Wednesday, February 29, 2012

Progress in number theoretic vision about TGD

During last weeks I have been writing a new chapter Quantum Adeles. The key idea is the generalization of p-adic number fields to their quantum counterparts and they key problem is what quantum p-adics and quantum adeles mean. Second key question is how these notions relate to various key ideas of quantum TGD proper. The new chapter gives the details: here I just list the basic ideas and results.

What quantum p-adics and quantum adeles really are?

What quantum p-adics are?

The first guess is that one obtains quantum p-adics from p-adic integers by decomposing them to products of primes l first and after then expressing the primes l in all possible manners as power series of p by allowing the coefficients to be also larger than p but containing only prime factors p1<p. In the decomposition of coefficients to primes p1<p these primes are replaced with quantum primes assignable to p.

One could pose the additional condition that coefficients are smaller than pN and decompose to products of primes l<pN mapped to quantum primes assigned with q= exp(i2π/pN). The interpretation would be in terms of pinary cutoff. For N=1 one would obtain the counterpart of p-adic numbers. For N>1 this correspondence assigns to ordinary p-adic integer larger number of quantum p-adic integers and one can define a natural projection to the ordinary p-adic integer and its direct quantum counterpart with coefficients ak<p in pinary expansion so that a covering space of p-adics results. One expects also that it is possible to assign what one could call quantum Galois group to this covering and the crazy guess is that it is isomorphich with the Absolute Galois Group defined as Galois group for algebraic numbers as extension of rationals.

One must admit that the details are not fully clear yet here. For instance, one can consider quantum p-adics defined in power series of pN with coefficients an<pN and expressed as products of quantum primes l<pN. Even in the case that only N=1 option works the work has left to surprisingly detailed understanding of the relationship between different pieces of TGD.

This step is however not enough for quantum p-adics.

  1. The first additional idea is that one replaces p-adic integers with wave functions in the covering spaces associated with the prime factors l of integers n. This delocalization would give a genuine content for the attribute "quantum" as it does in the case of electron in hydrogen atom.

    The natural physical interpretation for these wave functions would be as cognitive representations of the quantum states in matter sector so that momentum, spin and various internal quantum numbers would find cognitive representation in quantum Galois degrees of freedom.

    One could talk about self-reference: the unseen internal degrees of freedom associated with p-adic integers would make it possible to represent physical information. Also the ratios of infinite primes reducing to unity give rise to similar but infinite-dimensional number theoretical anatomy of real numbers and leads to what I call Brahman=Atman identity.

  2. Second additional idea is to replace numbers with sequences of arithmetic operations that is quantum sum +q and quantum product ×q represented as fundamental 3-vertices and to formulate basic laws of arithmetics as symmetries of these vertices give rise to additional selection rules from natural additional symmetry conditions. These sequences of arithmetics with sets of integers as inputs and outputs are analogous to Feynman diagrams and the factorization of integers to primes has the decomposition of braid to braid strands as a direct correlate. One can also group incoming integers to sub-groups and the hierarchy of infinite primes describes this grouping.

A beautiful physical interpretation for the number theoretic Feynman diagrams emerges.

  1. The decomposition of integers m and n of a quantum rational m/n to products of primes l correspond to the decomposition of two braids to braid strands labeled by primes l. TGD predicts both time-like and space-like braids having their ends at partonic 2-surfaces. These two kinds of braids would naturally correspond to the two co-prime integers defining quantum rational m/n.

  2. The two basic vertices +q and ×q correspond to the fusion vertex for stringy diagrams and 3-vertex for Feynman diagrams: both vertices have TGD counterparts and correspond at Hilbert space level direct sum and tensor product. Note that the TGD inspired interpretation of +q (direct sum) is different from string model interpretation (tensor product). The incoming and outgoing integers in the Feynman diagram corresponds to Hilbert space dimensions and the decomposition to prime factors corresponds to the decomposition of Hilbert space to prime Hilbert spaces as tensor factors.

  3. Ordinary arithmetic operations have interpretation as tensor product and direct sum and one can formulate associativity, commutativity, and distributivity as well as product and sum as conditions on Feynman diagrams. These conditions imply that all loops can be transformed away by basic moves so that diagram reduces to a diagram obtained by fusing only sum and product to initial state to produce single line which decays to outgoing states by co-sum and co-product. Also the incoming lines attaching to same line can be permuted and permutation can only induce a phase factor. The conjecture that these rules hold true also for the generalized Feynman diagrams is obviously extremely powerful and consistent with the picture provided by zero energy ontology. Also connection with twistor approach is suggestive.

  4. Quantum adeles for ordinary rationals can be defined as Cartesian products of quantum p-adics and of reals or rationals. For algebraic extensions of rationals similar definition applies but allowing only those p-adic primes which do not split to a product of primes or the extension. Number theoretic evolution means increasing dimension for the algebraic extension of rationals and this means that increasing number of p-adic primes drops from the adele. This means a selective pressure under which only the fittest p-adic primes survive. The basic question is why Mersenne primes and some primes near powers of two are survivors.

The connection with infinite primes

A beautiful connection with the hierarchy of infinite primes emerges.

  1. The simplest infinite primes at the lowest level of hierarchy define two integers having no common prime divisors and thus defining a rational number having interpretation in terms of time-like and space-like braids characterized by co-prime integers.

  2. Infinite primes at the lowest level code for algebraic extensions of rationals so that the infinite primes which are survivors in the evolution dictate what p-adic primes manage to avoid splitting. Infinite primes coding for algebraic extensions have interpretation as bound states and the most stable bound states and p-adic primes able to resist corresponding splitting pressures survive.

    At the n:th level of the hierarchy of infinite primes correspond to monic polynomials of n variables constructed from prime polymomials of n-1 variables constructed from.... The polynomials of single variable are in 1-1 correspondence with ordered collections of n rationals. This collection corresponds to n pairs of time-like and space-like braids. Thus infinite primes code for collections of lower level infinite primes coding for... and eventually everything boils down to collections rational coefficients for monic polynomials coding for infinite primes at the lowest level of the hierarchy. In generalized Feynman diagrams this would correspond to groups of groups of .... of groups of integers of incoming and outgoing lines.

  3. The physical interpretation is in terms of pairs time-like and space-like braids having ends at partonic 2-surfaces with strands labelled by primes and defining as their product integer: the rational is the ratio of these integers. From these basic braids one can form collections of braid pairs labelled by infinite primes at the second level of hierarchy, and so on and a beautiful connection with the earlier vision about infinite primes as coders of infinite hierarchy of braids of braids of... emerges. Space-like and time-like braids playing key role in generalized Feynman diagrams and representing rationals supporting the interpretation of generalized Feynman diagrams as arithmetic Feynman diagrams. The connection with many-sheeted space-time in which sheets containing smaller sheet define higher level particles, emerges too.

  4. Number theoretic dynamics for ×q conserves the total numbers of prime factors so that one can either talk about infinite number of conserved number theoretic momenta coming as multiples of log(p), p prime or of particle numbers assignable to primes p: pn corresponds to n-boson state and finite parts of infinite primes correspond to states with fermion number one for each prime and arbitrary boson number. The infinite parts of infinite primes correspond to fermion number zero in each mode. The two braids could also correspond to braid strands with fermion number 0 and 1. The bosonic and fermionic excitations would naturally correspond the generators of super-conformal algebras assignable to light-like and space-like 3-surfaces.

The interpretation of integers representing particles a Hilbert space dimensions

In number theoretic dynamics particles are labeled by integers decomposing to primes interpreted as labels for braid strands. Both time-like and space-like braids appear. The interpretation of sum and product in terms of direct sum and tensor product implies that these integers must correspond to Hilbert space dimensions. Hilbert spaces indeed decompose to tensor product of prime-dimensional Hilbert spaces stable against further decomposition.

Second natural decomposition appearing in representation theory is into direct sums. This decomposition would take place for prime-dimensional Hilbert spaces with dimension l with dimensions anpn in the p-adic expansion. The replacement of an with quantum integer would mean decomposition of the summand to a tensor product of quantum Hilbert spaces with dimensions which are quantum primes and of pn-dimensional ordinary Hilbert space. This should relate to the finite measurement resolution.

×q vertex would correspond to tensor product and +q to direct sum with this interpretation. Tensor product automatically conserves the number theoretic multiplicative momentum defined by n in the sense that the outgoing Hilbert space is tensor product of incoming Hilbert spaces. For +q this conservation law is broken.

Connection with the hierarchy of Planck constants, dark matter hierarchy, and living matter

The obvious question concerns the interpretation of the Hilbert spaces assignable to braid strands. The hierarchy of Planck constants interpreted in terms of a hierarchy of phases behaving like dark matter suggests the answer here.

  1. The enormous vacuum degeneracy of Kähler action implies that the normal derivatives of imbedding space coordinates both at space-like 3 surfaces at the boundaries of CD and at light-like wormhole throats are many-valued functions of canonical momentum densities. Two directions are necessary by strong form of holography implying effective 2-dimensionality so that only partonic 2-surfaces and their tangent space data are needed instead of 3-surfaces. This implies that space-time surfaces can be regarded as surfaces in local singular coverings of the imbedding space. At partonic 2-surfaces the sheets of the coverings co-incide.

  2. By strong form of holography there are two integers characterizing the covering and the obvious interpretation is in terms of two integers characterizing infinite primes and time-like and space-like braids decomposing into braids labelled by primes. The braid labelled by prime would naturally correspond to a braid strand and its copies in l points of the covering. The state space defined by amplitudes in the n-fold covering would be n-dimensional and decompose into a tensor product of state spaces with prime dimension. These prime-dimensional state spaces would correspond to wave functions in prime-dimensional sub-coverings.

  3. Quantum primes are obtained as different sum decompositions of primes l and correspond direct sum decompositions of l-dimensional state space associated with braid defined by l-fold sub-covering. What suggests itself strongly is a symmetry breaking. This breaking would mean the geometric decomposition of l strands to subsets with numbers of elements coming proportional to powers pn of p. Could anpn in the expression of l as ∑ akpk correspond to a tensor product of an-dimensional space with finite field G(p,n)? Does this decomposition to state functions localized to sub-braids relate to symmetries and symmetry breaking somehow? Why an-dimensional Hilbert space would be replaced with a tensor product of quantum-p1-dimensional Hilbert spaces? The proper understanding of this issue is needed in order to have more rigorous formulation of quantum p-adics.

  4. Number theoretical dynamics would therefore relate directly to the hierarchy of Planck constants. This would also dictate what happens for Planck constants in the two vertices. There are two options.

    1. For ×q vertex the outgoing particle would have Planck constant, which is product of incoming Planck constants using ordinary Planck constant as unit. For +q vertex the Planck constant would be sum. This stringy vertex would lead to generation of particles with Planck constant larger than its minimum value. For ×q two incoming particles with ordinary Planck constant would give rise to a particle with ordinary Planck constant just as one would expect for ordinary Feynman diagrams.

    2. Another possible scenario is the one in which Planck constant is given by hbar/hbar0= n-1. In this case particles with ordinary Planck constant fuse to particles with ordinary Planck constant in both vertices.

    For both options the feed of particles with non-standard value of Planck constant to the system can lead to a fusion cascade leading to a generation of dark matter particles with very large value of Planck constant. Large Planck constant means macroscopic quantum phases assumed to be crucial in TGD inspired biology. The obvious proposal is that inanimate matter transforms to living and thus also to dark matter by this kind of phase transition in presence of feed of particles - say photons- with non-standard value of Planck constant.


The work with quantum p-adics and quantum adeles and generalization of number field concept to quantum number field in the framework of zero energy ontology has led to amazingly deep connections between p-adic physics as physics of cognition, infinite primes, hierarchy of Planck constants, vacuum degeneracy of Kähler action, generalized Feynman diagrams, and braids. The physics of life would rely crucially on p-adic physics of cognition. The optimistic inside me even insists that the basic mathematical structures of TGD are now rather well-understood. This fellow even uses the word "breakthrough" without blushing. I have of course continually admonished him for his reckless exaggerations but in vain.

The skeptic inside me continues to ask how this construction could fail. A possible Achilles heel relates to the detailed definition of the notion of quantum p-adics. For N=1 it reduces essentially to ordinary p-adic number field mapped to reals by quantum variant of canonical identification. Therefore most of the general picture survives even for N=1. What would be lost are wave functions in the space of quantum variants of a given prime and also the crazy conjecture that quantum Galois group is isomorphic to Absolute Galois Group.

For details see the new chapter Quantum Adeles of "Physics as Generalized Number Theory".


At 7:36 PM, Blogger hamed said...

Deep connections between cognition, infinite primes, hierarchy of Planck constants, braids and … oh! , that’s really amazing, I don’t know my small capacity of mind will allow to understand them or not at future. This takes a long time process :)
I have best wishes for you.

At 11:25 PM, Blogger Ulla said...

Ye, but it didn't came at once...

"This fellow even uses the word "breakthrough" without blushing. I have of course continually admonished him for his reckless exaggerations but in vain."


I wish I am capable to put some light on the path :)

I really like the thought that biology would guide physicists :)
Why do we always think life was created secondary to matter? It could be a parallell, or even be at first place, and then the really odd question would be "How can there be any matter that not react quantum mechanically too?" "What force matter to stay in a classic position?" So the odd thing would be classic ordinary non-living matter :-)

At 11:33 PM, Anonymous said...

Dear Hamed,

no hurry. I took 34 years for me;-). We have not talked much about the dark side of TGD;-). I have been working since 1990 or so with TGD inspired theory of consciousness and quantum biology and the number theoretic vision about TGD is partially due to developments in this branch of TGD.

At 11:48 PM, Anonymous said...

The attitude of colleagues to biology and consciousness looks very strange to me. A 10 line "argument" claiming that biology involves no new physics and everything above weak boson length scale is understood is thought to be good enough justification for forgetting these branches of science altogether. M-theorists (who does still remember them;-)?) went even further and understood everything above Planck scale and much more;-).

The basic attitude of physicists is rather imperialistic. Lubos once described this attitude excellently from the point of believer on it. Physicists are builders of empire, they are invaders which mercilessly subdue new words under their power.

Biology and consciousness are troublesome regions of the big empire but officially they are under the control.

At 11:59 PM, Blogger Ulla said...

Tying the knot has been associated with longer life...

Also females live longer, and one reason is the caregiving for the kids, and oxytocin, biologists think.

What would be the TGD interpretation of this? Entanglement that yields more than everything? Negative negentropy? A big lottery win? Or just a jail of free will?

Hamed, in my mind you do a wonderful job. I wish you success. The problems begin when you start talking about these ideas, it won't be easy, but there was a late (2004) thesis also in this field. arXiv:gr-qc/0409123

At 10:46 AM, Blogger hamed said...

To Ulla,
So thanks this is a good theses and topics about geometrodynamics of Wheeler and review on other theories related to it in this theses will be useful for me at future certainly. I wonder that why I didn’t see the theses before, maybe it is your ability to searching :)

To Matti,
Although now I am struggling with Riemannian geometry and relation with Einstein equations but every week some feedback (as overview) to what I will learn in future is useful for me.
I have some obscure view about the role of Poincare in TGD. I wrote my misunderstanding of it, here:
In GRT there is Poincare invariance only locally, but in TGD one can have a global Poincare invariance at the level of entire WCW and local one at the level of classical space-time.
TGD speaks about two super-conformal symmetries that lead to union of symmetric spaces G/H. Each of symmetric spaces G/H has exact Poincare invariance. therefore at each of these spaces there is conservation energy and momentum and this leads to exact Poincare invariance at the level of entire H=M4*CP2.
In the Space of 3-surfaces in TGD each of 3-surfaces is a symmetric space and is in correspondence to each of symmetric spaces.
Classical space time in TGD as X4(X3) is like space-time of GRT and curvature is not constant for it and one can have only Poincare invariance locally.
Only classical TGD cannot solve problem of Poincare invariance and quantum TGD is needed? What is my misunderstanding in these sentences?

I feel my writing of English had improved in these weeks only because of these questions ;-) before these weeks it had been took time entire a morning to write a comment to you! Practice in writing in this manner was very useful for me.

At 1:15 PM, Blogger Ulla said...

Brian Josephson: Biological Observer-Participation and Wheeler’s
‘Law without Law’ okt 2011

...indicates that something analogous to cognitive development (including cultural development, assuming that cognitive development, in a social system, provides a basis for cultural emergence) can occur in a wider context, including that of our postulated primordial system....

A comment: Geometrodynamics is simply another name for general relativity. See, e.g., Misner, Thorne, and Wheeler's massive tome Gravitation from the 70's, where the two terms are used interchangeably.

But TGD is more like GR^2? (With Dark TGD?)

It seems they are not so far away after all?

I have written about the link on my blog where also a man named Prastaro is linked. He does much geometrodynamics, but his frame is a little otherwise. It can however be of good use?

At 1:53 PM, Blogger Ulla said...
On “law without law”
David Ritz Finkelstein

An answer to Josephson

At 10:00 PM, Anonymous said...

To Ulla:

Wheeler's Geometrodynamics cannot be identified with General Relativity. The infinite-D space of 3-metrics is the basic object and the dream is to quantize gravitation in this geometric framework generalizing Einstein's geometrization program. This is one of the deep ideas of Wheeler.

At least following two basic problems plague this approach.

The first problem is that one looses time: space-times are what we want. How to make 3-D of geometrodynamics to 4-D of general relativity? Semiclassical approximation to postulated path integral over 4-geometries is the obvious approach but has formidable mathematical difficulties.

[The mathematical non-existence of the path integral is quite general problem. QFT colleagues have done their best to forget this. Pretend that their is no problem when problem is too difficult: this has been the strategy of modern mainstream theoretical physics and guaranteed that nothing new has emerged for four decades;-).]

Second problem is that one does not obtain fermions. Fermions are the problem of classical general relativity too: space-time need not allow spin structure at all so that one cannot talk about spinor fields. Also this problem is much more general and plagues also string models and M-theory, would have been excellent hint that space-times must be replaced with 4-surfaces and spinor structure with induced spinor structure, has been put under the rug.

3-metrics are replaced with 3-surfaces in TGD framework. This solves both basic problems of geometrodynamics. WCW assigns to 3-surfaces space-time surfaces as analogs of Bohr orbits and classical physics becomes part of quantum physics. The problems with fermions and spin are circumvented via induced spinor structure. One fruit of labor is geometrization of fermionic statistics in terms of spinors of WCW.

To be continued....

At 10:08 PM, Anonymous said...

Continuation to Ulla:

Observer participancy is another very deep idea if Wheeler. Delayed choice experiment in which one changes geometric past is inspired by this idea.

Skeptic would react by saying that before we can talk about observer participancy we must have a physical definition for observer: we do not. Optimistic skeptic might try to imagine what this definition might be on basis of existing and maybe some new ideas. The notion of self is TGD inspired attempt to meet the challenge.

Evolution as a sequence of quantum jumps recreating the Universe repeatedly would realize observer-participancy in TGD Universe.

Most of us speak about cognitive, social, and cultural developments as something self evident. But theoretical physicist does not use these words. Very many words of biology and neuroscience are absent from his vocabulary: behavior, function, goal, homeostasis, punishments and rewards, evolution...: everything relating to intentionality, goal directedness, values is absent .., The brutal reason is that the existing mathematical tools do not allow even attempt to define these notions. New mathematics and new concepts are needed.

There is of course the easy way out: self-deception which is as easy as cheating the innocent laymen. Just say that all that is is nothing but dance of quarks and consciousness is illusion and life is nothing but complexity!

This is why I am talking about physics as generalized number theory, p-adic physics, infinite primes, hierarchy of Planck constants, etc.... I am observer wanting to participate the expansion of our understanding about the white regions of the map;-).

At 10:36 PM, Anonymous said...

Dear Hamed,

thank you for asking about Poincare invariance. This is a difficult subject although it looked trivial to me for 34 years ago;-). I will consider it first at space-time and imbedding space level.

a) Symmetries of space-time are identified usually as isometries. Distances between points are preserved and also angles between vectors. Everything related to size is preserved.

b) In General relativity isometries act as isometries of *space-time* and move space-time points. Problem: for curved space-times Poincare symmetries are lost. For Scwartschild metric for instance one has only rotations and time translations as symmetries. Translations are lost and this means that one cannot define the notion of 3-momentum as conserved Noether charge.

c) In TGD isometries act as isometries of the 8-D *imbedding space* (as opposite to space-time in GRT) and move the entire 3-surface or space-time. This is the big difference and tsolves the problems related to the loss of Poincare invariance. Space-time surfaces can be curved and can have even Euclidian signature of induced metric. The *induction of metric* is of course also a central notion.

Talking about space-times instead of space-time makes sense in quantum gravity where quantum states can be regarded as quantum superpositions of space-time surfaces so that the action of Poincare on quantum state is well-defined.

This idea looks good but in ZEO one encounters what looks like a problem, maybe more than a minor technical detail.

a) In ZEO one effectively replaces imbedding space M^4xCP_2 with CDxCP_2. CD is the intersection of future and past directed light-cones. Isn't this in conflict with Poincare invariance in global sense. For instance, translations for points of CD lead out of it near its boundaries? Is this lethal?

*Sloppy reaction: One might say that this problem is purely academic since only infinitesimal translations are needed to defined four-momenta. One has local Poincare invariance and this is enough. Certainly it is enough. One obtains local Noether charges and this invariance can be only broken by boundary conditions at the boundaries of CD. For instance, translation of space-time surface need not be anymore a preferred extremal of Kaehler action.

*Less sloppy reaction: Poincare symmetries are actually replaced with their local variants analogous to Kac Moody symmetries. The translation depends on the point of space-time surface and one can combine constant translation with space-time dependent translation which is such that it compensates the constant translation about boundaries of CD. Generalization of Kac-Moody type symmetries made possible by conformal invariance would be the way out of the problem at deeper level.

I could be also worry about non-conservation of gravitational momentum as opposed to conservation of inertial momentum in relation to Equivalence Principle but let us leave it to some other time;-)

I will continue about G/H etc in the next comment....

At 11:24 PM, Anonymous said...

Dear Hamed,

you make excellent questions with easily comprehensible English;-). Writing is very powerful tool of thinking: seeing one's own thoughts caught on paper as helpless victims of thinker's skeptic criticism is very effective feedback;-). Some people write mathematical formulas: this is one form of feedback. For me writing of less formal text is the feedback. It is amazing how fantastic ideas pop up when one just writes out what one does not understand!;-)

a) About G/H I more or less agree. Different G/H:s correspond to different values of zero modes not appearing in the metric of WCW as differentials but only as external parameters. The information about zero modes is partially coded by the values of the induced Kahler form at the partonic 2-surfaces. One can say that Kaehler E and B at partonic 2-surfaces are purely classical variables which do not quantum fluctuate. What does this mean is an interesting question in itself which I have not pondered.

b) The statement that 3-surfaces are symmetric spaces is wrong. Symmetric spaces are extremely restricted variety of spaces. They are basically coset spaces of type G/H for Lie groups. The only compact 3-D symmetric spaces would be SU(2) (3-D sphere S^3), SO(3) (S^3 with antipodal points identified). Very few others if any. WCW would contain very few points!

At 11:34 PM, Anonymous said...

Dear Hamed,

you asked also about 3-surfaces.

a) Without any further constraints 3-surfaces would be just intersections of space-time surfaces with boundary of CDxCP_2. Any 3-surface will do. One must assign to 3-surface a 4-surface to realize 4-D GCI: this is preferred extremal of Kaehler action whose value defines Kaehler function of WCW defining Kaehler metric of WCW Once one knows the 3-surfaces at the ends of CD, space-time surface as analog of Bohr orbit is fixed (forgetting now delicacies due to the failure of strict determinism). This is holography implied by GCI alone rather than postulated separately. This was my view for a long time.

b) There are however further constraints and they come from strong form of GCI and ZEO: these I have discovered during the last decade. Strong form of GCI says that both space-like 3-surfaces at the ends of space-time surface and light-like 3-surfaces code for the same physics. The only manner to realize this double coding is that the intersections of these two kinds of 3-surfaces do the coding. This implies effective 2-dimensionality stating that the partonic 2-surfaces and their tangent space data (briefly 2-data) code for physics. This means also strong form of holography. All follows from GCI which shows how incredibly powerful Einstein's postulates become in sub-manifold gravity.

c) Partonic 2-surfaces are defined as the intersections of the space-like 3 surfaces at ends of space-time surface with the light-like 3-D wormhole throats. Partonic 2-surfaces alone are not enough: if they were TGD would reduce to string model. Also tangent space data is needed. There is also a hierarchy of scales since strict determinism for Kaehler action fails and preferred extremal must be specified by giving CD plus sub-CDs plus sub-sub-CDs.... and specifying form them.

d) The building of preferred extremals would be rather practical business;-). Forming first a roughest frameset by giving 2-data at boundaries of CD. Then same for sub-CD:s and then for sub-sub-CDs and stopping when measurement resolution is satisfactory. After this one should construct numerically or otherwise space-time surfaces for all this CDs.

Classical space-time surface is preferrred extremal of Kahler action. The field equations defining it can be defined by varying Kahler action and "preferred" selects on the solutions of field equations when data about boundaries of CDs are given. The field equations are derived in "Basic Extremals of Kaehler action" at

Here also the known basic solutions of field equations deduced using symmetry arguments are discussed. A good future exercise would be the derivation of the field equations in general form from Kaehler action by varying it with respect to induced metric and Kahler form.

You asked also about space-time surfaces.

a) About X^4(X^3) you are right in the sense that Poincare invariances as isometries of X^4(X^3) are not symmetries. They act however as symmetries moving the partonic 2-surfaces at the ends of space-time surface and give rise to new space-time surface. This is key difference between TGD and GRT.

b) You are right when saying that classical TGD alone does not solve the problems related to symmetries. The very assumption that Poincare transformations affect entire space-time surface rather than only moving points along space-time surface and leaving it invariant as whole is inconsistent with classical gravity assuming fixed space-time. One must speak about quantum superpositions of space-times or strictly speaking- of partonic 2-surfaces and tangent space data- from beginning. For instance, quantum numbers like quark color and also momentum quantum numbers would not be possible without these quantum superpositions! In this sense TGD Universe is quantum multiverse.

At 11:32 AM, Blogger Ulla said...

The comment on Wheelers geometrodynamics was from Sarfattis FB, after I quoted there from wikipedia (Wheelers) about geometrodynamics. I have not read the 'bible' of gravitation. I was warned it was too heavy for me.

I wanted to test the statement, which indeed seemed peculiar. You talked of TGD as GR^2 about a year ago.

At 3:11 AM, Blogger Ulla said...

The wikipedia page about geometrodynamics is a bit unclear on this too. The quantum geometrodynamics link did not work, but look what I found when I searched it. It is hardly linked to time and QG.

An author named Claus Kiefer is doing interesting works, among them on geometrodynamics and quantum gravity

Cosmological constant as result of decoherence. This means non-commutative geometry?

Look: Does time exist in quantum gravity? Claus Kiefer.
Comments: 10 pages, second prize of the FQXi "The Nature of Time" essay contest

An earlier article (Adrian P. Gentle, Nathan D. George, Arkady Kheyfets, Warner A. Millerfrom 2004; Constraints in quantum geometrodynamics,

And about time and geometrodynamics, by the same authors

A geometric construction of the Riemann scalar curvature in Regge calculus. Jonathan R. McDonald, Warner A. Miller

A Discrete Representation of Einstein's Geometric Theory of Gravitation: The Fundamental Role of Dual Tessellations in Regge Calculus

Quantum Geometrodynamics of the Bianchi IX cosmological model
Arkady Kheyfets, Warner A. Miller, Ruslan Vaulin 2006

and from 1995,
Quantum Geometrodynamics I: Quantum-Driven Many-Fingered Time
Arkady Kheyfets, Warner A. Miller

All actually published.

At 3:12 AM, Blogger Ulla said...

hard linked , a typo :)

At 3:44 AM, Blogger Ulla said...

Isham was mentioned as one of the modern geometrodynamists on wikipedia. Look, he won the Dirac medal last year.

I get the strong feeling that Wikipedia tries to silence this all. Why is it not mentioned? And all these modern works?

At 3:09 AM, Blogger Ulla said...

How does the models differ? What is your opinion?

He talks of mapping of purely mental processes on mathematical spaces. We present various arguments - philosophic, mathematical, information, and neurophysiological - in favor of the p-adic model of mental space. p-adic spaces have structures of hierarchic trees and in our model such a tree hierarchy is considered as an image of neuronal hierarchy.


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