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TGD: Physics as Infinite-Dimensional Geometry
Note: Newest contributions are at the top!
Many steps of progress have occurred in TGD lately.
In the previous posting I explained how generalized braid diagrams emerge naturally as orbits of the minima of Higgs defined as a generalized eigenvalue of the modified Dirac operator.
The association of generalized braid diagrams to incoming and outgoing 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.
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.
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.
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".
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) ,
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 ,
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 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.
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.
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.
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 Configuration Space Spinor Structure.
Witten has been talking at Friday in string cosmology workshop in New York about his new ideas relating to 1+2 D quantum gravity. Peter Woit has been listening the talk and represents his understanding in Not-Even-Wrong. Lubos Motl gives a nice summary about the conformal field theoretic ideas involved. Why I got interested (even with this miserable technical background) is that Witten's talk gives an interesting perspective to quantum TGD, which reduces to almost topological QFT for light-like partonic surfaces defined by Chern-Simons action and its fermionic super counterpart.
1. Brief summary of main points
Very concisely, the message seems to be following.
There are very strong resemblances between Witten's model and the formulation quantum TGD at parton level.