Twistor Seminar
Today David Skinner entertained us with a talk about recent work on recursion relations in twistor theory, being careful to point out that the new geometrical aspects of this viewpoint on scattering amplitudes are quite different to those of the original string theory setting. For those of us who don't like integrals, there were some nice diagrams of triangulated polygons. Apparently Andrew Hodges put a new paper on the arxiv today explaining how amplitudes could be related to simple polytopes in a twistor space, but it hasn't materialised yet. Perhaps tomorrow.
Meanwhile, I have a nice pair of new shoes!
Meanwhile, I have a nice pair of new shoes!
posted by Kea | 5:13 AM
8 Comments:
Wow, of course Oxford is home to Sir Roger Penrose and thus home to the twistor theory! I had forgotten. I love the rotating toroidal illustration of a twistor in The Road to Reality 2004, and find it more physically appealing for how electrons spin and radiate gauge bosons to give their intrinsic dipole magnetic moments, than extradimensional strings.
Strings would be nice if their oscillations would predict numerical masses that could be checked, but of course the Calabi-Yau manifold to compactify the 6 extra spatial dimensions has about 100 unknown moduli that can together take maybe 10^500 or (more likely in my view) infinity combinations of values. An equation "predicting particle masses" with 100 unknowns describing 6 unobservably small Planck scale compactified dimensions in a Calabi-Yau manifold is as useless a thing imaginable, even if string theory could derive such an equation. I fact of course, it doesn't do that: the 100 or so moduli have to be stabilized with Rube-Goldberg machines of some kind and there are no certain proofs of what string theory really is, so even if we could probe the Planck scale and get data on the moduli of the compactified extra spatial dimensions, there would not be a cut-and-dried physical model with which to calculate particle masses. It would remain religion, even if we could see the Planck scale, as far as I can see.
At least twistor theory is less esoteric, it's assumptions are less speculative. It's not trying to reconcile a unification speculation with a quantum gravity speculation, using other speculations (extra spatial dimensions).
I hope that Dr Andrew Hodges' exciting new paper on twistors isn't censored and banned by the powers that be (Distler et al.) at arXiv headquarters.
(New shoes? I thought there was a dress code for all mathematicians in Oxford in the summer: sandals with socks?? One other question, does Sir Roger Penrose hang out around the maths department there ever, or is he permanently retired now he's in his late 70s???)
(Sorry for typos in my comment: "... In fact of course, ..." and for the apostrophe in "it's" in "less esoteric, its assumptions", if apostrophes are only permitted to signify a dropped letter. I keep forgetting the rules.)
Thanks also for the link to the paper by Lionel Mason and David Skinner, http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.2083v1.pdf.
I'm kind of surprised by the fact that they are doing scattering amplitudes with twistors. The SM already predicts scattering amplitudes very accurately, so why do that? Surely the thing to do is to use twistors to work on fundamental problems not addressed by the SM, e.g.
(1) Foundational issues in QM/QFT like alternatives to the picture of a particle as a piece of string;
(2) Predicting masses of particles (I know Carl Brannen is working on this with matrices, density operators and a generalized Koide formula, but despite progress I think there are still some unresolved interpretation issues);
and
(3) quantum gravity.
(Sorry for all the comments, and please accept apologies for my ignorance if I'm misguided.)
Marni, check out the matrices in this paper.
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 M^4xS type space so that the higher-dimensional space would not be dynamical. M^4 of course has also other marvelous properties: light-cone boundary in M^4 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 Kaehler action in M^4xCP_2 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 TxCP_2 or 10-D PTxCP_2.
The surprise was that for surfaces which are not representable as graphs of a map M^4-->CP_2 ("non-pertubative phase" for which QFT in M^4 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 from a friend (University of course cannot help!), 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".
This post reminded me of Penrose's book too. He is wisely skeptical of inflation and other "trendy" theories.
Congratulations on the shoes! Speaking of British; I must wear black leather boots with spike heels; a post about that is coming.
"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." - Matti Pitkanen
Thanks for those links Matti. I'm deeply interested in the application of twistors to spin-1 massless particles such as real and virtual photons. Feynman points out that from the success of path integrals, light uses a small core of space where the phase amplitudes for paths add together instead of cancelling out, so if that core overlaps two nearby slits the photon diffracts through both the slits:
‘Light ... uses a small core of nearby space. (In the same way, a mirror has to have enough size to reflect normally: if the mirror is too small for the core of nearby paths, the light scatters in many directions, no matter where you put the mirror.)’
– R. P. Feynman, QED, Penguin, 1990, page 54.
Feynman's approach is that any light source radiates photons in all directions, along all paths, but most of those cancel out due to interference. The amplitudes of the paths near the classical path reinforce each other because their phase factors, representing the relative amplitude of a particular path, exp(-iHT) = exp(iS) where H is the Hamiltonian (kinetic energy in the case of a free particle), and S is the action for the particular path measured in quantum action units of h-bar (action S is the integral of the Lagrangian field equation over time for a given path).
Because you have to integrate the phase factor exp(iS) over all paths to obtain the resultant overall amplitude, clearly radiation is being exchanged over all paths, but is being cancelled over most of the paths somehow. The phase factor equation models this as interferences without saying physically what process causes the interferences.
One simple guess would be that an electron when radiates sends out radiation in all directions, along all possible paths, but most of this gets cancelled because all of the other electrons in the universe around it are doing the same thing, so the radiation just gets exchanged, cancelling out in 'real' photon effects. (The electron doesn't lose energy, because it gains as much by receiving such virtual radiation as it emits, so there is equilibrium). Any "real" photon accompanying this exchange of unobservable (virtual) radiation is then represented by a small core of uncancelled paths, where the phase factors tend to add together instead of cancelling out.
Is the twistor nature of a particle like a photon compatible with this simple interpretation of the path integral for things like the double slit experiment, and virtual photons (the path integral for the coulomb force between charges)? I'm wondering whether the circulatory motion around the direction of propagation in twistors will cause the interferences and cancellation when they are exchanged in both directions between two charges, thus making virtual photons or gauge bosons invisibly apart from their role in causing forces?
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