Wednesday, December 15, 2010

Preferred extremals of Kähler action and perfect fluids

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 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 Does the Modified Dirac Equation Define the Fundamental Action Principle? of "Quantum TGD as Infinite-dimensional Spinor Geometry".


L. Edgar Otto said...


Thanks for the article. I am still posting sometimes on the other bloggers and if you do not mind I would like to hash out these differences of everyone's perceptions. pesla.blogspot com.

I am not a fan of Kaluza-Klein theory but some concept of it applies.

Maxwell's ideas had a residual flux of sorts- are you discussing this?

I cannot say I understand exactly what gravity, supergravity and the like are- especially when one considers information structures.

ThePeSla who is a fan and especially agrees with section 3 above. How does gravity fit into this?

Matti Pitkänen said...

Of course you can! I am flattered.

TGD has one common element with Kaluza Klein theory and this is the idea that symmetries of compact space define symmetries of particle physics: now electroweak and color symmetries. Otherwise the picture is totally different since space-time is surfaces rather than base space of bundle.

The action principle defining the dynamics of space-time surfaces is formally Maxwell action with Maxwell field replaced with Kahler form of CP_2 projected to tensor field at space-time surfaces. This means geometrization of U(1) part of classical em field. Quite generally, both classical electroweak and color gauge fields are geometrized in terms of sub-manifold geometry. Once you know the space-time surface you know all classical gauge fields.

Actually even more, you know also the classical gravitational field as induced metric which means that metric tensor is projection of metric tensor of imbedding space M^4xCP_2. You have sub-manifold gravity.

All this means that all classical boson fields are expressible in terms of 4 imbedding space coordinates and their gradients when 4 space-time coordinates are chosen to by some imbedding space coordinates. This is incredible reduction in the number of degrees of freedom.

The routine counter argument which allows the colleague to get rid of trouble reading further (15 books to read now, what you leave behind you find ahead;-)!) is that this approach cannot fully reproduce Einsteinian gravitation. This is true: the requirement of imbeddability of metric to M^4xCP_2 is an enormously powerful constraint. On the other hand, the basic problem of Einsteinian gravity is that it allows quite too many unphysical metrics: typical metric gives rise to non-physical energy momentum tensor by Einstein's equations.

The basic difference as compared to Maxwell's theory can be seen already in simplest cases. For instance, only a finite spacet-time region carrying constant magnetic field allows the imbedding as space-time surface. This I call topological field quantization and leads to many-sheeted space-time concept. For preferred extremals these space-time sheets would behave like perfect fluids and dissipation and viscosity would be due to splitting of space-time to these pieces.

Ulla said...

Poor Lubos, he is showing some opportunistic, chaotic behaviour. But he isn't yet in the same situation as you was some years ago.

Kea: LHC has begun to rule out popular ideas from string theory,

Borromean rings: This gives rise to a whole new hierarchy of possible states with Efimov states at the bottom.

The ship is slowly turning around.

Ulla said...

Quite off topic. Brain time travel.
When seems to be something external to the what, how, and where of the situation. But this creates a problem for neuroscientists.

We think we know how the fact that the brain could store the concept of "walking down the street" (or "walking" and "street"). Very roughly, simple sensory impressions are thought to get built up into more and more complex combinations, to cells that remember the face (Jennifer Aniston-cells in hippocampus)in medial temp.lobe. results suggest an invariant, sparse and explicit code...
But the fact that we can take any given scene, and effortlessly think of it as either "past", "present", or "future", is puzzling under this view because, as I said, the scene itself is the same in all cases. And it's not as if we have a sense devoted to time: the only time we're ever directly aware of, is "right now". brain activity associated with "mental time travel": Consciousness of subjective time in the brain.

Hippocampus is the back of the book = right now, implicit and explicit?

Tim said...

yes that is quite interesting wonder if the industrial sector, busy layering the mantel with long chain elastic polymers, may envision an actual photonic substrate.

Matti Pitkänen said...

To Ulla:

Sticking into belief system seems to be the worst disease of this profession. It is a pity if Lubos is not able to take a fresh start and forget M-theory. It is very difficult to get rid of strong beliefs of early youth because they have been adopted so emotionally.

They seem to have found a brain area activated differently during thinking of recent as compared to remembering and imagining future or past.

If brain areas are characterized by the time scale of memory and planned action coded by the value of Planck constant coding for the frequency scale of corresponding contribution EEG (in generalized sense involving much lower frequencies than 100 Hz), this would not be surprising.

An interesting question is whether both rememberig and imagination take place in future and past.

Matti Pitkänen said...

Beautiful graphics these rings inside rings inside... In nuclear string model one could consider linking and knotting of nuclear strings to form analogous structures.

Ulla said...

I have often seen the phenomen that people make regressions = goes back in times as instance to childhood and begins to act as a child. This would be explained by sides in the Big Book, labeled by hbar. This label would also be consciousness and time (memory). The consciousness level is much lower as child, and this could be seen as the decoherence level too. Consciousness would in that way be collapsed coherence :) Low decoherence means high quantum biological state and 'spirituality' or 'Life', what is charachteristic of children, and alsoof people with high age.

You talk of consciousness as quantum jumps, and those quantum jumps is then the same as decoherence or perceptions.

Thinking about the future is most energy demanding of all thinking, and is a projection of possibilities into future (frontal lobe) as a hierarchial flux tube?

Seen in this way the consciousness is coherent, and your quantum jumps need a small modification? Quantum jumps = entanglements, excitations, but it is not always about that? Self is the decoherent and differentiated state.

Sorry for posting this here. I need your opinion on it.

Ulla said...

Also a Plancks constant hierarchy