Wednesday, March 23, 2011

Summary about breakthrough in the construction of U-matrix and understanding of twistorialization

During last few days a dramatic development in the understanding of the notions of U-matrix, M-matrix and S-matrix- a trinity of matrices replacing in zero energy ontology the notion of S-matrix of positive energy ontology and allowing fusion of thermodynamics with quantum theory. Also twistorialization reduces to pure group theory-albeit infinite-dimensional: zero energy states define a Yangian algebra generalizing that of Grassmannian twistor approach. For this reason I feel it as appropriate to add a further posting trying to give bird's eye of view about the ideas since previous posting is rather long and detailed and reader easily loses the thread.

  1. The realization that the hermitian square roots of density matrices form infinite-D unitary algebra and that their commutativity with universal S-matrix implies that zero energy states define the generalization of Kac-Moody algebra became only after I had realized the possibility to construct U-matrix. The Hermitian sub-Lie-algebra commuting with S is large: for SU(N) it would correspond to SU(N-1)× U(1). It is this observation which reduces the construction of U-matrix (or matrices if they form algebra) to that for S is expected to correspond directly to the ordinary S-matrix. A possible interpretation of the algebra of U-matrices is in terms of scales of CDs coming as positive integer powers of two. Another possibility more in line with the usual interpretation of S-matrix as time evolution operator is that scales of CDs come as integers and these integers correspond to powers of S.

    What is so fascinating is that zero energy states themselves define the symmetry algebra of the theory and that this algebra can be interpreted as a generalization of Yangian responsible for the successes of Grassmannian twistor approach by replacing finite-dimensional conformal group of Minkowski space with infinite-dimensional super-conformal algebras associated with partonic 2-surfaces in accordance with the replacement of point-like particles with surfaces. The basic characteristic of Yangian algebra is the multilocality of its generators and zero energy states are indeed multilocal since they involve partonic surfaces at both light-like boundaries of CD. Quantum TGD reduces to pure group theory! Note only states but also dynamics is coded completely by symmetries since M-matrices code for quantum dynamics! This powerful aspect of zero energy ontology I have not realized before.

  2. In ordinary QFT Feynman diagrams are purely algebraic objects. In TGD framework they reduce to space-time topology and geometry with Euclidian regions of space-time surfaces having interpretation as generalized Feynman diagrams. At the vertices of generalized Feynman diagrams in coming partonic 2-surfaces meet just like in ordinary Feynman diagrams which means deep difference from string theory. A more general assumption is that entire 4-D lines of generalized Feynman diagram meet at vertices. This could apply to the Euclidian regions only.

    There is also a second kind of branching involved with the hierarchy of Planck constants. In Minkowskian regions similar meeting would take place for the branches of space-time sheets with same values of canonical momentum densities of Kähler action at the ends of CDs and have interpretation in terms of fractionization and hierarchy of Planck constants. The value of Planck constant for single branch would be effectively and integer multiple of the ordinary one. For the entire multi-sheeted structure describable naturally in terms of singular covering space of M4× CP2 it would be just the ordinary value.

  3. Zero energy ontology with massless external wormhole wormholes implies as such twistorialization of the theory although external wormhole momenta must be assumed to be massive bounds states of massless throats (see this). This also guarantees exact Yangian symmetry and the absence of IR divergences. If also virtual wormhole throats are massless, twistorialization takes place in strong sense. This is possible only in zero energy ontology and accepting the identification of wormhole throats as basic building blocks of particles.

  4. The notion of a bosonic emergence means that bosonic propagators emerge as radiative loops for wormhole contacts. The emergence generalizes to all states associated with wormhole contacts and also to flux tubes having wormhole contacts at their ends. What is nice that coupling constants emerge as normalization factors of propagators. Note that for single wormhole throat as opposed to wormhole contact having two throats bosonic propagator would result as a product of two collinear fermionic propagators and have the standard form. For states with higher total number of fermions and anti-fermions the propagator of wormhole throat behaves as pn, n>2. Here however p is replaced with what I call pseudo-momentum.

  5. Number theoretical universality suggest that at given level (CD) only finite sum of diagrams appears: otherwise there is a danger that one obtains sum of rational functions which is not rational anymore. This gives strong constraints on generalized Feynman diagrams at the lowest level of the hierarchy. This follows naturally if twistor diagrams are identified as sums of Feynman diagrams which are irreducible in the sense that they do not represent two subsequence scatterings. Only these diagrams contribute to twistor diagram and the number of these diagrams is finite if all particles have small mass (even photon which would eat the remaining Higgs component).

  6. Category theoretical approach to TGD based on planar operad proposed for few years ago fits nicely with the twistorial construction of amplitudes interpreting radiative corrections in terms of CDs within CDs picture. The generalized Feynman diagrams with radiative corrections define the analog of planar operad with disk containing within itself disks containing.... replaced with causal diamond containing causal diamonds containing....

To make this more concrete it deserves to describe TGD counterparts for the recursion formula and duality between descriptions using twistors and momentum twistors.

  1. The great victory of twistor approach is the recursion formula for the amplitudes (see also the representation in TGD framework) applying to all planar diagrams of N=4 SYM becoming an exact formula at the large N limit for gauge group SU(N). In the recent case the infinite-dimensional character of the Yangian symmetry algebra of S-matrix could be correlate for large N limit so that the planar limit should make sense. Also the fact that string worlds sheets are an essential aspect of TGD approach suggests that stringy picture deduced by t'Hooft for gauge theories at this limit implies planarity. What is relevant in the recent case is the general structure of the reduction formula, not the details which as such are of course extremely interesting also in TGD framework since Grassmannian amplitudes are claimed to provide a universal representation of Yangian invariants.

    The recursive formula expresses scattering amplitude with n external particles with k negative helicities up to l loops is expressible as a sum of two terms. The first term-referred to as classical contribution- involves a fusion of twistor amplitudes with smaller number of particles and with the number of loops not larger than l by a procedure used already for tree diagrams. Second term - called quantum contribution- involves l loops and is irreducible in the sense that it is not expressible as a fusion of lower amplitudes and is obtained from n+2 particle by a process eliminating two particles. The identification of the TGD counterparts of these terms is obvious. The "classical" term corresponds to the proposed fusion of the lower level amplitudes associated with polygons for sub-CDs. The "quantum" term corresponds to the contribution appearing at the level of CD itself and involves genuine loops in Feynman sense but only a finite number of them.

    Since zero energy states correspond to generators of Yangian algebra or rather- its Kac-Moody variant with integer power of phase factor identified as integer power of S, the recursion formula might allow an interpretation as a direct counterpart for the recursive definition of Yangian algebra in terms of relations allowing the construction of generators labeled by non-negative integers.

  2. One of the fascinating findings of twistor Grassmannian approach is that conformal invariance and its dual correspond in twistor approach to descriptions in terms of twistors in ordinary Minkowski space by starting from Feynman diagrams and in terms of momentum twistors in its dual by starting from Wilson loops. Also this duality has counterpart in TGD.

    String world sheets are an essential part of quantum TGD and the translation of Witten's work with knots to TGD context led to a precise identification of string world sheets and a deep connection between TGD and the theory of knots, braids, braid cobordisms, and 2-knots emerges (see this).

    Amusingly, the basic idea of this connection emerged from the model of DNA as topological quantum computer developed for few years ago. The braiding defining the quantum computation is time-like and can be illustrated using dance metaphor: the world lines of dancers define the running topological computation program. If you connect the feet of dancers to a wall with threads (dancers are lipids at cell membrane forming 2-D liquid, wall is represented by DNA nucleotide sequence, and threads are magnetic flux tubes), the threads entangle during dance and give rise to a space-like braiding and code the computer program to memory: a fundamental mechanism of memory. These braidings are clearly dual and this duality relates closely to the duality just to the duality between Feynman graphs and Wilson loops! The time evolution of this space-like braiding defines braid cobordism and also a 2-knot.

    The natural implication of strong form of holography made possible by preferred extremal (Bohr orbit in generalized sense) property of space-time surfaces is that the descriptions in terms of string world sheets and partonic 2-surfaces are dual. The twistorial representation of this duality is as the duality of descriptions in terms of Feynman diagrammatics in ordinary space-time and Wilson sheets-rather than loops- in the dual space-time assigned with region momenta.

To conclude, it seems that all the basic visions about quantum TGD combine to single coherent whole! Maybe the long march is in some sense over!

For background see the chapter Construction of Quantum Theory: M-matrix of "Towards M-Matrix". The recent results are summarized at the end of the chapter. I added to my homepage also an article summarizing the recent progress.

By the way, there is excellent video of a lecture by Nima Arkani-Hamed to mathematicians with title Grassmannians, Polytopes and Quantum Field Theory. If you identify yourself as a theoretician do not miss it! Just the opportunity to experience Nima's enthusiasm, joy, and energy convinces that theoretical physics can also be an exploration and adventure.


At 11:59 PM, Blogger Ulla said...

OH, my, you actually found it? Congrat.

This is the unification point of algebras and reals, the back of the BigBook giving LIFE, the oscillation point between entropy and negentropy, a world of dissipation and symmetry breaking?

So Higgs is completely transformed. The vacuum energy is what?

"To conclude, it seems that all the basic visions about quantum TGD combine to single coherent whole! Maybe the long march is in some sense over!"

I have waited for this day, eagerly. But CAN you really RELAX?
Now you have time?

At 12:10 AM, Blogger Ulla said...


At 12:28 AM, Blogger Ulla said...

I saw after it belonged to Intelligent Design. I am no fan of them, as neither you. Distotion of the truth is what they do, but this text hold some other properties too. I guess you see them?

At 1:32 AM, Blogger LEO VUYK said...

Hello Matti and Ulla,

I am enthusiastic about the participatory universe, but only if the observer relation is between entangled states of the same entities like quanta.
How could we achieve that?
My suggestion is: Mirror symmetry inside a multiverse.
As a weird consequence: The big bang produced only mirror symmetrical Quanta and vacuum structures which are entangled over huge distances, between opposing (anti) COPY Universes with the shape of a raspberry/ blackberry.

Best regards,

Leo Vuyk.

“This is another way of fathoming the role of the observer, the quantum discovery that the outcome of an experiment is affected, or even determined, when it is observed. “What we call reality, Wheeler writes coyly, “arises in the last analysis from the posing of yes-no questions. He adds, All things physical are information-theoretic in origin, and this is a participatory universe.—

At 2:24 AM, Anonymous Matti Pitkanen said...

To Ulla:

I have nothing against intelligent redesign. TGD Universe is recreating itself all the time and evolution which is usually regarded as a main argument against intelligent design (God creating the universe and then forgetting it) is a direct empirical proof for this.

To Leon:

I agree completely with Wheeler's view. Wheeler even discovered that past is changed in quantum jump although he did not yet formulate this rigorously as it is done in TGD framework where quantum jump takes place between entire time evolutions and one avoids the basic paradox of quantum measurement theory.

By the way, Wheeler regarded my thesis work as brilliant and that's why it became a thesis work! More than decade later and after I had written the first book about TGD two young finnish professors agreed that TGD fails to satisfy every possible imaginable criterion for what it is to be a scientific work. Curiously, these juvenile delinquents did not end up to jail: probably because in Finland also young professors are above law;-).

I do not see it as necessary to assume mirror symmetry in order to have entanglement in arbitrary long scales. The hierarchy of Planck constants makes possible long range entanglement and also time-like entanglement (zero energy ontology).

Also number theoretic entanglement entropy which can be negative, makes entanglement stable if one believes in negentropy maximization principle implying second law for ordinary entanglement entropy: this kind of entanglement might make living matter living.

At 2:49 AM, Blogger LEO VUYK said...

Mirror symmetry over long distances looks absurd indeed, we all should have entanglement relation with our own mirror self.

You bet it took some time for me to grasp that idea.
Hoewever still a logical simple idea?

At 3:23 AM, Blogger LEO VUYK said...

In addition about the shocking discovery of my mirror self, I made a simple image at my blog, based on the wheeler U image.

At 7:17 AM, Blogger Ulla said...

Leo, it is almost always shocking to discover the 'mirror' Self (magnetic body):)

Personality = entangled selves within the body, also in time, or eg, timeless (quantal)?
Collective mind = superposition of all minds or entanglement between minds (fish-stims or cultural evolution?)

The question what is consciousness is still unsolved, but if we take this picture of Matti, it is born in the moment of measurement, and then the anthropic principle is wrong. There are the rules of information only, which can be interpreted by entropy. So the asymmetry is a must. I think the consciousness must be reinterpreted as something much simpler than it has been until now.

If we take this picture even the question what make matter living may be solved, and then there is a quite NEW SITUATION.

In fact, this piece is the missing link to explain the LIVING MATTER. And it is almost already proved by LHC. Guess who opened my eyes? Lubos with his E8 :)

I am so thrilled with this.

At 8:34 AM, Blogger LEO VUYK said...


Yes asymmetry is needed in my opinion only in the asymmetry of the vacuum system.
A material universe could harbour a left handed rotating ( spiralling) vacuum lattice and an ant-matterial COPY universal vacuum could have the opposite way around.
see perhaps atoms of spacetime:


Leo Vuyk.

At 9:12 AM, Blogger Ulla said...

I think there is even galaxies spiralling the other way :)
NGC 4622

The asymmetry depends on the energylevel, and note the time mirror too :)

At 10:12 PM, Anonymous Matti Pitkanen said...

Zero energy states defined time-like analog of mirror symmetry. The positive and negative energy parts of zero energy state are not however identical and represent in standard positive energy ontology initial and final states of quantum jump having fractal quantum jumps within quantum jumps structure with a hierarchy of time scales characterizing the geometric duration assignable to it.

Almost mirror me as the me of the past which I see by remembering and with I commute by negative energy signals reflected back as positive energy signals (seeing in time direction) and with which I entangle in time-like manner.

At 2:19 AM, Blogger LEO VUYK said...

Still the idea of mirror Me is quite simple.
Th collapse of all wavefunctions in our material universe are entangled related to the COPY wavefunctions in our opposing anti-material universe.
see perhaps:

We are not unique! not in the center of the universe and our universe has at least One entangled Copy double

At 4:45 AM, Blogger Ulla said... or

Relative locality? What is this if not ZEO, or the CD-network. This is so OBVIOUS. Do as Einstein said and trust your senses.

Energy moves.

At 7:12 AM, Blogger LEO VUYK said...

I see no need to make the case more complex.
My only suggstion is that lots of problems with Cats and human consciousness can be solved by the introduction of one or more CPT symmetrical entangled Copy universes down to the quantum scale of particles and also the vacuum structure.
So local entanglement is supposed to be also entangled to the opposing copy universe.

At 1:55 AM, Blogger Ulla said...


I think you have messed up a little.
The cats and human difference is an hierarchial one. What you mean by symmetrical entangled copies is hard to know for me. Look at Vitiello's The Double, if it may help. He talks of ZEO.

Hierarchy, or creation of subsystems, is behind the difference between cats and humans, and in child - adult, body - parts of the body and their different consciousnesses. You take it to the particle-level, so you see also particles have a boundary or 'Self',and 'consciousness'.

As humans we have defined this consciousness wrongly. As the decoherence proceeds, we have to make choises, that is give away consciousness. Free will?

At 6:36 AM, Blogger LEO VUYK said...

Dear Ulla,
sorry for the delay.
However I messed up the mainstream interpretations on consciousness and free will that is right.
I found Vitiellos book on the internet, however I had not the time to study his ideas.
To go a bit more in detail however,
Indeed our conscious human responsibility is not the same as the collapse of a simple wave function but it is the end of more than one collapses in our multiple multiverse éntangled brains at huge distances.

Perhaps you will understand my point better by the next blog paper:

At 6:39 AM, Blogger Ulla said...

You can also look at this.
Nima Arkani-Hamed talks about Space-Time, Quantum Mechanics and Scattering Amplitudes.

Look at his enthusiasm when he discover TGD Universe, especially the Zero Energy Ontology.
He calls it "N-4 super Yang Mills gauge theory" for emergent spacetime and QM dichotomly, with infinitely many hidden symmetries, and one dichotomy is invisible (shows p-adic hierarchy with changing hbar).

Momentum conservation, parity invariant, k-plane<->kn-plane is similar= ZEO, but avery simple one, not at all as sophisticated as TGD yet.

Not string theory, nor Feynman diagrams, twistors, nor algebraic geometry alone but a momentum pointlike space :) with Wilson loops (Pythagoras geometry). Who is in the audience? Lee Smolin!

And enjoy his enthusiasm.

At 6:54 AM, Blogger Ulla said...

Sorry, Our posts met in the air.

Consciousness and responsibility is two different things too. In the latter is collective mind as a part.

Sorry, Matti, if I comment this much here, but you was silent. And this is so obvious. Everyone must be able to see.

At 1:40 AM, Blogger Ulla said...


Function does not follow form, but contrarily
Zero energy is the vacuum energy, not ZEO generally
CPT is not symmetrically broken?
Entanglement is outside time, usually

The idea with the U-figure is its asymmetry, you look at yourself, measure yourself. There is already two parts, body and magnetic body (mind)

'More consciousness' is the basically wrong assumption of human consciousness. 'Consciousness can only be lost' in the symmetry breaking (choice), but something new arises instead, intelligence and awareness.

Sorry to say this.

You can read my FQXI-essay, topic/938 to see my point.

I must point out my picture is a little different than Mattis. I use two selves, one bodily and one quantal.

At 3:42 AM, Blogger LEO VUYK said...

Thank you Ulla,
How can I get to FQXI papers?

in the meantime I translated my Vixra letters into an e-book, called;
Zero Point Energy Black Holes
published by

I will try to find FQXI papers.

At 3:58 AM, Blogger Ulla said...

At 12:40 AM, Blogger LEO VUYK said...

it did not come through sorry:

At 12:49 AM, Blogger LEO VUYK said...

Ok I made a mistake before , so I have to remember my last typing.

You proposal is that living Matter is in a mixed state between quantum and classical physics.

In my proposal is the quantum entanglement between mirror copy universes the origin of wavefunction collapse, but with a retardation system for living matter like DNA.
Our brains seem to be able to think our decissions over and over before we act.
Bhuddists argue that NOT doing is the best.
In my perspective this means that we have to abort much of our urges and intentions, and think it over to become responsible for our acting.
I Would ask attention for my proposal for an STP-ROM.
Solar Toroidal Read Only but also Adress memory:

At 1:30 AM, Blogger Ulla said...

The quantum entanglement between mirror copy universes the origin of wavefunction collapse, when decoherence happen. That is symmetry breaking, just replace the word 'universe' with 'self'. This means of course also that Multiverses cannot be excluded, but they are not interesting now..

A retardation of this symmetrybreaking? This is more entanglement than in normal matter. But in physics we have more decoherence, and exactly this decoherence is the trouble. Just to say it is not there doesn't solve anything.

DNA and nerves is more coherent, yes, but this is because of their structures favouring quantum entanglement, that is in connection with the information (entropy, reals) in our outer world. This happen only through some windows (the window-effect) and is algebra (p-adics). So you can say the outer field is the living (conscious) one and the body is just acting matter. This is why Matti says the magnetic body is the real Self. Every particle has a field.

The connection between the both worlds, bodily matter and outer field, is exactly what Matti now have found. This is the back of the BigBook of superimposed non-commutative matrices. Nima and Lee are far from it yet, but Nima maybe knows better what he is looking for.

These matrices are asymmetric together, through the back of the Book. They are of course behind all matter. This is the asymmetric supersymmetry.

This is also the reason why living matter may reveal much about the physics, because it has solved the problems of decoherence differently than ordinary matter.

Right, Matti? (I use different words than you, and maybe that is good :))

I think, of some reason you have not the right perspective. Too much of an architect? Try to think fields.

At 1:38 AM, Blogger Ulla said...

Look at this picture from art. It is schizo. They have often a paranormal sense, and can feel the magnetic body, and see it,like a 'fluid'. Here it is absent. This is what happen in symmetry breaking.

Maybe this makes you understand. Ordinary matter must be cooled down to avoid movements, but living matter must be flexible, reacting.

At 8:15 AM, Blogger LEO VUYK said...

Indeed an intriguing picture.

I needed time to think about your field suggestion to study, and in fact i did before:
photons seems to travel in my model in the form of Bunches of energetic oscillating Higgs inside the vacuum lattice or "spacetime atoms" (prof. Loll)
The size and form of these bunches seem to be related to polarisation and wavelength of the photon.
Atoms of spacetime:


Post a Comment

<< Home