Friday, November 02, 2007

Mandelbrot and Julia sets from fundamental physics?

Kea told at her blog about the Serpentine Gallery of mathematical formulas of twenty first century.

My favourite formula is very simple looking: z→z2+c of Mandelbrot. Mandelbrot set is defined as the set of values of c for which the iteration starting from z=0 does not lead to infinity. To me this is the equation of the last century which is richest of hidden information and beauty. The definition of Mandelbrot set involves two complex spaces. Julia set J(f), where f is holomorphic function in the most general case, looks very similar looking fractal, and its definition involves no parameters. If f is polynomial, Julia set is the boundary of the set of points whose orbits remain bounded under iteration.

One has some kind of mystic dejavu feeling as one looks at these sets: I must have seen them before! By just looking at this fractal you feel that in some profound sense it really contains Universes within Universes. This might make sense in some sense as will be found!

No wonder that any theoretician would be extremely happy if she or he could identify a generalization of Mandelbrot set or Julia set as something physical and concrete. I realized that in the case of TGD there might be some hopes for this.

1. Does partonic 2-surface as the fundamental 2-D space and light-like 3-surface the fundamental iteration map?

The boundary of the Mandelbrot set represents a critical set: the points of the complex plane (parameter c) for which the iteration leads and does not lead to infinity. Obviously also the Julia set is a critical set. Could quantum criticality of TGD Universe be able to produce these miracles from physics?

  1. The fundamental 2-D space would be the partonic 2-surface X2 at δM4+× CP2. Here δ M4+/- refers to the boundary of future/past directed lightcone M4+/-. These light-cones form a causal diamond containing the light-like partonic 3-surfaces. X2 is mapped by the light-like orbit X3l of X2 to the final state partonic 2-surface Y2 at δ M4-× CP2. This map followed by return to the lower end of the diamond should define the fundamental iteration step. If one could identify Julia set as a subset of X2 - the fundamental conformal dynamical object in TGD having arbitrarily large size - one could indeed say that Julia sets contain universes within universes!

  2. The iteration map would be defined by a braiding. Kähler magnetic flow lines or flow lines of Kähler potential define two candidates for the braiding at the light-like 3-surface. Only the latter can be considered for the partonic 3-surfaces whereas the first one could be important in the interior of space-time surfaces. Internal consistency requires that the braiding by Kähler gauge potential is homotopic with the braiding associated with the minima of Higgs potential. The ends of braid correspond to X2 and Y2 at the intersections of future and past directed lightcones forming a causal diamond. The iteration map could be defined by the braiding taking point z at the lower end of diamond to a point point z1 at the upper end of diamond. Then you would look what happens to z1 interpreted as a point at the lower end of the diamond. And so on...

  3. With a proper choice of complex coordinate for X2 the iteration map should be a holomorphic (or perhaps even rational or polynomial) map. Conformal invariance indeed encourages to think that this might be the case. The complex coordinate defined by the generalized eigenvalue of D is very natural candidate in this respect. Also the complex coordinates of the two geodesic spheres S2II subset CP2 and S2 subset δ M4+ are reasonable candidates.

2. A candidate for the TGD counterpart of Julia set

The identification of the candidate for the Julia set is easier since it is a subset of the space, where the iteration takes place rather than a subset of the parameter space.

  1. The number theoretic braid corresponds to the decomposition of the S2II projection of X2 to separate regions having interpretation as domains inside which conformal field theory applies. The conformal fields associated witgh different regions are independent dynamically so that X2 remains effectively 2-D in discretized sense. This decomposition can be interpreted as a decomposition of 2-D landscape to (complex) mountains. At the peaks of the mountains the complex eigenvalue of the modified Dirac operator D vanishes: this corresponds physically to the vanishing of Higgs and to quantum criticality.

  2. What is the counterpart for the non-convergence of the iteration? Each mountain is separated from the neighbouring mountains by a saddle point curve of Higgs modulus containing one or more Higgs minima. The escape of the iteration to infinity would naturally correspond to a situation in which the outcome of repeated iteration z→z1 does not belong at the original mountain. Thus one expects each mountain to decompose into regions separated by boundaries at which iteration ceases to lead to the original valley. One can also identify boundaries of regions in which iteration leads out in N steps and in this manner make the might-exist counterpart of Julia set colored.

3. A candidate for the TGD counterpart of Mandelbrot set

Concering the identification of the candidate for the TGD counterpart of Mandelbrot set the basic challenge is to identify the physical counterpart of the complex parameter c.

  1. The only identification of the complex parameter c that comes in mind and does not mean a modification of X2 is the choice of S2II defining the quantization axes in color degrees of freedom and parametrized by some complex coordinate c. One should show that different choices of S2II define a holomorphic iteration map. The counterpart of Mandelbrot set would exist at S2II.

  2. The point at which Higgs vanishes (peak of the complex mountain) is the obvious choice for the initial point z=0 of the iteration map. This favors complex "Higgs" as the coordinate in which iteration is represented by a holomorphic function.

  3. Again the open question is whether one really obtains holomorphic/rational/polynomial map with a proper choice of complex coordinate for X2 and for the space of geodesic spheres.


At 11:22 PM, Anonymous said...

You ask meaningful questions on a number of levels, thank you for the hope.

Is it reasonable to hypothesize that "we" are but mere self-similar subset to an overarching pattern consisting of matter and energy interconnected from the very beginning?

To elaborate on the question; the Copenhagan Hypothesis alludes to the viewpoint that the observer is intimately tied to the event. As such, an event with multiple observers are all connected. Given current thought on the origins of the Cosmos via the "Big Bang," this would mean that every observer has been connected since the beginning (albeit indirectly). If this is true, then the subset of reality consisting of our collective conscious is similar to structures akin to the Julia set because the thoughts within the horizen are accessible and have not yet escaped to infinity.

How would the above hypothesis be testable; with the results verifiable and reproducable?

Any connection would be greatly appreciated, thanks.


At 11:54 PM, Blogger Matti Pitkanen said...

Dear MCB,

the connectedness idea looks rather reasonable. Entanglement would be connectedness in quantum context and should have geometric correlates: my favorite is magnetic flux quanta. If one takes the hierarchy of Planck constants seriously and the hypothesis that the Planck constant associated with space-time sheets mediating gravitational interaction is gigantic, then macroscopic quantum coherence and entanglement are present in all length scales since quantum scales scale like hbar. By the way, these gravitational space-time sheets explain dark energy
since elementary particle at this kind space-time sheet has enormous Compton size so that energy density is constant in excellent accuracy.

State function reduction reduces the entanglement and would destroy it in standard quantum theory. In TGD framework one must replace standard quantum measurement theory based on von Neumann algebras known as factors of type I with that associated with hyperfinite factors of type II_1 or even III_1. This brings in the notion of measurement resolution. Measurements are always done with a finite measurement resolution and also entangelement is reduced in state function reduction to zero only in this resolution. Everything would be indeed entangled. The sharing and fusion of mental images by entanglement of subselfs of separate selves in resolution which is below that assignable to selves would be example of this. There would be cosmic pool of mental images, perhaps standardized, kind of quantum archetypes.

Self-similarity would be present but I think that the notion of fractality is much more general than that represented by Mandelbrot and Julia sets. p-Adic fractal hierarchy, fractal hierarchy of dark matters characterized by different Planck constants, fractal hierachy of selves,...

Your idea using Julia set as metaphor looks interesting. If causal diamonds (CDs) defined as intersections of future and past directed lightcones are identified as correlates of selves then one indeed ends up with this kind of picture. CD is however very boring structure as compared to Julia set: fractality is realize as CDs within CDs within ...(selves with subselves with...)

One class of tests would test the hypotesis about the hierarchy of Planck constants. There are anomalies supporting this hypothesis. Anomalously low dissipation, Bohr orbitology at astrophysical systems, anomalous gravitational effects such as Allais effect,... Quite recently I learned from the finding of a "hot Jupiter" which is planet revolving extremely near to its star. It should have spiralled long time ago to the star but has not done it. Similar observation about electron in atom led to Bohr model of atom for century ago.

At 2:06 PM, Blogger Nissim Levy said...

I don't have your technical savvy so please have patience with Me :-)

Would you say that Dark Energy is a Julia Set?

At 7:09 PM, Anonymous said...

Dear Nissim,

thank you for asking. I do not see any connection between dark energy and Julia sets.

Just in occasion I mention a misty idea which emerged during this year (2011, I wrote this posting 2007) about how Julia sets might emerge in TGD physics.

a) Suppose that the general solution of field equations is in given in terms of real octonion analytic map of M^4xCP_2 to itself whose restriction to partonic 2-surfaces gives ordinary complex real analyticity. This idea is based on an old romantic proposal which was dead and buried for years ago. Suddenly it however experienced resurrection in somewhat modified form and I was not able to kill it.

Space-time surface would be analogous to a curve obtained by requiring the vanishing of the imaginary part of complex real-analytic function.

b) The value of image octonion under this map can be expressed as sum of Q1="real part"=quaternion and "imaginary part"= IQ2= I*quaternion. I is additional octonionic uni.

The vanishing of "imaginary" part of Q1+IQ2 (Q_2=0) would define space-time surface and Q1 would defined for this surface quaternionic coordinate.

c) One can sum, subtract, multiply and divide these maps pointwise and also *function composition* is possible. The analog local field with function composition as additional operation is obtained.

d) In particular iteration of these maps is possible and map space-time surface to a new one. Iteration could be used to define the analogs of Julia sets also now and one would obtain 4-D analogs of Julia sets. By real-octonion analyticity maps would be induced by real-complex analytic maps so that they could have as "cores" ordinary Julia sets.

At 10:32 AM, Blogger Nissim Levy said...

Thank you very much for responding to me. As I said I don't have your technical expertise in Physics. I am a software developer by profession with a keen interest in matters relating to the origins of reality. The reason I suspected that Julia sets might explain Dark Energy is that Julia Sets are repellers and Dark Energy is causing the accelerated expansion of the universe.

My way of thinking is that the universe should be seen as a chaotic system ruled by the laws of non linear dynamics. if this is true then all of the concepts in Chaos Theory should have an isomorphic reality in the universe. If Dark Energy is not a Julia Set then what other Chaos Theory concept might be isomorphic to it?

At 4:44 AM, Anonymous said...

I think that fractality and chaos theory are realized in rather abstract manner. For instance, fractality allows many interpretations and Mandelbrot and Julia fractals represent only one- and extremely beautiful- example.
Therefore I find it difficult to imagine any direct identifications like Dark energy-Julia set.

At 12:41 PM, Blogger Nissim Levy said...

There are many varieties of non Euclidian geometries but Einstein found a beautiful ismorphism between one particular type of non Euclidian geometry and gravity. Might it not be possible that there is an isomorphism between one particular kind of fractal system and Gravity or Dark Energy?

I apologize if I am disturbing you and you need not reply to me. I would just like to express my strong intuition that there must be a correspondence between such things as gravity and Dark Energy and Chaos Theory.

At 9:12 PM, Anonymous said...

I am convinced that fractality is fundamental for physics and replaces the naive reductionism reducing everything to the dance of quarks.

One implication in TGD framework is the hierarchy of p-adic length scales and hierarchy of Planck constants involving also length and time scale hierarchy. This hierarchy serves as a borderline between other unified theories and TGD. Other unified theories typically assume Planck scale as the only fundamental scale.

I would be however very cautious in taking correspondences artificially to too detailed level.

At 4:35 PM, Blogger Nissim Levy said...

I've also been looking into isomorphisms between Godel's Incompleteness Theorems and Chaos Theory. Do Godel's theorems play any part in your thinking about TGD?

At 7:26 PM, Anonymous said...

TGD inspired theory of consciousness leads to a new view about mathematics.

The Universe of TGD is re-created in each quantum jump. The new Universe contains information about previous Universes and thus about quantum jumps too. Just the attempt to understand the Universe replaces the Universe with a more complex one since the physical representation of the information gained about the previous Universe changes it.

The evolution of mathematics is discovery of new unprovable truths which define new axioms in ever extending axiom system allowing mechanical deduction of their consequences by using mechanical logical deduction. The non-mechanical aspect of mathematics is discovery of an axiom, an unprovable conjecture.

Godel's incompleteness theorem in this framework is much what I said about continual recreation of the Universe. There is no last quantum jump giving rise to complete enlightment.

The endless re-creation allows also to circumvent the basic paradox which one ends up when one asks what it really means that I am conscious that I have certain conscious experience: such as seeing red which seems to be the only sensory percept that philosophers have;-). An infinite regression is the outcome.

The solution of the paradox is simple. I can be conscious that I *had* certain conscious experience. Very important distinction. The evolution of consciousness replaces infinite regression leading to a paradox.

If this sequence of re-creations is iteration like process, it leads asymptotically to fractal structures as fixed points of iteration. You probably know how simple iteration algorithms produce very complex fractals occurring in Nature as fixed point of iteration. A fractal representing forest is one example.


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