Wednesday, July 25, 2012

Quantum criticality and electro-weak gauge symmetries

Quantum criticality is one of the basic guiding principles of Quantum TGD. What it means mathematically is however far from clear.

  1. What is obvious is that quantum criticality implies quantization of Kähler coupling strength as a mathematical analog of critical temperature so that the theory becomes mathematically unique if only single critical temperature is possible. Physically this means the presence of long range fluctuations characteristic for criticality and perhaps assignable to the effective hierarchy of Planck constants having explanation in terms of effective covering spaces of the imbedding space. This hierarchy follows from the vacuum degeneracy of Kähler action, which in turn implies 4-D spin-glass degeneracy. It is easy to interpret the degeneracy in terms of criticality.

  2. At more technical level one would expect criticality to corresponds deformations of a given preferred extremal defining a vanishing second variation of Kähler action. This is analogous to the vanishing of also second derivatives of potential function at extremum in certain directions so that the matrix defined by second derivatives does not have maximum rank. Entire hierarchy of criticalities is expected and a good finite-dimensional model is provided by the catastrophe theory of Thom. Cusp catastrophe is the simplest catastrophe one can think of, and here the folds of cusp where discontinuous jump occurs correspond to criticality with respect to one control variable and the tip to criticality with respect to both control variables.

  3. I have discussed what criticality could mean for modified Dirac action (see this) and claimed that it leads to the existence of additional conserved currents defined by the variations which do not affect the value of Kähler action. These arguments are far from being mathematically rigorous and the recent view about the solutions of the modified Dirac equation predicting that the spinor modes are restricted to 2-D string world sheets requires a modification of these arguments.
In the following these arguments are updated. The unexpected result is that critical deformations induce conformal scalings of the modified metric and electro-weak gauge transformations of the induced spinor connection at X2. Therefore holomorphy brings in the Kac-Moody symmetries associated with isometries of H (gravitation and color gauge group) and quantum criticality those associated with the holonomies of H (electro-weak-gauge group) as additional symmetries.

The variation of modes of the induced spinor field in a variation of space-time surface respecting the preferred extremal property

Consider first the variation of the induced spinor field in a variation of space-time surface respecting the preferred extremal property. The deformation must be such that the deformed modified Dirac operator D annihilates the modified mode. By writing explicitly the variation of the modified Dirac action (the action vanishes by modified Dirac equation) one obtains deformations and requiring its vanishing one obtains

δ Ψ=D-1(δ D)Ψ .

D-1 is the inverse of the modified Dirac operator defining the analog of Dirac propagator and δ D defines vertex completely analogous to γkδ Ak in gauge theory context. The functional integral over preferred extremals can be carried out perturbatively by expressing Δ D in terms of δ hk and one obtains stringy perturbation theory around X2 associated with the preferred extremal defining maximum of Kähler function in Euclidian region and extremum of Kähler action in Minkowskian region (stationary phase approximation).

What one obtains is stringy perturbation theory for calculating n-points functions for fermions at the ends of braid strands located at partonic 2-surfaces and representing intersections of string world sheets and partonic 2-surfaces at the light-like boundaries of CDs. δ D- or more precisely, its partial derivatives with respect to functional integration variables - appear atthe vertices located anywhere in the interior of X2 with outcoming fermions at braid ends. Bosonic propagators are replaced with correlation functions for δ hk. Fermionic propagator is defined by D-1.

After 35 years or hard work this provides for the first time a reasonably explicit formula for the N-point functions of fermions. This is enough since by bosonic emergence(se this) these N-point functions define the basic building blocks of the scattering amplitudes. Note that bosonic emergence states that bosons corresponds to wormhole contacts with fermion and antifermion at the opposite wormhole throats.

What critical modes could mean for the induced spinor fields?

What critical modes could mean for the induced spinor fields at string world sheets and partonic 2-surfaces. The problematic part seems to be the variation of the modified Dirac operator since it involves gradient. One cannot require that covariant derivative remains invariant since this would require that the components of the induced spinor connection remain invariant and this is quite too restrictive condition. Right handed neutrino solutions delocalized into entire X2 are however an exception since they have no electro-weak gauge couplings and in this case the condition is obvious: modified gamma matrices suffer a local scaling for critical deformations:

δ Γμ = Λ(x)Γμ .

This guarantees that the modified Dirac operator D is mapped to Λ D and still annihilates the modes of νR labelled by conformal weight, which thus remain unchanged.

What is the situation for the 2-D modes located at string world sheets? The condition is obvious. Ψ suffers an electro-weak gauge transformation as does also the induced spinor connection so that Dμ is not affected at all. Criticality condition states that the deformation of the space-time surfaces induces a conformal scaling of Γμ at X2, It might be possible to continue this conformal scaling of the entire space-time sheet but this might be not necessary and this would mean that all critical deformations induced conformal transformations of the effective metric of the space-time surface defined by {Γμ, Γν}=2 Gμν. Thus it seems that effective metric is indeed central concept (recall that if the conjectured quaternionic structure is associated with the effective metric, it might be possible to avoid problem related to the Minkowskian signature in an elegant manner).

Note that one can consider even more general action of critical deformation: the modes of the induced spinor field would be mixed together in the infinitesimal deformation besides infinitesimal electroweak gauge transformation, which is same for all modes. This would extend electroweak gauge symmetry. Modified Dirac equation holds true also for these deformations. One might wonder whether the conjecture dynamically generated gauge symmetries assignable to finite measurement resolution could be generated in this manner.

Thus the critical deformations would induce conformal scalings of the effective metric and dynamical electro-weak gauge transformations. Electro-weak gauge symmetry would be a dynamical symmetry restricted to string world sheets and partonic 2-surfaces rather than acting at the entire space-time surface. For 4-D delocalized right-handed neutrino modes the conformal scalings of the effective metric are analogous to the conformal transformations of M4 for N=4 SYMs. Also ordinary conformal symmetries of M4 could be present for string world sheets and could act as symmetries of generalized Feynman graphs since even virtual wormhole throats are massless. An interesting question is whether the conformal invariance associated with the effective metric is the analog of dual conformal invariance in N=4 theories.

Critical deformations of space-time surface are accompanied by conserved fermionic currents. By using standard Noetherian formulas one can write

Jμi= Ψbar Γμδi Ψ + δi ΨbarΓμΨ .

Here δ Ψi denotes derivative of the variation with respect to a group parameter labeled by i. Since δ Ψi reduces to an infinitesimal gauge transformation of Ψ induced by deformation, these currents are the analogs of gauge currents. The integrals of these currents along the braid strands at the ends of string world sheets define the analogs of gauge charges. The interpretation as Kac-Moody charges is also very attractive and I have proposed that the 2-D Hodge duals of gauge potentials could be identified as Kac-Moody currents. If so, the 2-D Hodge duals of J would define the quantum analogs of dynamical electro-weak gauge fields and Kac-Moody charge could be also seen as non-integral phase factor associated with the braid strand in Abelian approximation (the interpretation in terms of finite measurement resolution is discussed earlier).

One can also define super currents by replacing Ψbar or Ψ by a particular mode of the induced spinor field as well as c-number valued currents by performing the replacement for both Ψbar and Ψ. As expected, one obtains a super-conformal algebra with all modes of induced spinor fields acting as generators of super-symmetries restricted to 2-D surfaces. The number of the charges which do not annihilate physical states as also the effective number of fermionic modes could be finite and this would suggest that the integer N for the supersymmetry in question is finite. This would conform with the earlier proposal inspired by the notion of finite measurement resolution implying the replacement of the partonic 2-surfaces with collections of braid ends.

Note that Kac-Moody charges might be associated with "long" braid strands connecting different wormhole throats as well as short braid strands connecting opposite throats of wormhole contacts. Both kinds of charges would appear in the theory.

What is the interpretation of the critical deformations?

Critical deformations bring in an additional gauge symmetry. Certainly not all possible gauge transformations are induced by the deformations of preferred extremals and a good guess is that they correspond to holomorphic gauge group elements as in theories with Kac-Moody symmetry. What is the physical character of this dynamical gauge symmetry?

  1. Do the gauge charges vanish? Do they annihilate the physical states? Do only their positive energy parts annihilate the states so that one has a situation characteristic for the representation of Kac-Moody algebras. Or could some of these charges be analogous to the gauge charges associated with the constant gauge transformations in gauge theories and be therefore non-vanishing in the absence of confinement. Now one has electro-weak gauge charges and these should be non-vanishing. Can one assign them to deformations with a vanishing conformal weight and the remaining deformations to those with non-vanishing conformal weight and acting like Kac-Moody generators on the physical states?

  2. The simplest option is that the critical Kac-Moody charges/gauge charges with non-vanishing positive conformal weight annihilate the physical states. Critical degrees of freedom would not disappear but make their presence known via the states labelled by different gauge charges assignable to critical deformations with vanishing conformal weight. Note that constant gauge transformations can be said to break the gauge symmetry also in the ordinary gauge theories unless one has confinement.

  3. The hierarchy of quantum criticalities suggests however entire hierarchy of electro-weak Kac-Moody algebras. Does this mean a hierarchy of electro-weak symmetries breakings in which the number of Kac-Moody generators not annihilating the physical states gradually increases as also modes with a higher value of positive conformal weight fail to annihilate the physical state?

    The only manner to have a hierarchy of algebras is by assuming that only the generators satisfying n mod N=0 define the sub-Kac-Moody algebra annihilating the physical states so that the generators with n mod N≠ 0 would define the analogs of gauge charges. I have suggested for long time ago the relevance of kind of fractal hierarchy of Kac-Moody and Super-Virasoro algebras for TGD but failed to imagine any concrete realization.

    A stronger condition would be that the algebra reduces to a finite dimensional algebra in the sense that the actions of generators Qn and Qn+kN are identical. This would correspond to periodic boundary conditions in the space of conformal weights. The notion of finite measurement resolution suggests that the number of independent fermionic oscillator operators is proportional to the number of braid ends so that an effective reduction to a finite algebra is expected.

    Whatever the correct interpretation is, this would obviously refine the usual view about electro-weak symmetry breaking.

These arguments suggests the following overall view. The holomorphy of spinor modes gives rise to Kac-Moody algebra defined by isometries and includes besides Minkowskian generators associated with gravitation also SU(3) generators associated with color symmetries. Vanishing second variations in turn define electro-weak Kac-Moody type algebra.

Note that criticality suggests that one must perform functional integral over WCW by decomposing it to an integral over zero modes for which deformations of X4 induce only an electro-weak gauge transformation of the induced spinor field and to an integral over moduli corresponding to the remaining degrees of freedom.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.


At 4:40 PM, Blogger ThePeSla said...

Rene Thom himself published his work under the general term "Universal Topology"

But is this not essentially the possibilities of Riemannian sphere space embedded in Euclidean planes?

At 7:58 PM, Anonymous Matti Pitkanen said...

To Zephir:

Coming respected physicists is not about this kind of tricks. I am respected physicist already know. Everyone with good understanding of basics of physics any intellectually honest colleague realizes how deep my vision is [and that it is actually very detailed theory rather than mere vision].

Official attitude is totally different thing and dictated by motivations which have much more to do with keeping the funding that with the progress of science.

We are going through a very painful turning point in the development of theoretical physics and perhaps the most painful outcome of it that colleagues eventually must admit that I am right. So many broken dreams, so many careers are wasted to work with ideas that failed. And signs about this were visible already 3-4 decades ago. No one can claim that I would not have done my best to tell what goes wrong. One should not underestimate the pain and hatred that jealousy induces in an arrogant mind.

There are of course well-defined basic principles. The whole process during the last 35 years has been an attempt to identify and formulate in precise mathematical terms these principles. I have talked about these principles and their implications in these posting. This posting is just one example of particular principle: mathematical realization of quantum criticality and its newly found connection to electroweak gauge symmetries emerging as dynamical symmetries.

At 7:36 PM, Anonymous matti Pitkanen said...

Dear All,

Zephir is one of the nightmares of blogger. During first comments he looks like emotionally balanced individual but soon loses his emotional control completely and begins to produce hate talk. I was therefore forced to move away. I have been much luckier than most bloggers: Zephir has been the only trouble maker in this blog.

I have been forced to throw Zephir away once earlier when he started to fill the blog with is weird views about relativity accompanied by all possible personal insults that he had managed to invent. I first tried to proceed by showing his claims wrong but found this hopeless.

I am sorry if other comments were lost in this process.

Finally a personal message to Zephir: You are *NOT* well-come here. If you have a blog, express your hatred and envy in it.

At 11:35 PM, Anonymous poisson distribution examples said...

The electro-gauge is a topic which is in the course of engineering and it has many applications.

At 12:38 AM, Anonymous Matti Pitkanen said...

I think that there is a misunderstanding here. Electro-weak gauge symmetry has nothing to do with "gauge" as it is understood in engineering. I noticed that I do not really know the motivation for the terminology: it could relate to measurement.

Gauge potentials define the rule for how various fields are translated from point A to B in such a manner that they can be compared with the fields at B. Parallel translation of vector field in Riemannian geometry was the predecessor of this notion. In plane everything is trivial but already in the case of sphere situation changes.

At 1:29 AM, Blogger Ulla said...

Qritique should be allowed, but it was the intention, maybe. Basically maybe Zephir had some message, but he was unable to formulate it?

Mostly these kinds of requests are just aimed to confuse and hurt.

At 8:44 PM, Anonymous Matti Pitkanen said...

To Ulla:

Zephir's comments certainly carry a clear message. The message is that he is a crackpot who hates bitterly people with respect to whom he feels himself intellectually inferior. I have encountered quite enough of these madmen to learn that the attempts to discuss with them are waste of time.

At 1:01 AM, Blogger Ulla said...

Sorry to say this, Matti, but I don't like what you say :) For not so long ago you was yourself considered a 'crackpot' (what I hate that word, not part of true science) and you should have compassion with fellows of other 'less evolved' theories. As Kea. She lives in big powerty, I have been told. You know how that feels. To pick at others don't make yourself better in any way. Live and let live is better? But also self-respect and dignity is good. Not arrogancy. Now you show tendencies to the latter.

And you discussed with me. I am 'not evolved' if anybody is. But I have an intuitive understanding? Compassion? I tried to pay back, but it went wrong. Was it a waste of your time? Is it a waste of time for me? No. Look at you now and then and compare. The only thing needed was someone who dared to speak in favour of you, and an outsider like me was good? I have no prestige to loose. I am not afraid. I can defend. But it may miscredit you, and I have asked many times if it is good, without answer. And I still need your forgiveness.

At 10:49 PM, Anonymous Matti Pitkanen said...

To Ulla:

Crackpotness and anticrackpotness are very real phenomena. Also labeling of those behind competing theories with crackpot label is a real phenomenon. This is avoided if all comments represented publicly are *about contents*. I have myself followed this rule: also in case of Zephir for a couple of years ago when I had to tell him that he is not well-come here I made clear that his claims where self-contradictory and demonstrate that he has not understood much about special relativity.

It is certainly tragic to be a real crackpot. To do something alone requires a lot of gifts and almost superhuman patience, determination, and devotion. Most of us do not have all this. It is actually strange that theoretical physics is so crackpotty area. No one at the age of 30 who has just bought a piano thinks that he can become a concert pianist. For some reason there are however lots of people who think that they can become followers of Einstein after having read some popular science book at this age.

I have thought a lot about what attitude to take to crackpots and anti-crackpots: should I be in-honest for humanitarian reasons or should I tell without hesitation if something is deadly wrong. I decided to choose the latter option since it only hurts the egos, something we all should bet rid of. Science does not feel mercy for egos.

And again: I am more than happy to receive criticism if it is about *contents*. Using this criterion one immediately sees whether the purpose of comment is a mere personal insult or whether it is meant to create the impression that the victim is wrong.

Zephir said absolutely nothing about contents (easy to guess why). Instead, he tried to play an older statesman giving "advices" to whom he was actually trying to give a label of - yes! - crackpot!

I have a rather liberal moderation policy: the only restriction is that people making comments behave like civilized human beings. Zephir failed badly in this respect. He did this already earlier.

At 12:37 AM, Blogger Ulla said...

But to become that concert pianist you need to practice a lot. In fact, I know of nobody that can learn me as much as you :) I hope I don't hurt you.

I do a lot of comments out of frame, but I try to give only the URL to keep it short. Not always.

At 2:33 AM, Anonymous Matti Pitkanen said...

Yes. It is to late to begin practicing at the age of 30;-).

Witten is an almost exception to the rule. He got interested in theoretical physics at the age of about 20: he studied humanistic sciences first.

At 3:18 AM, Blogger Ulla said...

If you start at 30 then, do you have anything meaningful to say at 50 or 60?

If you start at 20, then maybe at 40?

If you start earlier? As in your case, about 18?

Is the difference really that big? This seems ridiculous. Then my studies are not wort anything and a waste of time? Then why did you teach me?

At 7:44 AM, Anonymous Matti Pitkanen said...

Young brains are flexible and easily self-organizable. During aging spontaneity is gradually lost. Quantum jump by quantum jump. Less and less choices left. Thanks for God that this is just one particular life;-).

One can learn to enjoy Chopin at the age of 30 or even 60 but certainly not to play Chopin. One can make theoretical physics as a hobby after 30 years age but cannot become a practicing theoretical physics. Life is cruel but we cannot but forgive it;-).

At 2:21 PM, Blogger Ulla said...

This is your choice, we choose before the jump. You have chosen so much away.

My ambition is only to understand, and I dwelve at the roots of the tree, you swing in the top, and sometimes loose the connection to the roots, what Zephir maybe meant.

I would want to play on that piano :) but have no ambitions to be a concert pianist. I have come to understand the importance of the lightlike spinor space, but I need to build the root and trunk first. There are many blanks in your papers, silent knowledge, and the reason why TGD is badly understood, but your posts are really great :) This is no critique in that way.

At 8:30 PM, Anonymous Matti Pitkanen said...

Zephir refuses to realize that theoretical physics is elite sport. He has "simple" theory. Unfortunately, the "simplicity" of his theory reflects his own fatal mathematical limitations rather than simplicity of the real world rathe. This is the standard crackpot error.

The simplicity of Einstein's equations is not simplicity in this sense: these equations are extremely non-linear. The simplicity is at the level of guiding principles.

Blog democracy is a fantastic thing but the problem that a noisy enough person completely ignorant of even basic principles of science is treated by the general reader in the same position as a scientist who has devoted his entire life for physics and can provide fatherly advices to him! This is ridiculous and frustrating.

At 1:06 PM, Blogger Ulla said...

My mind is quite flexible :-)

At 2:37 AM, Anonymous Anonymous said...

thanks for sharing.


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