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TGD: Physics as Infinite-Dimensional Geometry

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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 Does the Modified Dirac Equation Define the Fundamental Action Principle?.

Could the notion of hyper-determinant be useful in TGD framework?

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.

  1. Why the equations stating the vanishing of n:th variation of Kähler action are interesting in TGD framework is due to the infinite vacuum degeneracy of Kähler action making possible an infinite hierarchy of criticalities: one can say that TGD Universe is quantum critical. Criticality means a hierarchy of vanishing n:th variations. Phase transitions inside phase transitions inside.... This property is responsible for a lot of new physics and mathematics involved with TGD.

  2. The equations for n:th variation of Kähler action formulated in terms of functional derivatives are formally of this form and the existence of solution means vanishing of a generalized hyper-determinant. In standard QFT vanishing n≥3:th variations are not terribly interesting and even their existence is questionable. Vanishing second variations correspond to zero modes and vanishing of Gaussian determinant.

  3. n:th variations correspond formally to infinite tensor product with same dimension for all tensor factors and in this case there should be no restrictions on the number of tensor factors. The definition of hyper-determinant in this case is of course highly non-trivial. Already functional (Gaussian) determinants are tricky objects. What makes hyper-determinant so interesting from TGD view point is that it applies to multilinear equations involving homogeneous polynomials. Something between linear and genuinely non-linear and solvable.

What hopes one has for genuine multilinearity, which seems to be almost synonymous to non-locality?

  1. In the general case multilinearity requires non-locality and in purely local non-linear field theories there are not must hopes about multilinearity. The field equations for n:th variation should not contain powers of the same imbedding space coordinate or same derivative of it at same point. This is certainly not the case for a typical action principle. If the equations are genuinely multilinear in some basis for the deformations of space-time surface they are solvable and generalized hyper-determinant should tell whether this is the case. Its vanishing would also code for criticality for a higher order phase transition.

  2. When one constructs perturbation theory for a functional integral using exponent of Kähler function, one considers Kähler function identified as Kähler action for a preferred extremal. Formally this is a non-local functional of the data about 3-surface but actually reduces to 3-D Chern-Simons Kähler action with constraints characterizing weak form of electric magnetic duality. By effective 2-dimensionality Chern-Simons action is however a non-local functional of data about partonic 2-surface and its tangent space. n:th variation for 3-surface and 4-surface reduce to a non-local function of n:th variation of partonic 2-surface and its tangent space data. This is just what genuine multilinearity means so that multilinearity seems to hold true!

  3. This also relates to the local divergences of quantum field theories. They are present just because of higher order purely local couplings. Now they are absent if non-locality implying multilinearity holds true so that the functional integral over partonic 2-surfaces plus tangent space data should be free of infinities. Hence multilinearity might be behind integrability and absence of divergences. Maybe this relates also to the Yangian algebras which are non-local.

For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.

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 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?.

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 Weak form of electric-magnetic duality, electroweak massivation, and color confinement and the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.

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 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?.

Entropic gravity and TGD

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.

  1. Time translation invariance is required in order to have the notions of conserved energy and thermodynamics. This assumption requires not only require time but also symmetry with respect to time translations. This is quite a powerful assumption and time translation symmetry not hold true in General Relativity- this was actually the basic motivation for quantum TGD.

  2. Holography is assumed. Information stored on surfaces, or screens and discretization is assumed. Again this means in practice the assumption of space-time since otherwise the notion of holography does not make sense. One could of course say that one considers the situation in the already emerged region of space-time but this idea does not look very convincing to me.

    Comment: In TGD framework holography is an essential piece of theory: light-like 3-surfaces code for the physics and space-time sheets are analogous to Bohr orbits fixed by the light-like 3-surfaces defining the generalized Feynman diagrams.

  3. The first law of thermodynamics in the form

    dE= TdS-Fdx

    Here F denotes generalized force and x some coordinate variable. In usual thermodynamics pressure P would appear in the role of F and volume V in the role of x. Also chemical potential and particle number form a similar pair. If energy is conserved for the motion one has

    Fdx= TdS.

    This equation is basic thermodynamics and is used to deduce Newton's equations.

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.

  1. Δ S= 2π kB states that entropy is quantized with a unit of 2π appearing as a unit. log(2) would be more natural unit if bit is the unit of information.

  2. The identification Δ x =hbar/mc involves Uncertainty principle for momentum and position. The presence of light velocity c in the formula means that Minkowski space and Special Relativity creeps in. At this stage I would not speak about emergence of space-time anymore.

    This gives T= FΔ x/Δ S= F×hbar/[2π×mc×kB]

    F has been exressed in terms of thermal parameters and mass.

  3. Next one feeds in something from General Relativity to obtain expression for acceleration in terms of thermal parameters. Unruh effect means that in an accelerted motion system measures temperate proportional to acceleration :

    kBT= hbar a/2π .

    This quantum effect is known as Unruh effect. This temperature is extremely low for accelerations encountered in everyday life - something like 10-16 K for free fall near Earth's surface.

    Using this expression for T in previous equation one obtains the desired F=ma, which would thus have a thermodynamical interpretation. At this stage I have even less motivations for talking about emergence of space-time. Essentially the basic conceptual framework of Special and General Relativities, of wave mechanics and of thermodynamics are introduced by the formulas containing the basic parameters involved.

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.

  1. The number of bits for the information represented on the holographic screen is assumed to be proportional to area.

    N =A/Ghbar.

    This means bringing in blackhole thermodynamics and general relativity since the notion of area requires geometry.

    Comment: In TGD framework the counterpart for the finite number of bits is finite measurement resolution meaning that the 2-dimensional partonic surface is effectively replaced with a set of points carrying fermion or antifermion number or possibly purely bosonic symmetry generator. The orbits of these points define braid giving a connection with topological QFTs for knots, links and braids and also with topological quantum computation.

  2. It is assumed that A=4π R2, where R is the distance between the masses. This means a very special choice of the holographic screen.

    Comment: In TGD framework the counterpart of the area would be the symplectic area of partonic 2-surfaces. This is invariant under symplectic transformations of light-cone boundary. These "partonic" 2-surfaces can have macroscopic size and the counterpart for blackhole horizon is one example of this kind of surface. Anyonic phases are second example of a phase assigned with a macroscopic partonic 2-surface.

  3. Special relativity is brought in via the bomb formula


    One introduces also other expression for the rest energy. Thermodynamics gives for non-relativistic thermal energy the expression

    E= 1/2N kBT.

    This thermal energy is identified with the rest mass. This identification looks to me completely ad hoc and I think that kind of holographic duality is assumed to justify it. The interpretation is that the points/bits on the holographic screen behave as particles in thermodynamical equilibrium and represent the mass inside the spherical screen. What are these particles on the screen? Do they correspond to gravitational flux?

    Comment: In TGD framework p-adic thermodynamics replaces Higgs mechanism and identify particle's mass squared as thermal conformal weight. In this sense inertia has thermal origin in TGD framework. Gravitational flux is mediated by flux tubes with gigantic value of gravitational Planck constant and the intersections of the flux tubes with sphere could be TGD counterparts for the points of the screen in TGD. These 2-D intersections of flux tubes should be in thermal equilibrium at Unruh temperature. The light-like 3-surfaces indeed contain the particles so that the matter at this surface represents the system. Since all light-like 3-surfaces in the slicing are equivalent means that one can choose the reresentation of the system rather freely .

  4. Eliminating the rest energy E from these two formulas one obtains NT= 2mc2 and using the expression for N in terms of area identified as that of a sphere with radius equal to the distance R between the two masses, one obtains the standard form for gravitational force.

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.

  1. In zero energy ontology quantum states become zero energy states consisting of pairs of the positive and negative energy states with opposite conserved quantum numbers and interpreted in the usual ontology as physical events. These states are located at opposite light-like boundaries of causal diamond (CD) defined as the intersection of future and past directed light-cones. There is a fractal hierarchy of them. M-matrix generalizing S-matrix defines time-like entanglement coefficients between positive and negative energy states. M-matrix is essentially a "complex" square root of density matrix expressible as positive square root of diagonalized density matrix and unitary S-matrix. Thermodynamics reduces to quantum physics and should have correlate at the level of space-time geometry. The failure of the classical determinism in standard sense of the word makes this possible in quantum TGD (special properties of Kähler action (Maxwell action for induced Kahler form of CP2) due to its vacuum degeneracy analogous to gauge degeneracy). Zero energy ontology allows also to speak about coherent states of bosons, say of Cooper pairs of fermions- without problems with conservation laws and the undeniable existence of these states supports zero energy ontology.

  2. Quantum classical correspondence is very strong requirement. For instance, it requires also that electrons traveling via several routes in double slit experiment have classical correlates. They have. The light-like 3-surfaces describing electrons can branch and the induced spinor fields at them "branch" also and interfere again. Same branching occurs also for photons so that electrodynamics has hydrodynamical aspect too emphasize in recent empirical report about knotted light beams. This picture explains the findings of Afshar challenging the Copenhagen interpretation.

    These diagrams could be seen as generalizations of stringy diagrams but do not describe particle decays in TGD framework. In TGD framework stringy diagrams are replaced with a direct generalization of Feynman diagrams in which the ends of 3-D lightlike lines meet along 2-D partonic surfaces at their ends. The mathematical description of vertices becomes much simpler since the 2-D manifolds describing vertices are not singular unlike the 1-D manifolds associated with string diagrams ("eyeglass" in fusion of closed strings).

  3. If entropy has a space-time correlate then also first and second law should have such and Verlinde's argument that gravitational force attraction follows from first law assuming energy correlation might identify this correlate. This of course applies only to the classical gravitation. Also other classical forces should allow analogous interpretation as space-time correlates for something quantal.

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.

  1. The basic structure of quantum TGD suggests a thermodynamic interpretation. The basic observation is that the vacuum functional identified as the exponent of Kähler function is analogous to a square root of partition function and Kähler coupling strength is analogous to critical temperature. Kähler function identified as Kähler action for a preferred extremal appears in the role of Hamiltonian. Preferred extremal property realizes holography identifying space-time surface as analog of Bohr orbit. One can interpret the exponent of Kähler function as the density of states in the world of classical worlds so that Kähler function would be analogous to entropy density. Ensemble entropy is average of Kähler function involving functional integral over the world of classical worlds. This exponent is the counterpart for the quantity Ω appearing in Verlinde's basic formula.

  2. The addition of a measurement interaction term to the modified Dirac action gives rise to a coupling to conserved charges. Vacuum functional is identified as Dirac determinant and this addition is visible as an addition of an interaction term to Kähler function. The interaction gives rise to forces coupling to various charges at classical level for quantum states with fixed quantum numbers for positive energy part of the state. These terms are analogous to chemical potential terms in thermodynamics fixing the average values of various charges or particle numbers. In ordinary non-relativistic thermodynamics energy is in a special role. In the recent case there is a complete quantum number democracy very natural in a framework with coordinate invariance and with four-momentum assigned with the isometries of the 8-D imbedding space. In Verlinde's formula there is exponential factor exp(-E/T- Fx) analogous to the measurement interaction term. In TGD however conserved charges multiplied by chemical potentials defining generalized forces appear in the exponent.

  3. This gives an analog of thermodynamics in the world of classical worlds (WCW) for fixed values of quantum numbers of the positive energy part of state. For zero energy states one however has also additional thermodynamics- or rather its square root. This thermodynamics is for the conserved quantum numbers whose averages are fixed. For general zero energy states one has sum over state pairs labelled by momenta and various other quantum numbers labelling the positive energy part of the state. The coefficients of the conserved quantities of the measurement interaction term linear in conserved quantum numbers define the analogs of temperature and various chemical potentials. The field equations defined by Kähler function and chemical potential terms have thermodynamical interpretation and give coupling to conserved charges and also to their thermal averages. What is important is that temperature and various chemical potentials assigned to positive and negative energy parts of the state allow a complete geometrization in a general coordinate invariant manner and allow explicit expressions in terms of functions expressible in terms of the induced geometry.

  4. The explicit expressions must be deduced from Dirac determinant defining exponent of Kähler function plus measurement interaction term, in which the conserved isometry charges of Cartan algebra (necessarily!) appearing in the exponent are contracted with the analogs of chemical potentials. One make two rather detailed educated guesses for the chemical potentials. For the modified Dirac action the measurement interaction term is 4-dimensional and essentially unique. For the Kähler action one can imagine two candidates for the measurement interaction term. For the first option the term is 4-dimensional and for the second one 3-dimensional.

5. Some details related to the measurement interaction term

As noticed, one can imagine two options for the measurement interaction term defining the chemical potentials in terms of the space-time geometry.

  1. For both options the M4 part of the interaction term is proportional to n(M4)G/R and CP2 part to a dimensionless constant n(CP2), and the condition that there is no dependence of hbar excludes the dependence on the dimensionless constant Ghbar/R2.

  2. One can consider two different forms of the measurement interaction part in Kähler function. For the first option the conserved Kähler current replaces fermion current in the modified Dirac action and defines a 4-dimensional interaction term highly analogous to that assigned with the modified Dirac action. The source term induced to the field equations corresponds to the variation of

    [(G/R)× n(M4)pq,A gAB(M4)jA,α +n(CP2)Qq,A gABJA,α(CP2)] Jα .

    Here Jα is Kähler current.

  3. For the second option the measurement interaction term in Kähler action is sum over contractions of quantum Cartan charges with corresponding classical Noether charges giving the sum of the term

    (G/R)× n(M4)pq,A pcl,A +n(CP2)Qq,A Qcl,A

    from both ends of the space-time sheet. For a general space-time sheet the classical charges are different at its ends so that the variation gives non-trivial boundary conditions equating the normal (time-like) component of the canonical momentum current with the contraction of the variation of classical Noether charges contracted with quantum charges. By the extremal property the measurement interaction terms at the ends of the space-time sheet cancel each other so that the effect on Kähler function is only via the boundary conditions in accordance with zero energy ontology. For this option the thermodynamics for conserved charges is visible at space-time level only via the appearence of the average quantal charges and universal chemical potentials.

  4. The vanishing of Kähler gauge current resp. classical Noether charges for the first resp. second option would suggest an interpretation in terms of infinite temperature limit. The fact that momenta and color charges are in completely symmetric position suggests however the vanishing of chemical potentials. One can in fact fix the value of the temperature to say T= R/G without loss of information and code thermodynamics in terms of the chemical potentials alone.

    The vanishing of the measurement interaction term occurs for the vacuum extremals. For CP2 type vacuum extemals with Euclidian signature of the induced metric interpretation in terms of vanishing chemical potentials is more natural. For vacuum extremals with Minkowskian signature of the induced metric fluctuations and consequently classical non-determinism are maximal so that the interpretation in terms of high temperature phase cannot be excluded. One must however notice that CP2 projection for vacuum extremals is 2-dimensional whereas high temperature limit would suggest 4-D projection so that the interpretation in terms of vanishing chemical potentials is more natural also now.

To sum up, TGD suggests two thermodynamical interpretations. p-Adic thermodynamics gives inertial mass squared as thermal conformal weight and also the basic formulation of quantum TGD allows thermodynamical interpretation. The thermodynamical structure of quantum TGD has of course been guiding principle for two decades. In particular, quantum criticality as the counterpart of thermal criticality has been extremely useful guide line and led to a breakthrough in the understanding of the modified Dirac equation during the last year. Also p-adic thermodynamics has been in the scene for more than 15 years and makes TGD a theory able to make precise quantitative predictions.

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".

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