Saturday, June 09, 2012

About deformations of known extremals of Kähler action

I have done a considerable amount of speculative guesswork to identify what I have used to call preferred extremals of Kähler action. The problem is that the mathematical problem at hand is extremely non-linear and that there is no existing mathematical literature. One must proceed by trying to guess the general constraints on the preferred extremals which look physically and mathematically plausible. The hope is that this net of constraints could eventually chrystallize to Eureka! Certainly the recent speculative picture involves also wrong guesses. The need to find explicit ansatz for the deformations of known extremals based on some common principles has become pressing. The following considerations represent an attempt to combine the existing information to achieve this.

The dream is to discover the deformations of all known extremals by guessing what is common to all of them. One might hope that the following list summarizes at least some common features.

Effective three-dimensionality at the level of action

  1. Holography realized as effective 3-dimensionality also at the level of action requires that it reduces to 3-dimensional effective boundary terms. This is achieved if the contraction jαAα vanishes. This is true if jα vanishes or is light-like, or if it is proportional to instanton current in which case current conservation requires that CP2 projection of the space-time surface is 3-dimensional. The first two options for j have a realization for known extremals. The status of the third option - proportionality to instanton current - has remained unclear.

  2. As I started to work again with the problem, I realized that instanton current could be replaced with a more general current j=*B∧J or concretely: jα= εαβγδBβJγδ, where B is vector field and CP2 projection is 3-dimensional, which it must be in any case. The contractions of j appearing in field equations vanish automatically with this ansatz.

  3. Almost topological QFT property in turn requires the reduction of effective boundary terms to Chern-Simons terms: this is achieved by boundary conditions expressing weak form of electric magnetic duality. If one generalizes the weak form of electric magnetic duality to J=Φ *J one has B=dΦ and j has a vanishing divergence for 3-D CP2 projection. This is clearly a more general solution ansatz than the one based on proportionality of j with instanton current and would reduce the field equations in concise notation to Tr(THk)=0.

  4. Any of the alternative properties of the Kähler current implies that the field equations reduce to Tr(THk)=0, where T and Hk are shorthands for Maxwellian energy momentum tensor and second fundamental form and the product of tensors is obvious generalization of matrix product involving index contraction.

Could Einstein's equations emerge dynamically?

For jα satisfying one of the three conditions, the field equations have the same form as the equations for minimal surfaces except that the metric g is replaced with Maxwell energy momentum tensor T.

  1. This raises the question about dynamical generation of small cosmological constant Λ: T= Λ g would reduce equations to those for minimal surfaces. For T=Λ g modified gamma matrices would reduce to induced gamma matrices and the modified Dirac operator would be proportional to ordinary Dirac operator defined by the induced gamma matrices. One can also consider weak form for T=Λ g obtained by restricting the consideration to sub-space of tangent space so that space-time surface is only "partially" minimal surface but this option is not so elegant although necessary for other than CP2 type vacuum extremals.

  2. What is remarkable is that T= Λ g implies that the divergence of T which in the general case equals to jβJβα vanishes. This is guaranteed by one of the conditions for the Kähler current. Since also Einstein tensor has a vanishing divergence, one can ask whether the condition to T= κ G+Λ g could the general condition. This would give Einstein's equations with cosmological term besides the generalization of the minimal surface equations. GRT would emerge dynamically from the non-linear Maxwell's theory although in slightly different sense as conjectured (see this)! Note that the expression for G involves also second derivatives of the imbedding space coordinates so that actually a partial differential equation is in question. If field equations reduce to purely algebraic ones, as the basic conjecture states, it is possible to have Tr(GHk)=0 and Tr(gHk)=0 separately so that also minimal surface equations would hold true.

    What is amusing that the first guess for the action of TGD was curvature scalar. It gave analogs of Einstein's equations as a definition of conserved four-momentum currents. The recent proposal would give the analog of ordinary Einstein equations as a dynamical constraint relating Maxwellian energy momentum tensor to Einstein tensor and metric.

  3. Minimal surface property is physically extremely nice since field equations can be interpreted as a non-linear generalization of massless wave equation: something very natural for non-linear variant of Maxwell action. The theory would be also very "stringy" although the fundamental action would not be space-time volume. This can however hold true only for Euclidian signature. Note that for CP2 type vacuum extremals Einstein tensor is proportional to metric so that for them the two options are equivalent. For their small deformations situation changes and it might happen that the presence of G is necessary. The GRT limit of TGD discussed in kenociteallb/tgdgrt kenocitebtart/egtgd indeed suggests that CP2 type solutions satisfy Einstein's equations with large cosmological constant and that the small observed value of the cosmological constant is due to averaging and small volume fraction of regions of Euclidian signature (lines of generalized Feynman diagrams).

  4. For massless extremals and their deformations T= Λ g cannot hold true. The reason is that for massless extremals energy momentum tensor has component Tvv which actually quite essential for field equations since one has Hkvv=0. Hence for massless extremals and their deformations T=Λ g cannot hold true if the induced metric has Hamilton-Jacobi structure meaning that guu and gvv vanish. A more general relationship of form T=κ G+Λ G can however be consistent with non-vanishing Tvv but require that deformation has at most 3-D CP2 projection (CP2 coordinates do not depend on v).

  5. The non-determinism of vacuum extremals suggest for their non-vacuum deformations a conflict with the conservation laws. In, also massless extremals are characterized by a non-determinism with respect to the light-like coordinate but like-likeness saves the situation. This suggests that the transformation of a properly chosen time coordinate of vacuum extremal to a light-like coordinate in the induced metric combined with Einstein's equations in the induced metric of the deformation could allow to handle the non-determinism.

Are complex structure of CP2 and Hamilton-Jacobi structure of M4 respected by the deformations?

The complex structure of CP2 and Hamilton-Jacobi structure of M4 could be central for the understanding of the preferred extremal property algebraically.

  1. There are reasons to believe that the Hermitian structure of the induced metric ((1,1) structure in complex coordinates) for the deformations of CP2 type vacuum extremals could be crucial property of the preferred extremals. Also the presence of light-like direction is also an essential elements and 3-dimensionality of M4 projection could be essential. Hence a good guess is that allowed deformations of CP2 type vacuum extremals are such that (2,0) and (0,2) components the induced metric and/or of the energy momentum tensor vanish. This gives rise to the conditions implying Virasoro conditions in string models in quantization:

    gξiξj=0 , gξ*iξ*j=0 , i,j=1,2 .


    Holomorphisms of CP2 preserve the complex structure and Virasoro conditions are expected to generalize to 4-dimensional conditions involving two complex coordinates. This means that the generators have two integer valued indices but otherwise obey an algebra very similar to the Virasoro algebra. Also the super-conformal variant of this algebra is expected to make sense.

    These Virasoro conditions apply in the coordinate space for CP2 type vacuum extremals. One expects similar conditions hold true also in field space, that is for M4 coordinates.

  2. The integrable decomposition M4(m)=M2(m)+E2(m) of M4 tangent space to longitudinal and transversal parts (non-physical and physical polarizations) - Hamilton-Jacobi structure- could be a very general property of preferred extremals and very natural since non-linear Maxwellian electrodynamics is in question. This decomposition led rather early to the introduction of the analog of complex structure in terms of what I called Hamilton-Jacobi coordinates (u,v,w,w*) for M4. (u,v) defines a pair of light-like coordinates for the local longitudinal space M2(m) and (w,w*) complex coordinates for E2(m). The metric would not contain any cross terms between M2(m) and E2(m): guw=gvw= guw* =gvw*=0.

    A good guess is that the deformations of massless extremals respect this structure. This condition gives rise to the analog of the constraints leading to Virasoro conditions stating the vanishing of the non-allowed components of the induced metric. guu= gvv= gww=gw*w* =guw=gvw= guw* =gvw*=0. Again the generators of the algebra would involve two integers and the structure is that of Virasoro algebra and also generalization to super algebra is expected to make sense. The moduli space of Hamilton-Jacobi structures would be part of the moduli space of the preferred extremals and analogous to the space of all possible choices of complex coordinates. The analogs of infinitesimal holomorphic transformations would preserve the modular parameters and give rise to a 4-dimensional Minkowskian analog of Virasoro algebra. The conformal algebra acting on CP2 coordinates acts in field degrees of freedom for Minkowskian signature.

Field equations as purely algebraic conditions

If the proposed picture is correct, field equations would reduce basically to purely algebraically conditions stating that the Maxwellian energy momentum tensor has no common index pairs with the second fundamental form. For the deformations of CP2 type vacuum extremals T is a complex tensor of type (1,1) and second fundamental form Hk a tensor of type (2,0) and (0,2) so that Tr(THk)= is true. This requires that second light-like coordinate of M4 is constant so that the M4 projection is 3-dimensional. For Minkowskian signature of the induced metric Hamilton-Jacobi structure replaces conformal structure. Here the dependence of CP2 coordinates on second light-like coordinate of M2(m) only plays a fundamental role. Note that now Tvv is non-vanishing (and light-like). This picture generalizes to the deformations of cosmic strings and even to the case of vacuum extremals.

For background see the chapter Basic Extremals of Kähler action of "Physics in Many-Sheeted Space-time". For details see the article About deformations of known extremals of Kähler action.



29 Comments:

At 8:46 PM, Anonymous Orwin said...

Bianchi Identity "freezes" GRT to 2+1 dynamics, which is now life after M-theory. But Bianchi matrix is common also to Poincare and Lioiuville gravity.

To me this flags it as marker of biological identity, as in immune surface-markings of cells. But is this not the signature of mind, transposed out of time, as for Bergson?

 
At 10:11 PM, Blogger hamed said...

This comment has been removed by the author.

 
At 10:18 PM, Blogger hamed said...

Effective three-dimensionality! First about Effective two dimensionality, I write what I understand:
A Lightlike 3-surface is metrically two dimensional. One can assign to each point of the 3-surface, 3 coordinates. Effective two-dimensionality means One can assign to each point of the 3-surface, only 2 coordinates? But a parton is 2D and we need another parameter to define how much is the orbit of the parton. Then 3 parameters is needed at end. why do you don't account last parameter in the metric of the 3-surface?

boundaries of the lightlike 3- surface correspond to a 2D parton. How is it possible to define each point of the 3-surface only going along an orbit of a point of the parton in some quantity?

 
At 9:43 AM, Blogger Ulla said...

Orwin,

"To me this flags it as marker of biological identity, as in immune surface-markings of cells. But is this not the signature of mind, transposed out of time, as for Bergson?"

Can you explain more what you mean? Thanks.

 
At 10:29 AM, Blogger ThePeSla said...

Matti,

I have been meaning to look up this K action but I think your approach is definitely an area to investigate and make more comprehensible.

Here is an interesting link today:\
http://www.sciencedaily.com/releases/2012/06/120608135725.htm which concerns symmetry breaking in the plane and crumpling. Now if someone reads my blog they will see that "I was there first" but it is not clear how long either of us worked on this. I mention this to show my intuition rather sound even when it is vague and synthetic first blush only- thus I stand behind your achievements regardless of who publishes what when- for yours is a living project.

I note Lubos was alerted to the black hole global warming analogy of which this other humble correspondent brought up before he did as a news issue of philosophic significance.

Now, it is ok to explore the complex aspects of these planes and the possibility we can unify things a little better despite the problems of extreme states so that we may find something complex but decidedly more linear. You are aware of the areas needing exploration.

In the existing cosmology or standard theories where the matices apply to the "flat" branes one has to choose between the red and blue pills as in the movie such that there is an impossible synthesis of the idea of a purple pill. Why does mathematics apply to physics even more so than we have yet imagined- but not rigidly?

Sooner or later all of you should realize that the 2+1 or 4 issue of representations (going back to Dirac) is not a choice to solve but a hint we need wider generalization of our idea of living planes.

Bergson considers "duration" as distinct from proper time and it is decidedly subjective and even considered second rate philosophy by other schools. Yet we have the hints of such distinction at the extreme conditions in general physics- in the remote implications of Einstein for example. Here the subjective and objective may be purple or impossibly confused (for us but evidently not for nature.)

The PeSla

 
At 4:55 PM, Anonymous Orwin said...

Ulla,

Any real computer has a clock and is managed by scheduler with time-bindings giving access to compute cycles, which are Carnot cycles of processing work/heat production.

Working brain needs rhythm generators, and Baldwin's closed loop was first try, but after Peirce, working with Cattell and Jastrow on reaction-times, he ditched it.

Cell clock/scheduler is not known/not asked after. Perhaps it is citric acid/Krebbs cycle producing ATP etc, as access tokens for decentralized execution! Time-bound for 40min = attention-span!

Matti now seems to have complementary vacuum and quadratic/energy tensors. Non-linear action proceeds per frequency/thermal balance, for distinctive momentum-flux. It is as if time factor is transposed into vector of momentum, but how the symmetries/conservation resolve is the big mystery.

 
At 10:37 PM, Anonymous matpitka@luukku.com said...

Dear Hamed,

I answered in separate email to your questions and lost the email. I do not bother to write again.

Matti

 
At 11:22 PM, Anonymous Orwin said...

Matti,

You wondered about different causality: cell does Carnot cycle upfront, so its not really like a computer, rather an 'imputer' which imputes, e.g. that virus is bad, when expressing antigen. This depends on catalysis of actual reactions, and something we don't understand: enzymes reaching docking sites ord. mag. above diffusion time: can this be due to a polarized or other phase-specific effect?

 
At 12:30 AM, Anonymous matpitka@luukku.com said...

To All,


some clarifications. Kahler action is basic element of TGD. Without it there would not be any TGD. The fundamental challenge is to understand what the preferred extremals of K are and this posting is related to what might be genuine progress in this respect. What emerges is strong resemblance with minimal surfaces (strings). Field equations reduce to purely algebraic conditions plus conditions stating that some components of the induced metric vanish.

Depending on whether the M^4 or CP_2 projection is 4-D these conditions state that space-time surface possess what I call Hamilton Jacobi structure or complex structure. Hamilton Jacobi structure for M^4 is the analogy of Hermitian structure for CP_2.

What the conditions state that TGD would be completely integrable theory just like string models and Euclidian pure gauge theories are. The naive view about dynamics as initial value problem would be wrong: the preferred extremals would be much more like "archetypal" field patterns having either Hamilton Jacobi or Hermitian structure in 4-D sense.

 
At 9:11 PM, Anonymous matpitka@luukku.com said...

To Hamed:

Sorry! I expressed myself in confusing manner. I gave an answer to your question by email but lost the email before I had attached it here.

 
At 12:14 AM, Blogger hamed said...

Dear Matti, Please don't sorry anytime to me, I be ashamed of the word of you:). There was not misunderstanding on your comment to me.

 
At 9:44 AM, Blogger Ulla said...

http://2012daily.com/?q=node%2F271

 
At 12:32 PM, Blogger Ulla said...

Can the 'clock' function be described as a 'sensing' too? I think of the difference in time/dissipation between the jump and the decay as instance.

 
At 12:55 PM, Anonymous Orwin said...

There is a direct sense of being 'energised,' which gets described as a 'tingling,' like a galvanic akin response: gap junctions tighten, increasing conductivity, conserving hear and enhancing skin sensitivity. So its strictly not local to the cell, but then the cAMP cascade works between cells! I think that's a very significant association.

 
At 3:06 PM, Anonymous Orwin said...

The Hamilton-Jacobi equation (http://beige.ucs.indiana.edu/B673/node48.html) is where control theory branches from physics. Now extends to estimating amplitude of a non-linear oscillator. So biological control systems can be paced by biological oscillators. This gives new life to Winfree's interest in oscillators, and there's a kind of networking of ideas here which can form a paradigm.

 
At 7:59 PM, Anonymous matpitka@luukku.com said...

To Orwin:

Getting attentive, getting alert would correspond to becoming energized rather concretely since ATP is the concomitant of attention- negentropic entanglement at flux tubes connecting the attendee to the target of attention - in TGD Universe.

Paying attention is strictly non-local process: the target of attention becomes part of attention during it. The division to attendee and target is made at the level of memory and the asymmetry is only apparent. Magnetic flux tubes bring non-locality to biology. There are really fascinating applications developing, which might modify our world view based on locality dogma.

Oscillators are one aspect to be living: I talked in some earlier posting about addressing system based on collections of frequencies which make possible remote control of gene activities (among other things).

 
At 10:48 PM, Blogger Ulla said...

I wanted to scale down this question to (living) molecules. If 'sensing' (k-coupling?) is what makes the life, then something must tell what is the difference in the messages. Is this 'something' just ATP as the max energetic message molecule, creating a string/dance of tensions? Then the homeostasis that prevent the system to loose all its negentropy (keep the tension) as some kind of memory (in compressed states made from superposition = folding, chrystals) is what?

If darkmatter controls the genome, can that control be made of other than magnetic extremals? Is 'sensing' (of couplings) by a resonator (the stack made of entropic-negentropic elements?) always electromagnetic? I know I have asked this before, but I got no answer.

What about induced em-fields that 'fall out' on some membrane/sheet, through creations of flux-tubes directly with environment? Is this not 'sensing'? Or stochastic resonance? As long as the drum is drumming, the matterstructure is dancing?

Then there is the question of what is death?

Many-body states in TGD? Are they made up of bits of three-body parts (triangular loops?)?

 
At 11:01 PM, Blogger Ulla said...

Is one kind of 'sensing' directly the deformations made of gravity? This requires some computations of expected structure and measurement of abbreviations from the expected structure?

Structure is information.

 
At 11:32 PM, Blogger Ulla said...

Sensing organ tilt? Starch as sensing organ?

Several lines of experimental evidence have demonstrated that starch is important but not essential for gravity sensing and have suggested that it is reasonable to regard plastids (containers of starch) as statoliths.

possess dynamic movement

movement is an intricate process involving vacuolar membrane structures and the actin cytoskeleton. This review covers current knowledge regarding gravity sensing, particularly gravity susception, and the factors modulating the function of amyloplasts for sensing the directional change of gravity. Specific emphasis is made on the remarkable differences in the cytological properties, developmental origins, tissue locations, and response of statocytes between root and shoot systems. Such an approach reveals a common theme in directional gravity-sensing mechanisms in these two disparate organs.

Other citations:
Root gravitropism is regulated by a transient lateral auxin gradient controlled by a tipping-point mechanism http://dx.doi.org/10.1073/pnas.1201498109

Molecular mechanisms of root gravity sensing and signal transduction http://dx.doi.org/10.1002/wdev.14
Models have been proposed to explain how mechanosensitive ion channels or ligand–receptor interactions could connect these events. Although their roles are still unclear, possible second messengers in this process include protons, Ca2+, and inositol 1,4,5-triphosphate.

Root cap angle and gravitropic response rate are uncoupled in the Arabidopsis pgm-1 mutant http://dx.doi.org/10.1111/j.1399-3054.2010.01439.x Our study suggests that the relationship between root cap angle and gravitropic response depends upon plastid sedimentation-based gravity sensing and supports the idea that there are multiple, overlapping sensory response networks involved in gravitropism.

Polarization of fluxes seems to be essential?

 
At 12:01 AM, Blogger Ulla said...

This may go too special? Sorry if this is of no use,

From: The mechanisms of root gravity sensings...
Gravity sensings and root curvature.

To be gravitropically competent, a plant organ must perceive a change in its orientation within the gravity field, convert this physical information into a biochemical signal, and respond to it by developing a curvature that brings it back to its GSPA. In roots, the columella cells in the root cap have been identified as a primary site of gravity perception in a variety of plant species (Figure 1(b) and (c)), and ablating these cells causes a strong gravitropism defect.2,3 These specialized cells are polarized, with the nucleus at the top, the ER at the periphery, and the starch-filled amyloplasts sedimented on the lower side.4 The centers of the columella cells have few organelles and instead contain highly dynamic actin microfilaments.5

 
At 8:23 PM, Anonymous Orwin said...

Ulla, the classic answer is lignin, taken to be distinctive of 'reaction wood' (q.v.) as in gnarled, wind-blown trees, especially those exposed to trade-winds. That's the story, from the old mercantile lore, with Perennial Philosophy.

Lignin makes me think of 'tin whiskers', crystal trees which grow in vacuum tubes and make them fail. Newton grew the crystal tree of antinomy, researching 'ferments' of metals in alchemy. Likely there is a field effect. BUT such are NOT 'ye gates' of consciousness, the theme from Newton that Aldous Huxley picked up.

In the tradition of Leibniz, the 'gate' becomes a device switching the state of a circuit, following a logical pattern. And with the logicism of Whitehead and Russell, consciousness is expected to reduce to this level. BUT a logical operator is NOT enough to make an assignment, assertion or statement! That involves binding a variable!

And operators are two a penny in QM. Reduce consciousness there and you have panpsychism, which makes a great assumption which proves nothing specific. And violates the authentic usage of 'psyche'.....

 
At 10:18 PM, Anonymous Orwin said...

More on sensation.

If Mandelbrot means anything to you, Gerard Duplantier is someone to watch. And he's saying that Euclidean forms scale in Liouville gravity! http://arxiv.org/abs/0808.1560

Now any indication of gravity within a cell is not going to be significant in itself: there's too much room for error, and all senses have thresholds of discrimination. So Euclidean extremals generated in the Kahler manifold would have to scale in this distinct process to appear as sensations!

That surely involves a phase change, and here's a view of the renormalization group theory which is unusually time-dependent: http://arxiv.org/abs/1201.3080

So it escapes the problem I flagged earlier that Ocedelet Theorem for Lyapunov exponent gives only time-average. Lyapunov exponent gives fractal dimesionality, which changes at phase boundary.

Since parameters diverge at boundary, one needs p-adics to track the change itself. And the paper reports new phenomena, involving Fermi stuff in nucleus.

The dependent variable here is the order-parameter, and that seems to feed back to Matti's new Kahler extremal solution. This could be the internal loop intuited by Baldwin when he was thinking Spinoza and natural theology, before he went naturalistic!!

But how a preferred growth direction is propagated to growth areas is a further problem, and could involve a field effect, and actually lignin and the like. You see Sheldrake hand-waving round here.

 
At 10:26 PM, Anonymous Orwin said...

Sorry, that's Bertrand Duplantier. I'm tired and I want to make time to write again.

 
At 1:47 AM, Blogger Ulla said...

Thanks,

The matrix sensor?

Actually also membranes are networks. Attatchments for flux tubes.

The matrix reminds me of Keas ideas.

If you follow the links you see that also gravity invokes piezoelectrically. So I ask if there is any other way to sense than electromagnetically? How can we sense the bendings of spacetime sheets? Through the stress they create on waves, strings and particles? Strings are flux tubes?

 
At 9:09 AM, Blogger Ulla said...

http://arxiv.org/pdf/1004.3747.pdf

STABILITY UNDER DEFORMATIONS OF EXTREMAL
ALMOST-K¨ AHLER METRICS IN DIMENSION 4.
MEHDI LEJMI
Given a path of almost-K¨ahler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost- K¨ahler metric is an extremal K¨ahler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-K¨ahler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.

A key observation, made by Fujiki [13] in the integrable case and by Donaldson [9] in the general almost-K¨ahler case, asserts that the natural action of the infinite
dimensional group Ham(M, w) of hamiltonian symplectomorphisms on the space AK_w of w-compatible almost-K¨ahler metrics is hamiltonian with moment map μ :
AK_w (Lie(Ham(M, w))) given by eq.. The critical points
of the norm are called extremal almost-K¨ahler metrics. It turns out that the symplectic gradient of sr of such metrics is a holomorphic vector field in the sense that its flow preserves the corresponding almost-complex structure. In particular, extremal K¨ahler metrics in the sense of Calabi [7] and almost-K¨ahler metrics with constant hermitian scalar curvature are extremal.

 
At 1:13 PM, Blogger Ulla said...

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.dmj/1337690405

the blowup of an extremal Kähler manifold

 
At 12:25 AM, Blogger Ulla said...

Well, if the answer is not lignin but the creation of 'holes'. Protons pop in and out of existence, depending on those deformations (gravity seen in pH)? Creating tensions on the hadron 'strings', making them wider or thinner, two or three phase = flat or curved? This set on all the other reactions?

 
At 11:47 AM, Blogger Ulla said...

http://www.johnwmoffat.com
http://www.johnwmoffat.com/pdfs/PICCPTalkl.pdf
+ and -energy and c.c.

 
At 12:14 AM, Blogger Ulla said...

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