Twistors, Integrable systems and the Lax Formalism

Having spent a little time talking over a new idea with someone who works in the area of integrable systems, I have found myself looking into twistor-theory. Now I must point out that I am primarily an (applied) mathematician, so my theoretical physics knowledge is fairly rudimentary.

The basic idea is to make an analytic continuation of the space/time coordinates in given integrable PDE to extend the notion of integrability to higher order systems (this involves some quite deep areas of Lax's formalism, a little functional analysis, some operator theory and so on - so we'll avoid the details!) but it at the crux of the matter is embedding a coordinate system into a higher dimensional, complex structure.

So, I was wondering if anyone here has any views on the structure (and elegance?) of the twistor formalism?

Twistor formalism is perfect and predicative, with the only subtle exception: it's solely unrealistic from physical point of view, being artificially constrained by 3D nested hierarchy. Unfortunately, it seems, the Universe by AWT knows nothing about human preference of three number....



This post has been edited by Zephir on Nov 25 2006, 01:15 AM

Except for the very obvious and very basic fact (mentioned in all descriptions of Twistor theory) it's based on 4 dimensions.

Having just attempted to get a bit of info about Twistor theory from Penrose's book and Wiki, I can safely say I've no idea about it other than it's (--++) signature form. It seems to make string theory look much nicer judging by the descriptions!

I would like to read more on this subject, especially how they were attempting to intergrate gravity into this framework.

Do you have links to provide?

Also, how versed are you in providing calculations based on Penrose's work?

Found something that might be of interest:

Spinors, Minimal Surfaces, Torsion, Helicity, Chirality, Spin, Twistors, Orientation, Continuity, Fractals, Point Particles, Polarization, the Light Cone and the Hopf Map.

Went right over my head, but you might find it relevant.

Well, I think you will need to start from the ground up. How about starting from here.

Woah, woah, woah. Slow down. This might be slightly better as a starting point for SSU

[removed]

So SSU admits this subject is way over his head and posts a bona fide link that may be of help to someone who is asking for it. Bona fide in the sense, that he is sincere in his attempt to be of help.

He then gets two responses from you know who and what's his name, that are nothing short of a childish slap in the face.

I think it's only appropriate, that he then posts, what I would call an answer to the question.

A "Twistor" that the two "Twistees" can take turns sharing. That would certainly represent an "integrable system", with the "lax formalism" left as a personal option.

Personally, I prefer a "lax infomalism", as there's just something so "symetric" about it.

He said "I would like to read more on this subject, especially how they were attempting to intergrate gravity into this framework." Twistor theory is extremely complex and even more abstract in it's methodology than general relativity. Other than the vaguest of vague descriptions (which Wikipedia can provide), you are not going to learn anything meaningful about Twistor theory without doing maths. A lot of maths. Two people I work with use twistor results (infact, they both went to a conference on it last Friday and met Penrose!) in their work in QCD processes and it's very complex quantum field theory.

As such, if SSU was truely interested in finding out the approaches Penrose used, a grasp of at least 1st year calculus is required. Euler provided a link which equates to pre-university mathematics revision which SSU is likely to be at, or below. Hence his link was actually a viable one, at least I get the distinct impression SSU doesn't know university level maths.

My reply was more aimed at taking a cheap shot at him, but I'm not bothered. I don't for a second think SSU would sit down and try to learn the maths required to do introductory twistor theory. It's too much like actual learning.

Yes, I did confuse your reaction with his other post. My bad.

I suppose my point otherwise, would be that SSU is not exactly "chump change" when it comes to being educated, in whatever manner his education has come. I'm not going to waste my time going back to the beginning of your discussions with him to find out directly, but I will ask... "when and how did the attitude between you begin"?

I have yet to see a good rationalization from you, for your overall behavior on the forum. What really is your motive for being here?

I know exactly what mine is. I also know that my work and my life require "character". I don't know what your family, prep school, or Cambridge taught you about character; but it is sorely behind in its development, in comparison to your other sudies.

I have an ex-friend who has an IQ of 180+. He was teaching computer science when he was 11 years old. He's in his early 30s now and his life has been one long train wreck. He's desparately alone and hanging by a thread. Why? Because he never "solved his ego".

Euler,

Insulting stuff; so much so in fact, SSU was positively vexed.

SSU,

Please try this:

Number Theory

When there seemed to be a sudden influx of cranks to these forums about 2 or 3 months ago. People who come to a physics discussion forum not to consider science or to learn but to advertise their latest theory with no interest in learning new results, doing actual physics or learning about verified models. As such, after my initial vague pleasantness, I stopped bothering.

When it's clear the other person/people are actually wanting to learn, I'm perfectly amicable. But given I know you, SSU, Nick, Zephir etc well enough now to know you aren't interested I don't bother with the initial pleasantries or to explain carefully results and models because I know you aren't interested. It's quicker and simpler for both sides if I just say "Wrong, because [short answer]." That way, should anyone else be reading who might not know you're wrong, at least they don't fall into the trap of thinking your comments are correct. At the very least, they hopefully look into it more.

For instance, Nick constantly says "Time stops at an event horizon". Initially, I replied with long explainations, including links and pictures, about why that wasn't true and it's a common misunderstanding. 8 months later, he's still doing it. He isn't interested in learning, he just comes to preach and state his ignorant guesses as fact. No justification, no logic, no clue.

Prep school?! I went to a standard secondary school thanks. Unlike the stereotypical view I'm sure plenty of people have, the VAST majority of people who get into Cambridge are normal people from average backgrounds.

As for my attitude, I was nice initially and I can be if I wish. For some people, I know not to bother.

I love how it's always my ego. I'm telling people verified, demonstrable FACTS about things and it's my ego because I'm not agreeing with them?! It is always the guy whose talking from a mainstream point of view whose the one accused of having an ego, never the one whose talking from a position of at best a lack of understanding and at worst complete ignorance whose the one whose got an ego for claiming he couldn't possibly be wrong, no it's always me

Do you actually think about that at all?! That for just one nanosecond, the one whose talking about the combined work of hundreds of thousands of people summing to millions of man-years of work might actually have a valid point compared to your 8 seconds of musing on a topic you don't know about?!

But no, it must be my ego My ego is fine thanks and when I'm talking with people who don't think their opinion is fact, I'm perfectly pleasant, I can explain things to people very well and I can learn things from them too. Shame you don't meet the criteria.

Time will tell.

It depends on the definition of time. The mainstreams science distinguishes between many time arrows: the radiative, imaginary, thermodynamic, cosmological of psychological time arrows, whereas the AWT is using even more general time model, based on the recursive phase interface energy spreading through the nested foam.



The radiative time stops at the even horizon, being based/derived on the light wave spreading, while the time derived from gravitational waves spreading persist. The discussion between you and Nick has completely no sense, until both of us doesn't define the time meaning. By such way, you cannot prove Nick is wrong by the same way, like the Nick cannot prove his truth.

Which facts do you mean, exactly? The black holes? The BigBang theory? The string theory? They're just interpretations of facts, not the facts as such. Even whole the Standard model or relativity theory is just an extrapolation of these facts. The true is, the particles aren't apparently pinpoint objects as Standard Model supposes, and the absolute reference frame for light spreading manifests sometimes (the Doppler shift for microwave background of Universe, gravitational lensing etc..), sometimes not. The theories are remaining a theories, face it.

This post has been edited by Zephir on Feb 24 2007, 10:57 AM