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Topological Geometrodynamics: an Overview
Note: Newest contributions are at the top!
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.
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.
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?
The following poor man's argument tries to make the idea about quantization a little bit more concrete.
For background see the chapter Overall View About TGD from Particle Physics Perspective.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
For more details see Topological Geometrodynamics: Three Visions.