Exploring Twistor Theory: What's the Value in Penrose's Approach?

[RiccardoVen]

Hi,
some years ago when I was studying Nuclear engineering in Turin ( Italy ), I attended
some courses with prof. Tullio Regge about groups theory, spinors and twistors.

I was too young to fully understand what he was telling us, but now, after some years
of private study about Einstein's gravity and a differential geometry, I gifted myself for
my birthday with a copy of both volumes of Spinors and Space-Time.
This is, of course, really different to the normal "tensorial" approach I was used to and I'm really
fascinated by this topic. Attending some extra courses in particle physics, I saw how spinors born
to life naturally in Dirac's theory, but I wasn't aware they could be used successfully in Einstein's gravity as well.
I've read the spinors approach is handier for describing some massless phenomena, and for other the tensor approach is more appropriate.
The real interest to me is how complex numbers theory comes in help in physics here and there.
Usually, when you are able to switch some theory rearranging it under the complex line of sight, you add to it some more physics content, and I'm curious about that.

So the question is: from your experience, does spinors/twistors Penrose's approach worth the pain to break my head on it? Or it just remain actually, after 40 years, a sterile theory, which destiny is to remain in the maths tool games?

Indeed, the first volume would worth the pain even only for its interpretation of Lorentz transformation as complex Mobius transformation. This connection is really, really interesting to me.

I know recently ( where recently I meant 10 years ago ) Ed Witten has given new life to it, encompassing it somehow in string theory, but the math is too advanced to me for now( I'm studying algebraic topology and knots theory for it, but it takes a long time to master it, as you may know ).

Thanks, regards

Ricky

EDIT: I'd like to stress my point here: since I'm a physics enthusiast any theory is actually fascinating me a lot,
twistors included, and I'm constantly wondering every day about its beauty. But my time is limited and I'm putting my efforts from since last year to have a better grasp on Einstein's gravity and differential geometry, mainly doing exercises or so. This is of course really time consuming and I'd avoid to jump into "exotic" theories before having a better grasp on the main ones. Nevertheless, from my experience about complex number theory, I'd bet this should open a new insight to me ( like Riemann sphere, which impressed me so much at the very first sight), just, for instance, about the "new" induced (2,2) metric signature from it.

 

[Naty1]

does spinors/twistors Penrose's approach worth the pain to break my head on it? Or it just remain actually, after 40 years, a sterile theory, which destiny is to remain in the maths tool games?

In the ROAD TO REALITY,2004, Penrose offers a chapter [#33] which may be of interest:
MORE RADICAL PERSPECTIVES: TWISTER THEORY

Under 33.14, about three and a half pages, 'The future of twistor theory' Penrose says:

The community of physicists who know much about the subject is rather small...twister theory could in no way be called a 'mainstream' activity of theoretical physicists today. Yet twister theory, like string theory, has had a significant influence on pure mathematics, and this has been regarded as one of its greatest strengths...

I have no personal opinion except that you would probably find the entire chapter of interest.

 

[WannabeNewton]

Spinors aren't as common in classical GR as they are in QFT; even today what you'll mostly see in classical GR is tensor calculus as opposed to 2-spinor calculus. If I had to dig deep, the most famous application of spinors in classical GR would have to be Witten's proof of the positive energy theorem. Needless to say, Penrose/Rindler's two texts are probably worth reading for a variety of reasons; for one, the first volume gives a very detailed and systematic exposition of the abstract index formalism.

 

[RiccardoVen]

Hi WannabeNewton,
I'm pleased to ear from you, since I've read a lot of your posts, they are always interesting to me.
thanks for your answer.
I know you have a big background on tensor calculus ( or at least it seems so from your posts ), so what
do you think about my point about complex treatment of GR...do you think you can have a sort of dual description
using 2-spinors instead of pure tensor calculus?

 

[WannabeNewton]

As far as I know, it does exist and is used in specific applications. To start with, see here: http://en.wikipedia.org/wiki/Newman–Penrose_formalism and here: http://www.lawrightproperties.com/images/SpinorGR-Penrose.pdf

 

[bahamagreen]

There may be a renewed interest in twistor theory with the 17-SEP-13 amplituhedron paper.
One of the PF form threads about it is here.

 

[RiccardoVen]

Thanks for this!

It's a pretty fantastic discovery, that all of the information about scattering amplitudes can be encoded in one geometric object, with no reference to space or time, locality or unitarity (which come out as emergent properties of the geometry). Granted, keep in mind that this model is built on maximally supersymmetric Yang-Mills Theory, and it will take a lot of work to generalize it to more complex QFTs (especially since we have found no evidence to support SUSY), but it looks pretty promising just in the beauty of the theory

I will look further to it deeply for sure.

even today what you'll mostly see in classical GR is tensor calculus as opposed to 2-spinor calculus

I was puzzling myself about why spinor theory has not become so popular in GR as well. I've not entered too much in it yet, but from Penrose's POV it seems giving more "natural" insights for GR as well ( of course he did the theory as first, so he was trying probably to push it a bit ). From my limited experience, through my studies, when I met complex numbers I've always seen a lot of new ideas from theories involving them, so I cannot a priori think Penrose's was too optimistic about that, i.e. usually complex structures have a richer content than "simple" theories on real fields. What's your opinion about that? the maths is too difficult compared to the insights it gives back?
I will study it for sure despite it's not popular, but I was just wondering about that.

Thanks

 

[WannabeNewton]

I think the simplest answer is that most things in classical GR, to put it lightly, can be described beautifully and perfectly using tensor calculus so there's no need for a formal development of spinor calculus in texts or elsewhere; spinor calculus (and the theory of spinors in general) requires more extensive background in mathematics and more care in development so if I had to guess, I would say that the tradeoffs in switching to a spinorial treatment (such as offering different geometric insights) aren't enough to completely replace tensor calculus, which has both ease of development and ease of use.

 

[RiccardoVen]

Thanks very much WannabeNewton,
I have not yet enough knowledge about both of them to open any further discussion about it.
I hope to have a deeper lead on them in future, carefully pushing them in parallel in my studies.

Regards

Ricky

 

[Naty1]

usually complex structures have a richer content than "simple" theories on real fields. What's your opinion about that?

it does seem that way.

some examples to consider...

The quantum fluctuations that are grown with accelerated cosmological expansion [dynamic geometry, dynamic spacetime] can be thought of as a mix of 'real and virtual particles' ...which are also field amplitudes...[analogous to the real and imaginary numbers of the Schrodinger wave equation]... So complex imaginary numbers and their operators are associated with virtual particles, which cannot be detected, while complex real numbers and their operators are associated with real [detectable] particles.

In Feynman-diagram language an external line corresponds to a physically real particle, whereas an internal line corresponds to a virtual particle. So it is common belief that only real particles (asymptotic states) can be detected.

more in this discussion:
What is a particle
https://www.physicsforums.com/showthread.php?t=386051

 

[WannabeNewton]

But that's QFT; spinors are extremely natural in QFT. The OP was asking about classical GR.

One thing at least is that once you get past the mathematical formalities of spinor theory and get to the actual computational aspects of spinor algebra/calculus, the mechanics is very similar to that of tensor algebra/calculus (with some caveats e.g. you can't just freely raise and lower indices with complete disregard of the ordering of indices like you can when dealing with just tensors).

 

[RiccardoVen]

I was wondering someone could address me eventually to some textbooks on spinor/twistor theory, which could help me in understanding better this really important ( to me ) aspect of them.

I landed to these Penrose's books from "Visual Complex Analysis" Needham book ( absolutely one of my favorites, covering almost all fields of complex analysis ) in which there's a very deep and important chapter about Mobius transformations and their analysis on Riemann sphere. There we can see how Penrose was suggesting in its first chapter how Lorentz transformations are really connected to Mobius transformations ( that's why, I guess, Riemann sphere is so important for Penrose, appearing in almost its slides on twistor theory conferences I found online ).

I'm already going to buy the original Cartan "Theory of Spinors", but this is quite more for collection purposes, since I consider it a "classic" for the argument.
May be you ( WBN, for instance ) could suggest a more modern approach to it.

Thanks, regards

EDIT: I actually this PF thread about twistor textbooks. I was more oriented towards spinors, first.