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Topological Geometrodynamics: an Overview

Note: Newest contributions are at the top!

Year 2010

Low viscosity liquids and preferred extremals of Kähler action as perfect fluid flows

Lubos Motl had an interesting article about Perfect fluids, string theory, and black holes. It of course takes some self discipline to get over the M-theory propaganda without getting very angry. Indeed, the article starts with

The omnipresence of very low-viscosity fluids in the observable world is one of the amazing victories of string theory. The value of the minimum viscosity seems to follow a universal formula that can be derived from quantum gravity - i.e. from string theory.

The first sentence is definitely something which surpasses all records in the recorded history of super string hype (for records see Not-Even Wrong). At the end of the propaganda strike Lubos however explains in an enjoyable manner some basic facts about perfect fluids, super-fluids, and viscosity and mentions the effective absence of non-diagonal components of stress tensor as a mathematical correlate for the absence of shear viscosity often identified as viscosity. This comment actually stimulated this posting.

In any case, almost perfect fluids seems to be abundant in Nature. For instance, QCD plasma was originally thought to behave like gas and therefore have a rather high viscosity to entropy density ratio x= η/s. Already RHIC found that it however behaves like almost perfect fluid with x near to the minimum predicted by AdS/CFT. The findings from LHC gave additional conform the discovery (see this). Also Fermi gas is predicted on basis of experimental observations to have at low temperatures a low viscosity roughly 5-6 times the minimal value (see this). This behavior is of course not a prediction of superstring theory but only demonstrates that AdS/CFT correspondence applying to conformal field theories as a new kind of calculational tool allows to make predictions in such parameter regions where standard methods fail. This is fantastic but has nothing to do with predictions of string theory.

In the following the argument that the preferred extremals of Kähler action are perfect fluids apart from the symmetry breaking to space-time sheets is developed. The argument requires some basic formulas summarized first. The detailed definition of the viscous part of the stress energy tensor linear in velocity (oddness in velocity relates directly to second law).

The physics oriented reader not working with hydrodynamics and possibly irritated from the observation that after all these years he actually still has a rather tenuous understanding of viscosity as a mathematical notion and willing to refresh his mental images about concrete experimental definitions as well as tensor formulas, can look the Wikipedia article about viscosity. Here one can find also the definition of the viscous part of the stress energy tensor linear in velocity (oddness in velocity relates directly to second law). The symmetric part of the gradient of velocity gives the viscous part of the stress-energy tensor as a tensor linear in velocity. This term decomposes to bulk viscosity and shear viscosity. Bulk viscosity gives a pressure like contribution due to friction. Shear viscosity corresponds to the traceless part of the velocity gradient often called just viscosity. This contribution to the stress tensor is non-diagonal.

  1. The symmetric part of the gradient of velocity gives the viscous part of the stress-energy tensor as a tensor linear in velocity. Velocity gardient decomposes to a term traceless tensor term and a term reducing to scalar.

    ivj+∂jvi= (2/3)∂kvkgij+ (∂ivj+∂jvi-(2/3)∂kvkgij).

    The viscous contribution to stress tensor is given in terms of this decomposition as

    σvisc,ij= ζ∂kvkgij+η (∂ivj+∂jvi-(2/3)∂kvkgij).

    From dFi= TijSj it is clear that bulk viscosity ζ gives to energy momentum tensor a pressure like contribution having interpretation in terms of friction opposing. Shear viscosity η corresponds to the traceless part of the velocity gradient often called just viscosity. This contribution to the stress tensor is non-diagonal and corresponds to momentum transfer in directions not parallel to momentum and makes the flow rotational. This term is essential for the thermal conduction and thermal conductivity vanishes for ideal fluids.

  2. The 3-D total stress tensor can be written as

    σij= ρ vivj-pgijvisc,ij.

    The generalization to a 4-D relativistic situation is simple. One just adds terms corresponding to energy density and energy flow to obtain

    Tαβ= (ρ-p) uα uβ+pgαβviscαβ .

    Here uα denotes the local four-velocity satisfying uαuα=1. The sign factors relate to the concentions in the definition of Minkowski metric ((1,-1,-1,-1)).

  3. If the flow is such that the flow parameters associated with the flow lines integrate to a global flow parameter one can identify new time coordinate t as this flow parametger. This means a transition to a coordinate system in which fluid is at rest everywhere (comoving coordinates in cosmology) so that energy momentum tensor reduces to a diagonal term plus viscous term.

    Tαβ= (ρ-p) gtt δtα δtβ+pgαβviscαβ .

    In this case the vanishing of the viscous term means that one has perfect fluid in strong sense.

    The existence of a global flow parameter means that one has

    vi= Ψ ∂iΦ .

    Ψ and Φ depend on space-time point. The proportionality to a gradient of scalar Φ implies that Φ can be taken as a global time coordinate. If this condition is not satisfied, the perfect fluid property makes sense only locally.

AdS/CFT correspondence allows to deduce a lower limit for the coefficient of shear viscosity as

x= η/s≥ hbar/4π .

This formula holds true in units in which one has kB=1 so that temperature has unit of energy.

What makes this interesting from TGD view is that in TGD framework perfect fluid property in approriately generalized sense indeed characterizes locally the preferred extremals of Kähler action defining space-time surface.

  1. Kähler action is Maxwell action with U(1) gauge field replaced with the projection of CP2 Kähler form so that the four CP2 coordinates become the dynamical variables at QFT limit. This means enormous reduction in the number of degrees of freedom as compared to the ordinary unifications. The field equations for Kähler action define the dynamics of space-time surfaces and this dynamics reduces to conservation laws for the currents assignable to isometries. This means that the system has a hydrodynamic interpretation. This is a considerable difference to ordinary Maxwell equations. Notice however that the "topological" half of Maxwell's equations (Faraday's induction law and the statement that no non-topological magnetic are possible) is satisfied.

  2. Even more, the resulting hydrodynamical system allows an interpretation in terms of a perfect fluid. The general ansatz for the preferred extremals of field equations assumes that various conserved currents are proportional to a vector field characterized by so called Beltrami property. The coefficient of proportionality depends on space-time point and the conserved current in question. Beltrami fields by definition is a vector field such that the time parameters assignable to its flow lines integrate to single global coordinate. This is highly non-trivial and one of the implications is almost topological QFT property due to the fact that Kähler action reduces to a boundary term assignable to wormhole throats which are light-like 3-surfaces at the boundaries of regions of space-time with Euclidian and Minkowskian signatures. The Euclidian regions (or wormhole throats, depends on one's tastes ) define what I identify as generalized Feynman diagrams.

    Beltrami property means that if the time coordinate for a space-time sheet is chosen to be this global flow parameter, all conserved currents have only time component. In TGD framework energy momentum tensor is replaced with a collection of conserved currents assignable to various isometries and the analog of energy momentum tensor complex constructed in this manner has no counterparts of non-diagonal components. Hence the preferred extremals allow an interpretation in terms of perfect fluid without any viscosity.

This argument justifies the expectation that TGD Universe is characterized by the presence of low-viscosity fluids. Real fluids of course have a non-vanishing albeit small value of x. What causes the failure of the exact perfect fluid property?

  1. Many-sheetedness of the space-time is the underlying reason. Space-time surface decomposes into finite-sized space-time sheets containing topologically condensed smaller space-time sheets containing.... Only within given sheet perfect fluid property holds true and fails at wormhole contacts and because the sheet has a finite size. As a consequence, the global flow parameter exists only in given length and time scale. At imbedding space level and in zero energy ontology the phrasing of the same would be in terms of hierarchy of causal diamonds (CDs).

  2. The so called eddy viscosity is caused by eddies (vortices) of the flow. The space-time sheets glued to a larger one are indeed analogous to eddies so that the reduction of viscosity to eddy viscosity could make sense quite generally. Also the phase slippage phenomenon of super-conductivity meaning that the total phase increment of the super-conducting order parameter is reduced by a multiple of 2π in phase slippage so that the average velocity proportional to the increment of the phase along the channel divided by the length of the channel is reduced by a quantized amount.

    The standard arrangement for measuring viscosity involves a lipid layer flowing along plane. The velocity of flow with respect to the surface increases from v=0 at the lower boundary to vupper at the upper boundary of the layer: this situation can be regarded as outcome of the dissipation process and prevails as long as energy is feeded into the system. The reduction of the velocity in direction orthogonal to the layer means that the flow becomes rotational during dissipation leading to this stationary situation.

    This suggests that the elementary building block of dissipation process corresponds to a generation of vortex identifiable as cylindrical space-time sheets parallel to the plane of the flow and orthogonal to the velocity of flow and carrying quantized angular momentum. One expects that vortices have a spectrum labelled by quantum numbers like energy and angular momentum so that dissipation takes in discrete steps by the generation of vortices which transfer the energy and angular momentum to environment and in this manner generate the velocity gradient.

  3. The quantization of the parameter x is suggestive in this framework. If entropy density and viscosity are both proportional to the density n of the eddies, the value of x would equal to the ratio of the quanta of entropy and kinematic viscosity η/n for single eddy if all eddies are identical. The quantum would be hbar/4π in the units used and the suggestive interpretation is in terms of the quantization of angular momentum. One of course expects a spectrum of eddies so that this simple prediction should hold true only at temperatures for which the excitation energies of vortices are above the thermal energy. The increase of the temperature would suggest that gradually more and more vortices come into play and that the ratio increases in a stepwise manner bringing in mind quantum Hall effect. In TGD Universe the value of hbar can be large in some situations so that the quantal character of dissipation could become visible even macroscopically. Whether this situation with large hbar is encountered even in the case of QCD plasma is an interesting question.

The following poor man's argument tries to make the idea about quantization a little bit more concrete.

  1. The vortices transfer momentum parallel to the plane from the flow. Therefore they must have momentum parallel to the flow given by the total cm momentum of the vortex. Before continuing some notations are needed. Let the densities of vortices and absorbed vortices be n and nabs respectively. Denote by vpar resp. vperp the components of cm momenta parallel to the main flow resp. perpendicular to the plane boundary plane. Let m be the mass of the vortex. Denote by S are parallel to the boundary plane.

  2. The flow of momentum component parallel to the main flow due to the absorbed at S is

    nabs m vpar vperp S .

    This momentum flow must be equal to the viscous force

    Fvisc = η (vpar/d)× S .

    From this one obtains

    η= nabsm vperpd .

    If the entropy density is due to the vortices, it equals apart from possible numerical factors to

    s= n

    so that one has

    η/s=mvperpd .

    This quantity should have lower bound x=hbar/4π and perhaps even quantized in multiples of x, Angular momentum quantization suggests strongly itself as origin of the quantization.

  3. Local momentum conservation requires that the comoving vortices are created in pairs with opposite momenta and thus propagating with opposite velocities vperp. Only one half of vortices is absorbed so that one has nabs=n/2. Vortex has quantized angular momentum associated with its internal rotation. Angular momentum is generated to the flow since the vortices flowing downwards are absorbed at the boundary surface.

    Suppose that the distance of their center of mass lines parallel to plane is D=ε d, ε a numerical constant not too far from unity. The vortices of the pair moving in opposite direction have same angular momentum mvperpD/2 relative to their center of mass line between them. Angular momentum conservation requires that the sum these relative angular momenta cancels the sum of the angular momenta associated with the vortices themselves. Quantization for the total angular momentum for the pair of vortices gives

    η/s= nhbar/ε

    Quantization condition would give

    ε =4π .

    One should understand why D=4π d - four times the circumference for the largest circle contained by the boundary layer- should define the minimal distance between the vortices of the pair. This distance is larger than the distance d for maximally sized vortices of radius d/2 just touching. This distance obviously increases as the thickness of the boundary layer increasess suggesting that also the radius of the vortices scales like d.

  4. One cannot of course take this detailed model too literally. What is however remarkable that quantization of angular momentum and dissipation mechanism based on vortices identified as space-time sheets indeed could explain why the lower bound for the ratio η/s is so small.

For background see the chapter Overall View About TGD from Particle Physics Perspective.

Topological explanation of family replication phenomenon

One of the basic ideas of TGD approach has been genus-generation correspondence: boundary components of the 3-surface should be carriers of elementary particle numbers and the observed particle families should correspond to various boundary topologies. Last summer meant quite a progress in the understanding of quantum TGD, which forced also the updating of the views about the topological explanation of family replication phenomenon.

With the advent of zero energy ontology the original picture changed somewhat. It is the wormhole throats identified as light-like 3-surfaces at with the induced metric of the space-time surface changes its signature from Minkowskian to Euclidian, which correspond to the light-like orbits of partonic 2-surfaces. One cannot of course exclude the possibility that also boundary components could allow to satisfy boundary conditions without assuming vacuum extremal property of nearby space-time surface. The intersections of the wormhole throats with the light-like boundaries of causal diamonds (CDs) identified as intersections of future and past directed light cones (CD × CP2 is actually in question but I will speak about CDs) define special partonic 2-surfaces and it is the comformal moduli of these partonic 2-surfaces which appear in the elementary particle vacuum functionals naturally.

The first modification of the original simple picture comes from the identification of physical particles as bound states of pairs of wormhole contacts and from the assumption that for generalized Feynman diagrams stringy trouser vertices are replaced with vertices at which the ends of light-like wormhole throats meet. In this picture the interpretation of the analog of trouser vertex is in terms of propagation of same particle along two different paths. This interpretation is mathematically natural since vertices correspond to 2-manifolds rather than singular 2-manifolds which are just splitting to two disjoint components. Second complication comes from the weak form of electric-magnetic duality forcing to identify physical particles as weak strings with magnetic monopoles at their ends and one should understand also the possible complications caused by this generalization.

These modifications force to consider several options concerning the identification of light fermions and bosons and one can end up with a unique identification only by making some assumptions. Masslessness of all wormhole throats- also those appearing in internal lines- and dynamical SU(3) symmetry for particle generations are attractive general enough assumptions of this kind. This means that bosons and their super-partners correspond to wormhole contacts with fermion and antifermion at the throats of the contact. Free fermions and their superpartners could correspond to CP2 type vacuum extremals with single wormhole throat. It turns however that dynamical SU(3) symmetry forces to identify massive (and possibly topologically condensed) fermions as (g,g) type wormhole contacts.

Do free fermions correspond to single wormhole throat or (g,g) wormhole?

The original interpretation of genus-generation correspondence was that free fermions correspond to wormhole throats characterized by genus. The idea of SU(3) as a dynamical symmetry suggested that gauge bosons correspond to octet and singlet representations of SU(3). The further idea that all lines of generalized Feynman diagrams are massless poses a strong additional constraint and it is not clear whether this proposal as such survives.

  1. Twistorial program assumes that fundamental objects are massless wormhole throats carrying collinearly moving many-fermion states and also bosonic excitations generated by super-symplectic algebra. In the following consideration only purely bosonic and single fermion throats are considered since they are the basic building blocks of physical particles. The reason is that propagators for high excitations behave like p-n, n the number of fermions associated with the wormhole throat. Therefore single throat allows only spins 0,1/2,1 as elementary particles in the usual sense of the word.

  2. The identification of massive fermions (as opposed to free massless fermions) as wormhole contacts follows if one requires that fundamental building blocks are massless since at least two massless throats are required to have a massive state. Therefore the conformal excitations with CP2 mass scale should be assignable to wormhole contacts also in the case of fermions. As already noticed this is not the end of the story: weak strings are required by the weak form of electric-magnetic duality.

  3. If free fermions corresponding to single wormhole throat, topological condensation is an essential element of the formation of stringy states. The topological condensation of fermions by topological sum (fermionic CP2 type vacuum extremal touches another space-time sheet) suggest (g,0) wormhole contact. Note however that the identification of wormhole throat is as 3-surface at which the signature of the induced metric changes so that this conclusion might be wrong. One can indeed consider also the possibility of (g,g) pairs as an outcome of topological conensation. This is suggested also by the idea that wormhole throats are analogous to string like objects and only this option turns out to be consistent with the BFF vertex based on the requirement of dynamical SU(3) symmetry to be discussed later. The structure of reaction vertices makes it possible to interpret (g,g) pairs as SU(3) triplet. If bosons are obtained as fusion of fermionic and antifermionic throats (touching of corresponding CP2 type vacuum extremals) they correspond naturally to (g1,g2) pairs.

  4. p-Adic mass calculations distinguish between fermions and bosons and the identification of fermions and bosons should be consistent with this difference. The maximal p-adic temperature T=1 for fermions could relate to the weakness of the interaction of the fermionic wormhole throat with the wormhole throat resulting in topological condensation. This wormhole throat would however carry momentum and 3-momentum would in general be non-parallel to that of the fermion, most naturally in the opposite direction.

    p-Adic mass calculations suggest strongly that for bosons p-adic temperature T=1/n, n>1, so that thermodynamical contribution to the mass squared is negligible. The low p-adic temperature could be due to the strong interaction between fermionic and antifermionic wormhole throat leading to the "freezing" of the conformal degrees of freedom related to the relative motion of wormhole throats.

  5. The weak form of electric-magnetic duality forces second wormhole throat with opposite magnetic charge and the light-like momenta could sum up to massive momentum. In this case string tension corresponds to electroweak length scale. Therefore p-adic thermodynamics must be assigned to wormhole contacts and these appear as basic units connected by Kähler magnetic flux tube pairs at the two space-time sheets involved. Weak stringy degrees of freedom are however expected to give additional contribution to the mass, perhaps by modifying the ground state conformal weight. A nice implication is that all elementary particles -not only gravitons- correspond to pairs of wormhole throats connected by magnetic flux tubes to form "weak strings". This has obvious implications at LHC.

Dynamical SU(3) fixes the identification of fermions and bosons and fundamental interaction vertices

For 3 light fermion families SU(3) suggests itself as a dynamical symmetry with fermions in fundamental N=3-dimensional representation and N× N=9 bosons in the adjoint representation and singlet representation. The known gauge bosons have same couplings to fermionic families so that they must correspond to the singlet representation. The first challenge is to understand whether it is possible to have dynamical SU(3) at the level of fundamental reaction vertices.

This is a highly non-trivial constraint. For instance, the vertices in which n wormhole throats with same (g1,g2) glued along the ends of lines are not consistent with this symmetry. The splitting of the fermionic worm-hole contacts before the proper vertices for throats might however allow the realization of dynamical SU(3). The condition of SU(3) symmetry combined with the requirement that virtual lines resulting also in the splitting of wormhole contacts are always massless, leads to the conclusion that massive fermions correspond to (g,g) type wormhole contacts transforming naturally like SU(3) triplet. This picture conformsl with the identification of free fermions as throats but not with the naive expectation that their topological condensation gives rise to (g,0) wormhole contact.

The argument leading to these conclusions runs as follows.

  1. The question is what basic reaction vertices are allowed by dynamical SU(3) symmetry. FFB vertices are in principle all that is needed and they should obey the dynamical symmetry. The meeting of entire wormhole contacts along their ends is certainly not possible. The splitting of fermionic wormhole contacts before the vertices might be however consistent with SU(3) symmetry. This would give two a pair of 3-vertices at which three wormhole lines meet along partonic 2-surfaces (rather than along 3-D wormhole contacts).

  2. Note first that crossing gives all possible reaction vertices of this kind from F(g1)Fbar(g2)→ B(g1,g2) annihilation vertex, which is relatively easy to visualize. In this reaction F(g1) and Fbar(g2) wormhole contacts split first. If one requires that all wormhole throats involved are massless, the two wormhole throats resulting in splitting and carrying no fermion number must carry light-like momentum so that they cannot just disappear. The ends of the wormhole throats of the boson must glued together with the end of the fermionic wormhole throat and its companion generated in the splitting of the wormhole. This means that fermionic wormhole first splits and the resulting throats meet at the partonic 2-surface.

    This requires that topologically condensed fermions correspond to (g,g) pairs rather than (g,0) pairs. The reaction mechanism allows the interpretation of (g,g) pairs as a triplet of dynamical SU(3). The fundamental vertices would be just the splitting of wormhole contact and 3-vertices for throats since SU(3) symmetry would exclude more complex reaction vertices such as n-boson vertices corresponding the gluing of n wormhole contact lines along their 3-dimensional ends. The couplings of singlet representation for bosons would have same coupling to all fermion families so that the basic experimental constraint would be satisfied.

  3. Both fermions and bosons cannot correspond to octet and singlet of SU(3). In this case reaction vertices should correspond algebraically to the multiplication of matrix elements eij: eij ekl = δjk eil allowing for instance F(g1,g2) +Fbar(g2,g3)→ B(g1,g3) . Neither the fusion of entire wormhole contacts along their ends nor the splitting of wormhole throats before the fusion of partonic 2-surfaces allows this kind of vertices so that BFF vertex is the only possible one. Also the construction of QFT limit starting from bosonic emergence led to the formulation of perturbation theory in terms of Dirac action allowing only BFF vertex as fundamental vertex.

  4. Weak electric-magnetic duality brings in an additional complication. SU(3) symmetry poses also now strong constraints and it would seem that the reactions must involve copies of basic BFF vertices for the pairs of ends of weak strings. The string ends with the same Kähler magnetic charge should meet at the vertex and give rise to BFF vertices. For instance, FFbarB annihilation vertex would in this manner give rise to the analog of stringy diagram in which strings join along ends since two string ends disappear in the process.

If one accepts this picture the remaining question is why the number of genera is just three. Could this relate to the fact that g≤ 2 Riemann surfaces are always hyper-elliptic (have global Z2 conformal symmetry) unlike g>2 surfaces? Why the complete bosonic de-localization of the light families should be restricted inside the hyper-elliptic sector? Does the Z2 conformal symmetry make these states light and make possible delocalization and dynamical SU(3) symmetry? Could it be that for g>2 elementary particle vacuum functionals vanish for hyper-elliptic surfaces? If this the case and if the time evolution for partonic 2-surfaces changing g commutes with Z2 symmetry then the vacuum functionals localized to g≤ 2 surfaces do not disperse to g>2 sectors.

These and many other questions are discussed in the chapters of p-Adic length scale hypothesis and dark matter hierarchy, in particular in the chapter Elementary Particle Vacuum Functionals.

By the way, I have performed and updating of several books about TGD in order to achieve a more coherent representation. I have also added three new chapters to the book Topological Geometrodynamics: an Overview discussing TGD from particle physics perspective (see this, this, and this).

Also the chapters of p-Adic length scale hypothesis and dark matter hierarchy are heavily updated.

Considerable progress in generalized Feynman diagrammatics

The following is expanded and somewhat edited response in Kea's blog. For reasons that should become obvious the response deserves to be published also here although I have done this implicitly via links to pdf files in earlier postings. My sincere hope is that at least single really intelligent reader might realize what is is involved;-). This might be enough.

I have been working with twistor program inspired ideas in TGD framework for a couple of years. The basic conceptual elements are following.

  1. The notion of generalized Feyman diagram defined by replacing lines of ordinary Feynman diagram with light-like 3-surfaces (elementary particle sized wormhole contacts with throats carrying quantum numbers) and vertices identified as their 2-D ends - I call them partonic 2-surfaces. Speaking somewhat loosely, generalized Feynman diagrams plus background space-time sheets define the "world of classical worlds" (WCW).

  2. Zero energy ontology (ZEO) and causal diamonds (intersections of future and past directed lightcones). The crucial observation is that in ZEO it is possible to identify off mass shell particles as pairs of on mass shell particles at throats of wormhole contact since both positive and negative signs of energy are possible. The propagator defined by modified Dirac action does not diverge (except for incoming lines) although the fermions at throats are on mass shell. In other words, the generalized eigenvalue of the modified Dirac operator containing a term linear in momentum is non-vanishing and propagator reduces to G=i/λγ , where γ is modified gamma matrix in the direction of stringy coordinate. This means opening of the black box of off mass shell particle-something which for some reason has not occurred to anyone fighting with the divergences of QFTs.

  3. Representation of 8-D gamma matrices in terms of octonionic units and 2-D sigma matrices. Modified gamma matrices at space-time surfaces are quaternionic/associative and allow a genuine matrix representation. As a matter fact, TGD and WCW can be formulated as study of associative local sub-algebras of the local Clifford algebra of 8-D imbedding space parameterized by quaternionic space-time surfaces. Central conjecture is that quaternionic 4-surfaces correspond to preferred extremals of Kähler action identified as critical ones (second variation of Kähler action vanishes for infinite number of deformations defining super-conformal algebra) and allow a slicing to string worldsheets parametrized by points of partonic 2-surfaces.

  4. Number theoretic universality requiring the existence of Feynman amplitudes in all number fields when one allows suitable algebraic extensions: roots of unity are certainly required in order to realize plane waves. Also imbedding space, partonic 2-surfaces and WCW must exist in all number fields and their extensions. These constraints are enormously powerful and the attempts to realize this vision have dominated quantum TGD for last 20 years.

  5. As far as twistors are considered, the first key element is the reduction of the octonionic twistor structure to quaternionic one at space-time surfaces and giving effectively 4-D spinor and twistor structure for quaternionic surfaces.

Quite recently quite a dramatic progress took place in this approach. It was just the simple observation -I should have made if for already half year ago!- that on mass shell property puts enormously strong kinematic restrictions on the loop integrations. With mild restrictions on the number of parallel fermion lines appearing in vertices (there can be several since fermionic oscillator operator algebra defining SUSY algebra generates the parton states)- all loops are manifestly finite and if particles has always mass -say small p-adic thermal mass also in case of massless particles and due to IR cutoff due to the presence largest CD- the number of diagrams is finite. Unitarity reduces to Cutkosky rules automatically satisfied as in the case of ordinary Feynman diagrams.

This is about momentum space aspects of Feynman diagrams but not yet about the functional (not path-) integral over small deformations of the partonic 2-surfaces. It took some time to see that also the functional integrals over WCW can be carried out at general level both in real and p-adic context.

  1. The p-adic generalization of Fourier analysis allows to algebraize integration- the horrible looking technical challenge of p-adic physics- for symmetric spaces for functions allowing the analog of discrete Fourier decomposion. Symmetric space property is indeed essential also for the existence of Kähler geometry for infinite-D spaces as was learned already from the case of loop spaces. Plane waves and exponential functions expressible as roots of unity and powers of p multiplied by the direct analogs of corresponding exponent functions are the basic building bricks and key functions in harmonic analysis in symmetric spaces. The physically unavoidable finite measurement resolution corresponds to algebraically unavoidable finite algebraic dimension of algebraic extension of p-adics (at least some roots of unity are needed). The cutoff in roots of unity is very reminiscent to that occurring for the representations of quantum groups and is certainly very closely related to these as also to the inclusions of hyper-finite factors of type II1 defining the finite measurement resolution.

  2. WCW geometrization reduces to that for a single line of the generalized Feynman diagram defining the basic building brick for WCW. Kähler function decomposes to a sum of "kinetic" terms associated with its ends and interaction term associated with the line itself. p-Adicization boils down to the condition that Kähler function, matrix elements of Kähler form, WCW Hamiltonians and their super counterparts, are rational functions of complex WCW coordinates just as they are for those symmetric spaces that I know of. This allows straightforward continuation to p-adic context. Incredibly simple!
  3. As far as diagrams are considered, everything is manifestly finite as the general arguments (non-locality of Kähler function as functional of 3-surface) developed two decades ago indeed allow to expect. General conditions on the holomorphy properties of the generalized eigenvalues λ of the modified Dirac operator can be deduced from the conditions that propagator decomposes to a sum of products of harmonics associated with the ends of the line and that similar decomposition takes place for exponent of Kähler action identified as Dirac determinant. This guarantees that the convolutions of propagators and vertices give rise to products of harmonic functions which can be Glebsch-Gordanized to harmonics and only the singlet contributes to the WCW integral in given vertex. The still unproven central conjecture is that Dirac determinant equals the exponent of Kähler function.

Ironically, twistors which stimulated all these development do not seem to be absolutely necessary in this approach although they are of course possible. Situation changes if one does not assumes small p-adically thermal mass due to the presence of massless particles and one must sum infinite number of diagrams. Here a potential problem is whether the infinite sum respects the algebraic extension in question.

For a more detailed representation see the short articles Weak form of electric-magnetic duality, electroweak massivation, and color confinement and How to define generalized Feynman diagrams?. See also the chapter The Geometry of the World of Classical Worlds.

How to perform WCW integrations in generalized Feynman diagrams?

The formidable looking challenge of quantum TGD is to calculate the M-matrix elements defined by the generalized Feynman diagrams. Zero energy ontology (ZEO) has provided profound understanding about how generalized Feynman diagrams differ from the ordinary ones. The most dramatic prediction is that loop momenta correspond to on mass shall momenta for the two throats of the wormhole contact defining virtual particles: the energies of the energies of on mass shell throats can have both signs in ZEO. This predicts finiteness of Feynman diagrams in the fermionic sector. Even more: the number of Feynman diagrams for a given process is finite if also massless particles receive a small mass by p-adic thermodynamics. The mass would be due to IR cutoff provided by the largest CD (causal diamond) involved.

The basic challenges are following.

  1. One should perform the functional integral over world of classical worlds (WCW) for fixed values of on mass shell momenta appearing in internal lines. After this one must perform integral or summation over loop momenta.

  2. One must achieve this also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

The realization that p-adic integrals could be defined if the manifold is symmetric space as the world of classical world (WCW) is proposed to be raises the hope that the WCW integration for Feynman amplitudes could be carried at the general level using Fourier analysis for symmetric spaces. Even more, the possibility to define p-adic intergrals for symmetric spaces suggests that the theory could allow elegant p-adicization. This indeed seems to be the case. It seems that the dream of transforming TGD to a practical calculational machinery does not look non-realistic at all.

I do not bother to type more but give a link to a short article summarizing the basic formulas. For more background see also the article and the chapter The Geometry of the World of Classical Worlds.

Weak form of electric-magnetic duality, particle concept, and Feynman diagrammatics

The notion of electric magnetic duality emerged already two decades ago in attempts to formulate the Kähler geometric of world of classical worlds. Quite recently a considerable step of progress took place in the understanding of this notion. Every new idea must be of course taken with a grain of salt but the good sign is that this concept leads to an identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic flux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement. The fascinating prediction is that the stringy view about elementary particles should become visible at LHC energies.

Zero energy ontology in turn inspires the idea that virtual particles correspond to pairs of on mass shell states assignable to the opposite throats of wormhole contacts: in TGD framework the propagators do not diverge although particles are on mass shell in standard sense. This assumption leads to powerful constraints on the generalized Feynman diagrams giving excellent hopes about the finiteness of loops. Finiteness has been obvious on basis of general arguments but has been very difficult to demonstrate convincingly in the fermionic sector of the theory. In fact, there are good arguments supporting that only a finite number of diagrams contributes to a given reaction: something inspired by the vision about algebraic physics (infinite sums lead out of the algebraic extension used). The reason is that the on mass shell conditions on states at wormhole throats reduce the phase space dramatically, and already in the case of four-vertex loops leave only a discrete set of points under consideration. This implies also finiteness. This wisdom can be combined with the new stringy view about particles to build a very concrete stringy view about generalized Feynman diagrams.

The outcome of the opening of the black box of virtual particle -an idea forced by the twistorial approach and made possible by zero energy ontology- is something which I dare to regard as a fulfillment of 32 year old dream.

For a more detailed representation of weak electric-magnetic duality see the last section of the pdf article Weak form of electric-magnetic duality, electroweak massivation, and color confinement. For Feynman diagrams and WCW integration see the article How to define generalized Feynman diagrams? summarizing the basic formulas. See also the chapter The Geometry of the World of Classical Worlds.

Topological Geometrodynamics: Three Visions

I have replaced the two-year old brief overview about quantum TGD with a new one and titled it as "TGD: Three Visions". Here is the abstract of the new chapter.

In this chapter I will discuss three basic visions about quantum Topological Geometrodynamics (TGD). It is somewhat matter of taste which idea one should call a vision and the selection of these three in a special role is what I feel natural just now.

  1. The first vision is generalization of Einstein's geometrization program based on the idea that the K\"ahler geometry of the world of classical worlds (WCW) with physical states identified as classical spinor fields on this space would provide the ultimate formulation of physics.

  2. Second vision is number theoretical and involves three threads. The first thread relies on the idea that it should be possible to fuse real number based physics and physics associated with various p-adic number fields to single coherent whole by a proper generalization of number concept. Second thread is based on the hypothesis that classical number fields could allow to understand the fundamental symmetries of physics and and imply quantum TGD from purely number theoretical premises with associativity defining the fundamental dynamical principle both classically and quantum mechanically. The third threadrelies on the notion of infinite primes whose construction has amazing structural similarities with second quantization of super-symmetric quantum field theories. In particular, the hierarchy of infinite primes and integers allows to generalize the notion of numbers so that given real number has infinitely rich number theoretic anatomy based on the existence of infinite number of real units.

  3. The third vision is based on TGD inspired theory of consciousness, which can be regarded as an extension of quantum measurement theory to a theory of consciousness raising observer from an outsider to a key actor of quantum physics.

For more details see Topological Geometrodynamics: Three Visions.

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