Twistors allow an impressive organization of ordinary Feynman diagrams of gauge theories. Instead of calculating an immense number of individual diagrams you get their sum as single twistor diagram. The minimal function for twistor diagrams would be this kind of organization.
Twistor diagrams inspire also more ambitious ideas. The notion of plane wave is usually taken as given but twistors suggest as basic objects the analogs of light-rays which are waves completely localized in directions transverse to momentum direction. These are perfectly ok quantum objects since de-localization still takes place in the direction of momentum. Parton picture in QCD strongly suggest them physically. Also quantum classical correspondence becomes especially clear for them: quantum states in particle experiment would really look what they do look in laboratory. There are excellent reasons to expect that IR divergences of gauge theories are eliminated by transverse localization.
The condition that twistor structure exists in space-time is also quite a constraint and suggests strongly that higher dimensional theories should use M4× S type space so that the higher-dimensional space would not be dynamical. M4 of course has also other marvelous properties: light-cone boundary in M4 is metrically 2-D and allows generalized conformal invariance (I wonder how many times I have said this without absolutely any effect on colleagues: they simply cannot take me seriously for the fraction of minute needed to realize "Hey, this guy is right!").
In spirit of twistorialization program of Penrose I proposed some time ago how space-time surfaces representing preferred extremals of Kähler action in M4× CP2 and coding locally basic data for light rays (local momentum direction and polarization essential for twistor concept) could be lifted to holomorphic surfaces in 12-D T× CP2 or 10-D PT× CP2.
The surprise was that for surfaces which are not representable as graphs of a map M4→CP2 ("non-pertubative phase" for which QFT in M4 description does not make sense) the surfaces would have dimension higher than 4: D=6,8,10. Maybe there is a connection with branes of M-theory and TGD.
Twistors are also highly powerful idea generators. Twistor concept led through a rather funny interlude to the realization that QFT limit of TGD must be based on Dirac action coupled to gauge bosons without any YM action. The counterpart of YM action is generated radiatively so that all gauge couplings are predicted provided the loop integration can be carried out so that divergences disappear. Gauge boson propagator would have standard form apart for normalization factor which represents square of gauge coupling.
The basic problem is definition of the cutoff of momentum integration and zero energy ontology and p-adic length scale hypothesis force this cutoff physically and allow a geometric interpretation for it in terms of fractal hierarchy of causal diamonds within causal diamonds. Theory produces realistically the basic aspects of coupling constant evolution for standard model gauge couplings apart from gauge boson loops. The values of fine structure constant at electron and intermediate boson length scale fix the two parameters - call them a and b, characterizing the cutoff in hyperbolic angle to two very natural values. b is exponent and exactly equal to b=1/3 by argument based on analyticity (no fractional powers of logarithms). Second one is coefficient equal to a=0.22050469512552 if fine structure constant is required exactly in electron length scale (this means of course over accuracy). Taking analyticity argument seriously, one can say that fine structure constant is predicted in intermediate gauge boson length scale.
It turned out that massivation of gauge bosons occurs unless the hyperbolic cutoffs for time-like and space-like momenta are related in a unique manner. The hyperbolic cutoff is the ad hoc element of the model, and the next project is to find whether the proposed model in which quantum criticality would fix the UV cutoff in hyperbolic angle really does it and whether it leads to the hyperbolic cutoff forced by the values of fine structure constant at electron and intermediate gauge boson length scale.
This involves rather heavy numerical calculations using rather primitive tools [just MATLAB (afforded by a friend since Helsinki University long time ago found it impossible to help by providing program packages like Mathematica), no symbol manipulation packages, no young left-brainy students] and represents quite a challenge for my 58 year old badly right-halved brain.
I have organized the work on twistors and emergence of gauge boson propagators to two new chapters: Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD and Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".
8 comments:
To my friend - a poem of mine.
When I close my eyes
I see a man
perhaps 85, all grey hair
laughing, joking
And he is proud
so proud.
I see him in younger years
with long, black hear
all in black
shy, sensitive, desperate
He had seen the light
he was one of those who knew
but nobody believed
it was the truth.
Long, long years
of struggle
joy sometimes
loneliness most of the time
sacrifying for the idea
Big sacrifice
And he went into bitterness
he lost his faith
in humankind.
I saw him then
bitter and lonely
I took him by his hands
led him out into sunshine
where birds were singing
with joy
a green flowering meadow
all in white.
Everything
got its solution
like a puzzle, all universe
piece by piece
a beautiful place
our mother Earth
and father Heaven, or Cosmos
together
Our new time in being
when the old time had ceased.
Relieved, he was so relieved
they finally, at last
believed in what he said
gradually, bit by bit
When he was 75 they called
from the Nobel committé
He was finally one of them
the next after Einstein
a hero - my hero
the knight of the truth.
Nice poem but I am not doing this to become a Nobel hero or a hard copy of Einstein. The passion and joy of good scientist is to understand. Heros have a more natural place in entertainment industry.
Of course you are not doing your work for Nobel, nobody does. I think you are too modest. I know you hate the talk of making your work public, but still I think you should do it. Not for your own, but for us all, if that sounds better. And heroes can be of many kinds. Also a lifelong struggle can be seen as heroic. I'm sure many agree with me in this.
Matti and anonymous:
both of you are right. Matti if you don't share your musings with others, however can someone continue your work? This is why you should try to make it digestible from time to time.You never know who is interested and what they might bring to the table.
Dear Mahndisa,
numerical work makes me rather non-communicative;-).
Dear Mahndisa,
numerical work makes me rather non-communicative;-). I have been working with the numerical realization for the model of coupling constant evolution based on quantum criticality. Numerical work is is not easy at this age and would require a hard wired brain at any age. To make challenge even more difficult, I am forced to use MATLAB without compiler and there are a lot of loops. The basic challenge is how to make things fast by making as much as possible analytically. This kind of problems sound of course childish and crackpottish in the ears of anyone allowed to use the computational resources of any physics department of any University. Perhaps I should be ashamed;-)!
There are also other reasons for being not so communicative. Last week MATLAB went completely mad: probably I became a victim of a clever virus attack, nothing standard against which I am well-shielded. This was not the first time.
Add to this jeremiad the problems of everyday life (for a week I have had nothing at my bank account and getting support social office is slow nowadays since there are long ques) and you begin to understand why the working conditions are not very inspiring. We are however in Finland and in the academic circles of this country thinkers are regarded next to criminals and the best manner to treat them has been found to be the academic equivalent of Siberia.
In any case I have made a lot of progress in understanding coupling constant evolution. The question whether the proposed realization of quantum criticality works is still open. In any case, at ultrahigh energies the behavior of em couplings strength would be like that for asymptotic free theory if criticality is accepted. For low energies the criticality is consistent with standard model behavior for fine structure constant (its value at electron and intermediate boson scale are the constraints). I do not yet know whether the low energy and high energy behaviors are consistent with each other or not. The calculations are desperately slow.
This problem led to the ask whether p-adicization of the theory is necessary to realize criticality. Within two days this led to a rather precise recipe for how to p-adicize the theory in terms of p-adic fractals- creatures which I discovered within first year of p-adic TGD but for which I have not found direct application in TGD hitherto. The recipe was very simple.
*Consider real Lorentz invariant amplitudes.
*Map Lorentz invariant kinematic quantities to their p-adic counterparts by some variant of canonical identification to get p-adic calued functions with same functional form.
*Carry out arithmetic operations such as the summation of perturbative contributions using p-adic arithmetics, and map the result back to reals to get a p-adic fractal.
You just go to p-adicity, perform arithmetics there and return to reality to see what you got!
In this manner the difficulties related to p-adicization such as the non-existence of p-adic definite integral, and the problems with minus sign and imaginary unit can be circumvented and the outcome cannot differ too much from real physics prediction.
I hope that I can write about this in blog within few days. The detailed situation as it is now is described in th e new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".
Best Regards,
Matti
I know of nobody who has such an capacity to work as you, Matti. Frenetically almost. And we live in a very interesting time.
Still I think of the same as Mahndisa. Somehow your work should live forever and not pass away with you. (I hope of course that you will live for a long, long time, and in good vigour.) It is too important for such a fate. I have thought of some kind of foundation, which could support the money for science and for publication. But I don't know how that would be done in practice. Some rich millionaire?
I also think that making the theory more easily understood is very important. A popular version? Of course you Matti can't do both the thinking and the writing. Let someone else do the popular version. Show the stupid Finns:-) I could try, if I'm allowed to. You know me:-)
Sorry, Matti. I put my nose again where it shouldn't be? A bad habit?
In think that time for a popularization is ripe when the work is generally recognized. And the people who have the capacity to develop TGD further do not read popular books nor anything else unless it relates to superstrings, M theory, F theory, trilobite theory or something even more baroque and non-predictive. Due to this elephant disease of theoretical physics the situation does not look very encouraging at this moment.
I am not at all sure whether I know you;-). Perhaps you could send some info to my email address.
Elephant disease is an interesting name...
The people who would develope TGD further, do you know them? How could you? I think Mahndisa hit the nail.
People who already "knows" that only say M theory is right and nothing else, will not develope TGD. You must create an easy text for students and others who don't surely "know" yet. They are the followers. And you can't expect them to digest your texts, which not even experts understand, as you has told me.
Asking the right questions is much more efficient to get pupils than serving an excellent dinner. And I can ask many questions. But I don't want to argue with you about this. TGD is yours.
You know me, but for different reasons I want to stay anonymous. I have told you my id.
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