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 Kähler 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) ,
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 Kähler 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 ,
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.
- 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.
- 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.