flux. ν=1/2 correspond to k=1,nb=2 and to a non-vanishing magnetic flux unlike in the case of composite fermion model.

  5. The understanding of the thermal stability is not trivial. The original FQHE was observed in 80 mK temperature corresponding roughly to a thermal energy of T≈ 10-5 eV. For graphene the effect is observed at room temperature. Cyclotron energy for electron is (from fe= 6× 105 Hz at B=.2 Gauss) of order thermal energy at room temperature in a magnetic field varying in the range 1-10 Tesla. This raises the question why the original FQHE requires so low a temperature? The magnetic energy of a flux tube of length L is by flux quantization roughly e2B2S≈ Ec(e)meL(hbar0=c=1) and exceeds cyclotron energy roughly by factor L/Le, Le electron Compton length so that thermal stability of magnetic flux quanta is not the explanation.

    A possible explanation is that since FQHE involves several values of Planck constant, it is quantum critical phenomenon and is characterized by a critical temperature. The differences of the energies associated with the phase with ordinary Planck constant and phases with different Planck constant would characterize the transition temperature. Saturation of magnetic field strength would be energetically favored.


[Laughlin] R. B. Laughlin (1983), Phys. Rev. Lett. 50, 1395.
[Jain] J. K. Jain (1989), Phys. Rev. Lett. 63, 199.

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

A further generalization of the notion of imbedding space

The 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 4-dimensional 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=M4×CP2. Choose submanifold M2×S2, where S2 is homologically non-trivial geodesic sub-manifold of CP2. 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 sub-manifold remains invariant under the transformations leaving the quantization axes invariant.

Form spaces M4= M4\M2 and CP2 = CP2\S2 and their Cartesian product. Both spaces have a hole of co-dimension 2 so that the first homotopy group is Z. From these spaces one can construct an infinite hierarchy of factor spaces M4/Ga and CP2/Gb where Ga is discrete group of SU(2) leaving quantization axis invariant. In case of Minkowski factor this means that the group in question acts essentially as a combination reflection and to rotations around quantization axies of angular momentum. The generalized imbedding space is obtained by gluing all these spaces together along M2×S2.

The hypothesis is that Planck constant is given by the ratio hbar= na/nb, where ni is the order of maximal cyclic subgroups of Gi. The hypothesis states also that the covariant metric of the Minkowski factor is scaled by the factor (na/nb)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. Ga× Gb 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 na where Zna is the maximal cyclic subgroup of Ga.

One can however imagine obtaining fractionization at the level of imbedding space for space-time sheets, which are analogous to multi-sheeted Riemann surfaces (say Riemann surfaces associated with z1/n since the rotation by 2π understood as a homotopy of M4 lifted to the space-time sheet is a non-closed 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.

  1. This is certainly not possible for M4, CP2, or H since their fundamental groups are trivial. On the other hand, the fixing of quantization axes implies a selection of the sub-space H4= M2× S2subset M4× CP2, where S2 is a geodesic sphere of CP2. M4=M4\M2 and CP2=CP2\S2 have fundamental group Z since the codimension of the excluded sub-manifold is equal to two and homotopically the situation is like that for a punctured plane. The exclusion of these sub-manifolds defined by the choice of quantization axes could naturally give rise to the desired situation.

  2. H4 represents a straight cosmic string. Quantum field theory phase corresponds to Jones inclusions with Jones index M:N<4. Stringy phase would by previous arguments correspond to M:N=4. Also these Jones inclusions are labelled by finite subgroups of SO(3) and thus by Zn identified as a maximal Abelian subgroup.

    One can argue that cosmic strings are not allowed in QFT phase. This would encourage the replacement M4×CP2 implying that surfaces in M4×S2 and M2×CP2 are not allowed. In particular, cosmic strings and CP2 type extremals with M4 projection in M2 and thus light-like geodesic without zitterwebegung essential for massivation are forbidden. This brings in mind instability of Higgs=0 phase.

  3. The covering spaces in question would correspond to the Cartesian products M4na× CP2nb of the covering spaces of M4 and CP2 by Zna and Znb with fundamental group is Zna× Znb. One can also consider extension by replacing M2 and S2 with its orbit under Ga (say tedrahedral, octahedral, or icosahedral group). The resulting space will be denoted by M4×Ga resp. CP2×Gb. Product sign does not signify for Caretsian product here.

  4. One expects the discrete subgroups of SU(2) emerge naturally in this framework if one allows the action of these groups on the singular sub-manifolds M2 or S2. This would replace the singular manifold with a set of its rotated copies in the case that the subgroups have genuinely 3-dimensional action (the subgroups which corresponds to exceptional groups in the ADE correspondence). For instance, in the case of M2 the quantization axes for angular momentum would be replaced by the set of quantization axes going through the vertices of tedrahedron, octahedron, or icosahedron. This would bring non-commutative homotopy groups into the picture in a natural manner.

    Also the orbifolds M4/Ga× CP2/Gb can be allowed as also the spaces M4/Ga× (CP2×Gb) and (M4×GaCP2/Gb. Hence the previous framework would generalize considerably by the allowance of both coset spaces and covering spaces.

4. Do factor spaces and coverings correspond to the two kinds of Jones inclusions?

What could be the interpretation of these two kinds of spaces?

  1. Jones inclusions appear in two varieties corresponding to M:N<4 and M:N=4 and one can assign a hierarchy of subgroups of SU(2) with both of them. In particular, their maximal Abelian subgroups Zn label these inclusions. The interpretation of Zn as invariance group is natural for M: N< 4 and it naturally corresponds to the coset spaces. For M:N=4 the interpretation of Zn has remained open. Obviously the interpretation of Zn as the homology group defining covering would be natural.

  2. M:N=4 should correspond to the allowance of cosmic strings and other analogous objects. Does the introduction of the covering spaces bring in cosmic strings in some controlled manner? Formally the subgroup of SU(2) defining the inclusion is SU(2) would mean that states are SU(2) singlets which is something non-physical. For covering spaces one would however obtain the degrees of freedom associated with the discrete fiber and the degrees of freedom in question would not disappear completely and would be characterized by the discrete subgroup of SU(2).

    For anyons the non-trivial homotopy of plane brings in non-trivial connection with a flat curvature and the non-trivial dynamics of topological QFTs. Also now one might expect similar non-trivial contribution to appear in the spinor connection of M2×Ga and CP2×Gb. In conformal field theory models non-trivial monodromy would correspond to the presence of punctures in plane.

  3. For factor spaces the unit for quantum numbers like orbital angular momentum is multiplied by na resp. nb and for coverings it is divided by this number. These two kind of spaces are in a well defined sense obtained by multiplying and dividing the factors of H by Ga resp. Gb and multiplication and division are expected to relate to Jones inclusions with M:N< 4 and M:N=4, which both are labelled by a subset of discrete subgroups of SU(2).

  4. The discrete subgroups of SU(2) with fixed quantization axes possess a well defined multiplication with product defined as the group generated by forming all possible products of group elements as elements of SU(2). This product is commutative and all elements are idempotent and thus analogous to projectors. Trivial group G1, two-element group G2 consisting of reflection and identity, the cyclic groups Zp, p prime, and tedrahedral, octahedral, and icosahedral groups are the generators of this algebra.

    By commutativity one can regard this algebra as an 11-dimensional module having natural numbers as coefficients ("rig"). The trivial group G1, two-element group G2 generated by reflection, and tedrahedral, octahedral, and icosahedral groups define 5 generating elements for this algebra. The products of groups other than trivial group define 10 units for this algebra so that there are 11 units altogether. The groups Zp generate a structure analogous to natural numbers acting as analog of coefficients of this structure. Clearly, one has effectively 11-dimensional commutative algebra in 1-1 correspondence with the 11-dimensional "half-lattice" N11 (N denotes natural numbers). Leaving away reflections, one obtains N7. The projector representation suggests a connection with Jones inclusions. An interesting question concerns the possible Jones inclusions assignable to the subgroups containing infinitely manner elements. Reader has of course already asked whether dimensions 11, 7 and their difference 4 might relate somehow to the mathematical structures of M-theory with 7 compactified dimensions.

  5. How do the Planck constants associated with factors and coverings relate? One might argue that Planck constant defines a homomorphism respecting the multiplication and division (when possible) by Gi. If so, then Planck constant in units of hbar0 would be equal to na/nb for H/Ga× Gb option and nb/na for H×(Ga× Gb) with obvious formulas for hybrid cases. This option would put M4 and CP2 in a very symmetric role and allow much more flexibility in the identification of symmetries associated with large Planck constant phases.
For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.

Does the quantization of Kähler coupling strength reduce to the quantization of Chern-Simons 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 light-like 3-surfaces. This almost TQFT involves Abelian Chern-Simons action for the induced Kähler form.

This raises the question whether the integer valued quantization of the Chern-Simons 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 3-D 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 Chern-Simons coupling k: 1/αK=4k if all factors are correct. k=26 is forced by the comparison with some physical input. Also p-adic temperature could be identified as Tp=1/k.

1. Quantization of Chern-Simons coupling strength

For Chern-Simons action the quantization of the coupling constant guaranteing so called holomorphic factorization is implied by the integer valuedness of the Chern-Simons coupling strength k. As Witten explains, this follows from the quantization of the first Chern-Simons class for closed 4-manifolds plus the requirement that the phase defined by Chern-Simons action equals to 1 for a boundaryless 4-manifold obtained by gluing together two 4-manifolds along their boundaries. As explained by Witten in his paper, one can consider also "anyonic" situation in which k has spectrum Z/n2 for n-fold 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 4-D Kähler action defining Kähler function for the world of classical worlds and conjectured to result as a Dirac determinant. The conjecture has been that gK2 has only single value. With some physical input one can make educated guesses about this value. The connection with the quantization of Chern-Simons coupling would however suggest a spectrum of values. This spectrum is easy to guess.

  1. The U(1) counterpart of Chern-Simons action is obtained as the analog of the "instanton" density obtained from Maxwell action by replacing J wedge *J with J wedge J. This looks natural since for self dual J associated with CP2 extremals Maxwell action reduces to instanton density and therefore to Chern-Simons term. Also the interpretation as Chern-Simons action associated with the classical SU(3) color gauge field defined by Killing vector fields of CP2 and having Abelian holonomy is possible. Note however that instanton density is multiplied by imaginary unit in the action exponential of path integral. One should find justification for this "Wick rotation" not changing the value of coupling strength and later this kind of justification will be proposed.

  2. Wick rotation argument suggests the correspondence k/4π = 1/4gK2 between Chern-Simons coupling strength and the Kähler coupling strength gK appearing in 4-D Kähler action. This would give

    gK2=π/k .

    The spectrum of 1/αK would be integer valued


    The result is very nice from the point of number theoretic vision since the powers of αK appearing in perturbative expansions would be rational numbers (ironically, radiative corrections might vanish but this might happen only for these rational values of αK!).

  3. It is interesting to compare the prediction with the experimental constraints on the value of αK. The basic empirical input is that electroweak U(1) coupling strength reduces to Kähler coupling at electron length scale (see this). This gives αK= αU(1)(M127)≈ 104.1867, which corresponds to k=26.0467. k=26 would give αK= 104: the deviation would be only .2 per cent and one would obtain exact prediction for αU(1)(M127)! This would explain why the inverse of the fine structure constant is so near to 137 but not quite. Amusingly, k=26 is the critical space-time dimension of the bosonic string model. Also the conjectured formula for the gravitational constant in terms of αK and p-adic prime p involves all primes smaller than 26 (see this).

  4. Note however that if k is allowed to have values in Z/n2, the strongest possible coupling strength is scaled to n2/4 if hbar is not scaled: already for n=2 the resulting perturbative expansion might fail to converge. In the scalings of hbar associated with M4 degrees of freedom hbar however scales as 1/n2 so that the spectrum of αK would remain invariant.

3. Justification for Wick rotation

It is not too difficult to believe to the formula 1/αK =qk, q some rational. q=4 however requires a justification for the Wick rotation bringing the imaginary unit to Chern-Simons 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.

  1. To see what this symmetry might be consider the generalization of the Montonen-Olive duality obtained by combining theta angle and gauge coupling to single complex number via the formula

    τ= θ/2π+i4π/g2.

    What this means in the recent case that for CP2 type vacuum extremals (see this) Kähler action and instanton term reduce by self duality to Kähler action obtained by the replacement g2 with -iτ/4π. The first duality τ→τ+1 corresponds to the periodicity of the theta angle. Second duality τ→-1/τ corresponds to the generalization of Montonen-Olive duality α→ 1/α. These dualities are definitely not symmetries of the theory in the recent case.

  2. Despite the failure of dualities, it is interesting to write the formula for τ in the case of Chern-Simons theory assuming gK2=π/k with k>0 holding true for Kac-Moody representations. What one obtains is

    τ= 4k(1-i).

    The allowed values of τ are integer spaced along a line whose direction angle corresponds to the phase exp(i2π/n), n=4. The transformations τ→ τ+ 4(1-i) generate a dynamical symmetry and as Lorentz transformations define a subgroup of the group E2 leaving invariant light-like momentum (this brings in mind quantum criticality!). One should understand why this line is so special. One should understand why this line is so special.


  3. This formula conforms with the number theoretic vision suggesting that the allowed values of τ belong to an integer spaced lattice. Indeed, if one requires that the phase angles are proportional to vectors with rational components then only phase angles associated with orthogonal triangles with short sides having integer valued lengths m and n are possible. The additional condition that the phase angles correspond to roots of unity! This leaves only m=n and m=-n>0 into consideration so that one would have τ= n(1-i) from k>0.

  4. Notice that theta angle is a multiple of 8kπ so that a trivial strong CP breaking results and no QCD axion is needed (this of one takes seriously the equivalence of Kähler action to the classical color YM action).

4. Is p-adicization needed and possible only in 3-D sense?

The action of CP2 type extremal is given as S=π/8αK= 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 p-adicize also in 4-D sense, this raises a problem.

Before considering this problem, consider first the 4-D p-adicization more generally.

  1. The definition of Kähler action and Kähler function in p-adic case can be obtained only by algebraic continuation from the real case since no satisfactory definition of p-adic definite integral exists. These difficulties are even more serious at the level of configuration space unless algebraic continuation allows to reduce everything to real context. If TGD is integrable theory in the sense that functional integral over 3-surfaces reduces to calculable functional integrals around the maxima of Kähler function, one might dream of achieving the algebraic continuation of real formulas. Note however that for lightlike 3-surface the restriction to a category of algebraic surfaces essential for the re-interpretation of real equations of 3-surface as p-adic equations. It is far from clear whether also preferred extremals of Kähler action have this property.

  2. Is 4-D p-adicization the really needed? The extension of light-like partonic 3-surfaces to 4-D space-time surfaces brings in classical dynamical variables necessary for quantum measurement theory. p-Adic physics defines correlates for cognition and intentionality. One can argue that these are not quantum measured in the conventional sense so that 4-D p-adic space-time sheets would not be needed at all. The p-adic variant for the exponent of Chern-Simons action can make sense using a finite-D algebraic extension defined by q=exp(i2π/n) and restricting the allowed lightlike partonic 3-surfaces so that the exponent of Chern-Simons form belongs to this extension of p-adic numbers. This restriction is very natural from the point of view of dark matter hierarchy involving extensions of p-adics by quantum phase q.

If one remains optimistic and wants to p-adicize also in 4-D sense, the transcendental value of the vacuum functional for CP2 type vacuum extremals poses a problem (not the only one since the p-adic norm of the exponent of Kähler action can become completely unpredictable).

  1. One can also consider extending p-adic numbers by introducing exp(π) and its powers and possibly also π. This would make the extension of p-adics infinite-dimensional which does not conform with the basic ideas about cognition. Note that ep is not p-adic transcendental so that extension of p-adics by powers e is finite-dimensional and if p-adics are first extended by powers of π then further extension by exp(π) is p-dimensional.

  2. A more tricky manner to overcome the problem posed by the CP2 extremals is to notice CP2 type extremals are necessarily deformed and contain a hole corresponding to the lightlike 3-surface or several of them. This would reduce the value of Kähler action and one could argue that the allowed p-adic deformations are such that the exponent of Kähler action is a p-adic number in a finite extension of p-adics. This option does not look promising.

5. Is the p-adic temperature proportional to the Kähler coupling strength?

Kähler coupling strength would have the same spectrum as p-adic temperature Tp apart from a multiplicative factor. The identification Tp=1/k is indeed very natural since also gK2 is a temperature like parameter. The simplest guess is

Tp= 1/k.

Also gauge couplings strengths are expected to be proportional to gK2 and thus to 1/k apart from a factor characterizing p-adic 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 αK=1/104, this would give 1/Tp≈ 1/26. This means that photon, graviton, and gluons would be massless in an excellent approximation for say p=M89, which characterizes electroweak gauge bosons receiving their masses from their coupling to Higgs boson. For fermions one has Tp=1 so that fermionic lightlike wormhole throats would correspond to the strongest possible coupling strength αK=1/4 whereas gauge bosons identified as pairs of light-like wormhole throats associated with wormhole contacts would correspond to αK=1/104. Perhaps Tp=1/26 is the highest p-adic temperature at which gauge boson wormhole contacts are stable against splitting to fermion-antifermion pair. Fermions and possible exotic bosons created by bosonic generators of super-canonical algebra would correspond to single wormhole throat and could also naturally correspond to the maximal value of p-adic 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 αK makes sense only if the 4-D space-time sheets associated with fermions and bosons can be regarded as disjoint space-time regions. Gauge bosons correspond to wormhole contacts connecting (deformed pieces of CP2 type extremal) positive and negative energy space-time sheets whereas fermions would correspond to deformed CP2 type extremal glued to single space-time sheet having either positive or negative energy. These space-time sheets should make contact only in interaction vertices of the generalized Feynman diagrams, where partonic 3-surfaces are glued together along their ends. If this gluing together occurs only in these vertices, fermionic and bosonic space-time 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 αK, allows to identify the inverse of p-adic 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.

S-matrix as a functor and the groupoid like structure formed by S-matrices

In zero energy ontology S-matrix can be seen as a functor from the category of Feynman cobordisms to the category of operators. S-matrix can be identified as a "complex square root" of the positive energy density matrix S= ρ1/2+S0, where S0 is a unitary matrix and ρ+ is the density matrix for positive energy part of the zero energy state. Obviously one has SS*+. S*S=ρ- gives the density matrix for negative energy part of zero energy state. Clearly, S-matrix can be seen as a matrix valued generalization of Schrödinger amplitude. Note that the "indices" of the S-matrices 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 hyper-finite factor of II1 it is not strictly speaking possible to speak about indices since the matrix elements are traces of the S-matrix multiplied by projection operators to infinite-dimensional subspaces from right and left.

The functor property of S-matrices 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-1f are defined but not identical in general, and one has fgg-1=f and f-1fg= g.

The reason for the groupoid like property is that S-matrix 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= ρf,-g, and the conditions ff*+ and f*f=ρ- are satisfied. Here ρf+/- 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-1L=f*ρf,+-1 satisfies ff-1L= Id+ and fR-1f,--1f* satisfies f-1Rf= Id-.

There are good reasons to believe that also tensor product of its appropriate generalization to the analog of co-product makes sense with non-triviality characterizing the interaction between the systems of the tensor product. If so, the S-matrices would form very beautiful mathematical structure bringing in mind the corresponding structures for 2-tangles and N-tangles. 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 S-matrix as a generalized Schrödinger amplitude would suggest Ψ-matrix. Since the interaction with Kea's M-theory blog (with M denoting Monad or Motif in this context) helped to realize the connection with density matrix, also M-matrix might work. S-matrix 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 M-theory in the stringy sense of the word but wouldn't it be fair if stringy M-theory 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 freedom

Dark matter hierarchy corresponds to a hierarchy of conformal symmetries Zn of partonic 2-surfaces with genus g≥ 1 such that factors of n define subgroups of conformal symmetries of Zn. By the decomposition Zn=∏p|n Zp, where p|n 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 sub-hierarchy Zp, Zp2=Zp× Zp, etc... such that the moduli at n+1:th level are contained by n:th level. In the similar manner the moduli of Zn are sub-moduli 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 non-commutative discrete group having Zn 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 Zn one could have tedrahedral and icosahedral groups plus cyclic group Z2n with reflection added but not Z2n+1 nor the symmetry group of cube. The conjecture is wrong. Consider the orbit of the subgroup of rotational group on standard sphere of E3, 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 ruler-and-compass integers having as factors only first powers of Fermat primes and power of 2 would define a physically preferred sub-hierarchy of quantum criticality for which subsequent levels would correspond to powers of 2: a connection with p-adic 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 CP2→ CP2/U(2) consisting of geodesic spheres of CP2. In this case the discrete subgroup might correspond to a selection of a subgroup of SU(2)subset SU(3) acting non-trivially on the geodesic sphere. Cosmic strings X2× Y2 subset M4×CP2 having geodesic spheres of CP2 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 families

One 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 Zn 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 space-time correlate for fermionic 2-valuedness 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 resolution

I 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 M-theory as something, which is not the 11-D something making me rather unhappy as a physicist with second foot still deep in the muds of low energy phenomenology.

Kea talks about topos, n-logos,... 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 space-time resolution. In TGD framework discretization by number theoretic braids replaces partonic 2-surface 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 p-adic physics by gluing real and p-adic number fields to single super-structure 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 hyper-finite factors of type II1 (HFF, see this).

  • Could topology with particular algebraic points have a generalization allowing a category theoretic formulation of the quantum measurement theory without states identified as complex rays?

  • How to achieve this? In the transition of ordinary Boolean logic to quantum logic in the old fashioned sense (von Neuman again!) the set of subsets is replaced with the set of subspaces of Hilbert space. Perhaps this transition has a counterpart as a transition from Sierpinski topology to a structure in which sub-spaces of Hilbert space are quantum sub-spaces with complex rays replaced with the orbits of subalgebra defining the measurement resolution. Sierpinski space {0,1} would in this generalization be replaced with the quantum counterpart of the space of 2-spinors. Perhaps one should also introduce q-category theory with Heyting algebra being replaced with q-quantum logic.

3. Fuzzy quantum logic as counterpart for Sierpinksi space

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 2-spinor 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?).

Q-spinors and qqbits: For q-spinors the two components a and b are not commuting numbers but non-Hermitian 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 q-logic is inherently fuzzy: there are no absolute truths or falsehoods. One can actually predict the spectrum of eigenvalues of probabilities for say 1. q-Spinors 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.

Q-locale: Could one think of generalizing the notion of locale to quantum locale by using the idea that sets are replaced by sub-spaces of Hilbert space in the conventional quantum logic. Q-openness would be defined by identifying quantum spinors as the initial object, q-Sierpinski space. a (resp. b for dual category) would define q-open set in this space. Q-open 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 q-counterpart of rather uninteresting topology in which all sets are open and every map is continuous.

Q-locale and HFFs: The q-Sierpinski character of q-spinors 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 2-spinors and corresponding Clifford algebra serves as basic building brick although tensor powers of any matrix algebra provides a representation of HFF.

Q-measurement theory: Finite measurement resolution (q-quantum measurement theory) means that complex rays are replaced by sub-algebra 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 q-topology and q-spinors. Fuzzyness of qqbits of course correlates with the finite measurement resolution.

Q-n-logos: For other q-representations 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 n-logos, quantum generalization of n-valued logic. All of these would be however less fundamental and induced by q-morphisms to the fundamental representation in terms of spinors of the world of classical worlds. What would be however very nice that if these q-morphisms are constructible explicitly it would become possible to build up q-representations 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 Zn associated with Jones inclusions and leaving the choice of quantization axes invariant, bring in mind the n-point 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 p-adic variants of algebraic partonic 2-surface 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 M4×CP2→ M4×CP2/Ga×Gb 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 matter

Every 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 Bose-Einstein condensates of ordinary gravitons. This suggests that Bose-Einstein 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 Bose-Einstein condensate as characterized of Planck constant

The following picture is the simplest I have been able to imagine hitherto.

  1. Suppose that darkness corresponds to the darkness of the field bodies (em, Z0,W,...) of the elementary particle so that the elementary particle proper is not affected in the transition to large hbar phase. This stimulates the idea that some Bose-Einstein condensate associated with the field body provides a microscopic description for the darkness and that one can relate the value of hbar to the properties of Bose-Einstein condensate.

  2. Since the spin of the particle is not affected in the transition, it would seem that the bosons in question are Lorentz scalars. Hence a Bose-Einstein condensate of Higgs suggests itself as the relevant structure. Higgs would have a double role since the coherent state of Higgs bosons associated with the field body would be responsible for or at least closely relate to the contribution to the mass of fermion identified usually in terms of a coupling to Higgs. The ground state would correspond to a coherent state annihilated by the new annihilation operators unitarily related to the original ones. Bose-Einstein condensate would be obtained as a many-Higgs state obtaining by applying these creation operators and would not be an eigen state of particle number in the old basis.

  3. As a rule, quantum classical correspondence is a good guideline. Suppose that the field body corresponds to a pair of positive and negative energy MEs connected by wormhole contacts representing the bosons forming the Bose-Einstein condensate. This structure could be more or less universal. In the general case MEs carry light-like gauge currents and light-like Einstein tensor. These currents can also vanish and should do so for the ground state. MEs could carry both coherent states of gauge bosons and gravitons but would not be present in the ground state. The CP2 part of the trace of second fundamental form transforming as SO(4) vector and doublet with respect to the groups SU(2)L and SU(2)R, is the only possible candidate for the classical Higgs field. The Fourier spectrum of CP2 coordinates has only light-like longitudinal momenta so that four-momenta are slightly tachyonic for non-vanishing transverse momenta. This state of facts might be a space-time correlate for the tachyonic character of Higgs.

  4. The quantum numbers of the particle should not be affected in the transition changing the value of Planck constant. The simplest explanation is that Higgs bosons have a vanishing net energy. This is possible since in the case of bosons the two wormhole throats have different sign of energy. Indeed, if the energies, spins, and em charges of fermion and antifermion at wormhole throats are of opposite sign, one is left with a coherent state of zero energy Higgs particles as a microscopic description for constant value of Higgs field.

  5. How do the properties of the Bose-Einstein condensate of Higgs relate to the value of Planck constant? MEs should remain invariant under the discrete groups Zna and Znb and the bosons at the sheets of the multiple covering should be in identical state. The number na× nb of zero energy Higgs bosons in the Bose-Einstein condensate would characterize the darkness at microscopic level.

2. How this affects the view about particle massivation?

This scenario would allow to add some details to the general picture about particle massivation reducing to p-adic thermodynamics plus Higgs mechanism, both of them having description in terms of conformal weight.

  1. The mass squared equals to the p-adic thermal average of the conformal weight. There are two contributions to this thermal average. One from the p-adic thermodynamics for super conformal representations, and one from the thermal average related to the spectrum of generalized eigenvalues λ of the modified Dirac operator D. Higgs expectation value appears in the role of a mass term in the Dirac equation just like λ in the modified Dirac equation. For the zero modes of D λ vanishes.

  2. There are good motivations to believe that λ is expressible as a superposition of zeros of Riemann zeta or some more general zeta function. The problem is that λ is complex. Since Dirac operator is essentially the square root of d'Alembertian (mass squared operator), the natural interpretation of λ would be as a complex "square root" of the conformal weight.

    Confession: The earlier interpretation of lambda as a complex conformal weight looks rather stupid in light of this observation. It seems that there is again some updating to do;-)!

    This encourages to consider the interpretation in terms of vacuum expectation of the square root of Virasoro generator, that is generators G of super Virasoro algebra, or something analogous. The super generators G of the super-conformal algebra carry fermion number in TGD framework, where Majorana condition does not make sense physically. The modified Dirac operators for the two possible choices t+/- of the light-like vector appearing in the eigenvalue equation DΨ = λ tk+/-ΓkΨ could however define a bosonic algebra resembling super-conformal algebra.

    The p-adic thermal expectation values of contractions of t-kΓkD+ and t+kΓkD- should co-incide with the vacuum expectations of Higgs and its conjugate. This makes sense if the two generalized eigenvalue spectra of D are complex conjugates. Note that D+ and D- would be same operator but with different definition of the generalized eigenvalue and hermitian conjugation would map these two kinds of eigen modes to each other. The real contribution to the mass squared would thus come naturally as <λλ*>. Of course, < H>=<λ> is only a hypothesis encouraged by the internal consistency of the physical picture, not a proven mathematical fact.

3. Questions

This leaves still some questions.

  1. Does the p-adic thermal expectation < λ> dictate < H> or vice versa? Physically it would be rather natural that the presence of a coherent state of Higgs wormhole contacts induces the mixing of the eigen modes of D. On the other hand, the quantization of the p-adic temperature Tp suggests that Higgs vacuum expectation is dictated by Tp.

  2. Also the phase of <λ> should have physical meaning. Could the interpretation of the imaginary part of < λ> make possible the description of dissipation at the fundamental level?

  3. Is p-adic thermodynamics consistent with the quantal description as a coherent state? The approach based on p-adic variants of finite temperature QFTs associate with the legs of generalized Feynman diagrams might resolve this question neatly since thermodynamical states would be genuine quantum states in this approach made possible by zero energy ontology.

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

Could also gauge bosons correspond to wormhole contacts?

The developments in the formulation of quantum TGD which have taken place during the period 2005-2007 (see this, this, and this) suggest dramatic simplifications of the general picture about elementary particle spectrum. p-Adic 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 S-matrix and by the almost topological QFT property of quantum TGD at parton level (see this and this). Or more concretely: do partonic 2-surfaces carry only free many-fermion states or can they carry also bound states of fermions and anti-fermions identifiable as bosons?

What is known that Higgs boson corresponds naturally to a wormhole contact (see this). The wormhole contact connects two space-time sheets with induced metric having Minkowski signature. Wormhole contact itself has an Euclidian metric signature so that there are two wormhole throats which are light-like 3-surfaces and would carry fermion and anti-fermion number in the case of Higgs. Irrespective of the identification of the remaining elementary particles MEs (massless extremals, topological light rays) would serve as space-time correlates for elementary bosons. Higgs type wormhole contacts would connect MEs to the larger space-time 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 anti-fermions. 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 2-surfaces carrying fermion-anti-fermion pair such that either fermion or anti-fermion has a non-physical polarization. For this option CP2 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.

  1. This option fails if the theory at partonic level is free field theory so that anti-fermions and elementary bosons cannot be identified as bound states of fermion and anti-fermion with either of them having non-physical polarization.

  2. Mathematically oriented mind could also argue that the asymmetry between Higgs and elementary gauge bosons is not plausible whereas asymmetry between fermions and gauge bosons is. Mathematician could continue by arguing that if wormhole contacts with net quantum numbers of Higgs boson are possible, also those with gauge boson quantum numbers are unavoidable.

  3. Physics oriented thinker could argue that since gauge bosons do not exhibit family replication phenomenon (having topological explanation in TGD framework) there must be a profound difference between fermions and bosons.

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 space-time sheet and propagating with light velocity. For this option there would be no need to assume the presence of non-physical fermion or anti-fermion polarization since fermion and anti-fermion would reside at different wormhole throats. Only the definition of what it is to be non-physical would be different on the light-like 3-surfaces defining the throats.

The difference would naturally relate to the different time orientations of wormhole throats and make itself manifest via the definition of light-like operator o=xkγk appearing in the generalized eigenvalue equation for the modified Dirac operator (see this and this). For the first throat ok would correspond to a light-like tangent vector tkof the partonic 3-surface and for the second throat to its M4 dual tdk in a preferred rest system in M4 (implied by the basic construction of quantum TGD). What is nice that this picture non-asks the question whether tkor tdkshould 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 CP2 type extremals topologically condensed on MEs and for bosons as pieces of CP2 type extremals connecting ME to the larger space-time sheet. For fermions topological condensation is possible to either space-time sheet.

2.2 Phase conjugate states and matter-antimatter asymmetry

By fermion number conservation fermion-boson and boson-boson couplings must involve the fusion of partonic 3-surfaces 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 anti-fermion at the same space-time sheet would take place via the basic vertex at which the 2-dimensional ends of light-like 3-surfaces 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 anti-fermion resulting in the decay.

Also a candidate for a new kind interaction vertex emerges. The splitting of bosonic wormhole contact would generate fermion and time-reversed anti-fermion 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 matter-antimatter asymmetry in the sense that fermionic antimatter would consist dominantly of negative energy anti-fermions at space-time 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 Z0 boson (see this).

3. Graviton and other stringy states

Fermion and anti-fermion 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 fermion-anti-fermion pairs and boson-anti-boson pairs with coefficients determined by the coupling of the parton to graviton. Graviton-graviton pairs might emerge in higher orders. Fermion and anti-fermion would reside at the same space-time sheet and would have a non-vanishing relative angular momentum. Also bosons could have non-vanishing relative angular momentum and Higgs bosons must indeed possess it.

Gravitons are stable if the throats of wormhole contacts carry non-vanishing 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 2-surfaces 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 Lp2, where Lp is p-adic length scale, and that physical graviton corresponds to p=M127=2127-1. 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 p-adic 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 CP2 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 ≈ Lp2.

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 non-vanishing electro-weak 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 M127quark and anti-quark 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 Tc super-conductivity involves stringy states connecting the space-time 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 non-stringy states

The 1-throat character of fermions is consistent with the generation-genus correspondence. The 2-throat character of bosons predicts that bosons are characterized by the genera (g1,g2) 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 wave-function would be expressible as a matrix Mg1,g2 and ordinary gauge bosons would correspond to a diagonal matrix Mg1,g2g1,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 2-throat character of bosons should relate to the low value T=1/n<< 1 for the p-adic 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 23=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 non-stringy 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 electro-weak 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 Z0 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 non-rectilinearity 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 S-matrix and U matrix

TGD 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 U-matrix characterizing the unitary process associated with the quantum jump (and followed by state function reduction and state preparation) and the S-matrix defining time-like 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 time-like entanglement.

1. S-matrix

In zero energy ontology S-matrix characterizes time like entanglement of zero energy states (this is possible only for HFFs for which Tr(SS+)=Tr(Id)=1 holds true). S-matrix 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 sub-factor.

The finite number theoretic braid having Galois group G as its symmetries is the space-time correlate for both the finite measurement resolution and the effective reduction of HFF to that associated with a finite-dimensional quantum Clifford algebra M/N. SU(2) inclusions would allow angular momentum and color quantum numbers in bosonic degrees of freedom and spin and electro-weak 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 sub-spaces 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 2-surface 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 S-matrix. S-matrix would reduce to a direct integral of S-matrices associated with various collections of Galois groups characterizing the number theoretical properties of partonic 2-surfaces. It is not difficult to criticize this picture.

  1. Why time like entanglement should be always characterized by a unitary S-matrix? Why not some more general matrix? If one allows more general time like entanglement, the description of particle reaction rates in terms of a unitary S-matrix must be replaced with something more general and would require a profound revision of the vision about the relationship between experiment and theory. Also the consistency of the zero energy ontology with positive energy ontology in time scales shorter than the time scale determined by the geometric time interval between positive and negative energy parts of the zero energy state would be lost. Hence the easy way to proceed is to postulate that the universe is self-referential in the sense that quantum states represent the laws of physics by coding S-matrix as entanglement coefficients.

  2. Second objection is that there might a huge number of unitary S-matrices so that it would not be possible to speak about quantum laws of physics anymore. This need not be the case since super-conformal symmetries and number theoretic universality pose extremely powerful constraints on S-matrix. A highly attractive additional assumption is that S-matrix is universal in the sense that it is invariant under the inclusion sequences defined by Galois groups G associated with partonic 2-surfaces. Various constraints on S-matrix might actually imply the inclusion invariance.

  3. One can of course ask why S-matrix should be invariant under inclusion. One might argue that zero energy states for which time-like entanglement is characterized by S-matrix invariant in the inclusion correspond to asymptotic self-organization patterns for which U-process and state function reduction do not affect the S-matrix in the relabelled basis. The analogy with a fractal asymptotic self-organization pattern is obvious.

2. U-matrix

In a well-defined sense U process seems to be the reversal of state function reduction. Hence the natural guess is that U-matrix means a quantum transition in which a factor becomes a sub-factor whereas state function reduction would lead from a factor to a sub-factor.

Various arguments suggest that U matrix could be almost trivial and has as a basic building block the so called factorizing S-matrices for integrable quantum field theories in 2-dimensional Minkowski space. For these S-matrices particle scattering would mean only a permutation of momenta in momentum space. If S-matrix is invariant under inclusion then U matrix should be in a well-defined 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

U-matrix 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 p-adic-to-real 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 sub-factor to a factor is followed by a reduction to a sub-factor 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 sub-systems with a fixed state space could be cyclic breathing like process N→ M supset N → N1 subset M→ .. extending and reducing the state space of the sub-system by entanglement followed by de-entanglement.

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" 2-surfaces 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 p-adic 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 S-matrix and translational invariance in the lattice defined by sub-factors

Fractality realized as the invariance of the S-matrix in Jones inclusion means that the S-matrices of N and M relate by the projection P: M→N as SN=PSMP. SN should be equivalent with SM with a trivial re-labelling of strands of infinite braid.

Inclusion invariance would mean translational invariance of the S-matrix with respect to the index n labelling strands of braid defined by the projectors ei. Translations would act only as a semigroup and S-matrix elements would depend on the difference m-n only. Transitions can occur only for m-n≥ 0, that is to the direction of increasing label of strand. The group G leaving N element-wise 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?.

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