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TGD: Physics as Infinite-Dimensional Geometry
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 Does the Modified Dirac Equation Define the Fundamental Action Principle?.
Hyperdeterminants have stimulated interesting discussions in viXra blog and also Kea has talked about them. The notion is new to me but so interesting from TGD point of view that I cannot resist the temptation of making fool of myself by declaring why it looks so interesting. This gives also an excellent opportunity to demonstrate my profound ignorance about the notion;-). Instead of typing all my ignorance in html, I give a link to pdf article Could the notion of hyper-determinant be useful in TGD framework?.
Addition: I decided to glue the response to a comment by Phil Gibbs summarizing my motivations for getting interested in hyper-determinants.
What hopes one has for genuine multilinearity, which seems to be almost synonymous to non-locality?
For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.
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 of generalized Feynman diagrammatics 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 Does the Modified Dirac Equation Define the Fundamental Action Principle?.
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.
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 Weak form of electric-magnetic duality, electroweak massivation, and color confinement and the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.
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 coutcome 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 Does the Modified Dirac Equation Define the Fundamental Action Principle?.
Eric Verlinde has posted an interesting eprint titled On the Origin of Gravity and the Laws of Newton to arXiv.org. Lubos has commented the article here and also here. What Linde heuristically derives is Newton's F=ma and gravitational force F= GMm/R2 from thermodynamical considerations plus something else which I try to clarify (at least to myself!) in the following.
1. Verlinde's argument for F=ma
The idea is to deduce Newton's F=ma and gravitational force from thermodynamics by assuming that space-time emerges in some sense. There are however various assumptions involved which more or less impy that both special and general relativity has been feeded in besides quantum theory and thermodynamics.
After this some quantum tricks -a rather standard game with Uncertainty Principle and quantization when nothing concrete is available- are needed to obtain F=ma which as such does not involve hbar nor Boltzmann constant kB. What is needed are thermal expression for acceleration and force and identifying these one obtains F=ma.
2. Verlinde's argument for F= GMm/R2
The next challenge is to derive gravitational force from thermodynamic consideration. Now holography with a very specially chosen screen is needed.
Comment: In TGD framework light-like 3-surfaces (or equivalently their space-like duals) represent the holographic screens and in principle there is a slicing of space-time surface by equivalent screens. Also Verlinde introduces a slicing of space-time surfaces by holographic screens identified as surfaces for which gravitational potential is constant. Also I have considered this kind of identification.
It is difficult to say whether the outcome is something genuinely new or just something resulting unavoidably by feeding in basic formulas from general thermodynamics, special relativity, and general relativity and using holography principle in highly questionable and ad hoc manner.
3. In TGD quantum classical correspondence predicts that thermodynamics has space-time correlates
From TGD point of view entropic gravity is a misconception. On basis of quantum classical correspondence - the basic guiding principle of quantum TGD - one expects that all quantal notions have space-time correlates. If thermodynamics is a genuine part of quantum theory, also temperature and entropy should have the space-time correlates and the analog of Verlinde's formula could exist. Even more, the generalization of this formula is expected to make sense for all interactions.
Zero energy ontology makes thermodynamics an integral part of quantum theory.
4. The simplest identification of thermodynamical correlates in TGD framework
The first questions that pop up are following. Inertial mass emerges from p-adic thermodynamics as thermal conformal weight. Could the first law for p-adic thermodynamics, which allows to calculate particle masses in terms of thermal conformal weights, allow to deduce also other classical forces? One could think that by adding to the Hamiltonian defining partition function chemical potential terms characterizing charge conservation it might be possible to obtain also other forces.
In fact, the situation might be much simpler. The basic structure of quantum TGD allows a very natural thermodynamical interpretation.
As noticed, one can imagine two options for the measurement interaction term defining the chemical potentials in terms of the space-time geometry.
Some conclusions drawn from Verlinde's argument is that gravitation is entropic interaction, that gravitons do not exist, and that string models and theories introducing higher-dimensional space-time are a failure. TGD view is different. Only a generalization of string model allowing to realize space-time as surface is needed and this requires fixed 8-D imbedding space. Gravitons also exist and only classical gravitation as well as other classical interactions code for thermodynamical information by quantum classical correspondence. In any case, it is encouraging that also colleagues might be finally beginning to get on the right track although the path from Verlinde's arguments to quantum TGD as it is now will be desperately long and tortuous if colleagues continually refuse to receive the helping hand.
For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle? of "Quantum TGD as Infinite-dimensional Spinor Geometry".