Sunday, May 23, 2010

Twistors and Mass

Already by the early 1980s, some progress had been made on understanding massive states within the twistor program of Roger Penrose. In particular, the mass parameter of the Klein Gordon equation could be seen as arising from configurations of not one, but several twistors. For example, Andrew Hodges, the main developer of twistor diagram methods, wrote a paper about this way back in 1984.

The pivotal mathematical idea is discussed in this 1981 paper by Hughston and Hurd. That is, the description of rest mass requires higher cohomology, which may be built from the one dimensional cohomology associated to massless fields.

In my mind, this cohomological fact was always one of the strongest motivations for the study of category theory, and abstract cohomology, in the foundations of physics. Now, with the new success of twistor theory in scattering amplitudes for QFT, one can only agree with Penrose's sentiment that it is high time we revisited cohomological studies of rest mass.


  1. I have been working with twistor program inspired ideas in TGD framework for a couple of years. The basic elements are following.

    a) The notion of generalized Feyman diagram defined by replacing lines of ordinary Feynman diagram with light-like 3-surfaces (elementary particle sized wormhole contacts with throats carrying quantum numbers) and vertices identified as their 2-D ends - I call them partonic 2-surfaces.

    b) Zero energy ontology and causal diamonds (intersections of future and past directed lightcones). The crucial observation is that in ZEO it is possible to identify off mass shell particles as pairs of on mass shell particles at throats of wormhole contact since both positive and negative signs of energy are possible. The propagator defined by modified Dirac action does not diverge although the fermions at throats are on mass shell. This means opening of the black box of off mass shell particle-something which for some reason has not occurred to anyone fighting with the divergences of QFTs.

    c) Representation of 8-D gamma matrices in terms of octonionic units and 2-D sigma matrices),

    d) Number theoretic universality requiring the existence of Feynman amplitudes in all number fields allowing suitable algebraic extensions. Also imbedding space, partonic 2-surfacesm and WCW must exist in all number fields and their extensions.

    e) As far as twistors are considered the first key element is the reduction of the octonionic spinor structure to quaternionic one and giving effectively 4-D spinor and twistor structure for quaternionic surfaces. Modified gamma matrices at space-time surfaces are surfaces quaternionic/associative and allow a matrix representation. As a matter fact, TGD and WCW can be formulated as study of associative local sub-algebras of the local Clifford algebra of 8-D imbedding space parameterized by quaternionic space-time surfaces.

    Quite recently quite a dramatic progress took place in this approach. It was just the simple observation that on mass shell property puts enormously strong kinematic restrictions on the loop integrations. All loops are manifestly finite and if particles has always mass -say small p-adic thermal mass also in case of massless particles and due to IR cutoff due to the presence largest CD- the number of diagrams is finite. Unitarity reduces to Cutkosky rules automatically satisfied as in the case of ordinary Feynman diagrams.

    This is about momentum space aspects of Feynman diagrams but not yet about the functional integral over small deformations of the partonic 2-surfaces. It took some time to see that also the functional integrals over the "world of classical worlds" (WCW) can be carried out at general level both in real and p-adic contex.

    a) The p-adic generalization of Fourier analysis allows to algebraize integration- the basic technical problem of p-adic physics- for symmetric spaces for functions allowing the analog of discrete Fourier decomposion. Symmetric space property is essential also for the existence of Kaehler geometry for infinite-D spaces as was learned already from the case of loop spaces.

    b) Everything is also now manifestly finite and general conditions on holomorphy properties of the generalized eigenvalues of modified Dirac action can be deduced and WCW geometrization reduces to that for single line of generalized Feynman diagram.

    Ironically, twistors which stimulated all these development do not seem to be absolutely necessary in this approach although they are of course possible.

  2. Matti, on shell restrictions are well appreciated today by people working in twistor QFT.

    As you know, I do not believe that any classical continuum geometry is fundamental, and that includes twistors and Kaehler geometries. However, since it remains to rigorously recover GR, I believe that twistors are the right thing to look for as limits to the categorical (and number theoretic) 'p-adic' axiomatic structures.

  3. Maybe this one?
    from a search on arXiv:

    Lubos and Nima?

  4. Dear Kea, you missed my point. Of course I am well aware that twistor people appreciate the on mass shell restriction. The challenge is to overcome them. And even more, we must ask what are the deeper structures behind Feynman diagrams. To open the black box of Feynman diagram and understand what goes wrong when we get the divergences.

    Zero energy ontology makes possible description of off mass shell states as pairs of on mass shell states: different signs of energy are possible and this makes possible also space-like virtual net momenta. The strong kinematical constraints allow to get rid of divergences and to have unitary in terms of Cutkoski rules. This something totally new and extremely non-trivial. My humble hope was to get this message through since still after these years I have the stubborn feeling that the divergence problem might interest theoretical physicists.

    It is good to make clear what one means with classical- which is of course synonyme for boring;-). Does manifold geometry extended by generalizing the notion of number by fusing reals and p-adic number fields together still represent an example of this boring classical continuum? Is the generalization of manifold geometry to that of generalized Feynman diagrams singular as manifolds still something totally uninteresting? Is infinite-D world of classical worlds similar hopelessly classical stuff? Is the dicsrete intersection real and p-adic geometries (at WCW level and space-time level) just the same old classical stuff? Does the generalization of imbedding space to a book-like structure to describe hierarchy of Planck constants belong to the same category?

    To make my point clear: the success of Einstein's geometrization program has been incredible and it is very strange that it has not been generalized by others than me to infinite-D context. Categories are important but their natural discreteness means that they cannot be the whole story: excellent auxiliary tool when one describes things in finite measurement resolution which is essential notion also in quantum TGD. Physical intuition is the best guideline and the mathematics used must be outcome of this intuition rather than the axiomatic starting point.

  5. To Matti: generalizing the notion of number by fusing reals and p-adic number fields together - Is the p-adic then functioning as cutoff, creating the different bookpages? How could otherwise the fusion be made? Is the division of the real numbers real or only an intermediate outcome? Ask me that can't math at all:)

    Look what Lubos write with collaborators (Nima et al):

    A sharp form of the conjecture is that there are always light “elementary” electric and magnetic objects with a mass/charge ratio smaller than the corresponding ratio for macroscopic extremal black holes, allowing extremal black holes to decay. This conjecture is supported by a number of non-trivial examples in string theory. It implies the necessary presence of new physics beneath the Planck scale, not far from the GUT scale, and explains why some apparently natural models of inflation resist an embedding in string theory.

    there is a hidden ultraviolet scale A ∼ gMPl, where the effective field theory breaks
    down, and that there are light charged particles with mass smaller than A. While this statement is completely unexpected to an effective field theorist, it resonates nicely with the impossibility of having global symmetries in quantum gravity
    ...ability for large charged black holes to dissipate their charge in evaporating down to the Planck scale.

    A should be an open A.

    Where goes that dissipation of charges go? Isn't it information?

    ...for the lightest charged particle along the direction of some basis vectors in charge space, the (M/Q) ratio is smaller than for extremal black holes. Such an assumption allows all extremal black holes to decay into these states. The weaker statement says that there should exist some state with mass/charge ratio smaller than for extremal black holes. In all the examples we have seen, this state has a “reasonably small” charge, so it is light; however, the weaker form allows the possibility that the smallest M/Q is realized
    for some large charge Q∗ and objects that are “nearly” extremal black holes. While the number of exactly stable states would be finite in this case, it would still be extremely large.

    our conjecture seems to at least parametrically exclude apparently natural models for inflation based on periodic scalars with super-Planckian decay constants, which seem perfectly sensible from the point of view of a consistent effective theory.

    ...things like a small cosmological constant are taken to be non-generic, tuned, but possible. But it is extremely interesting that phenomena of clear physical interest, like inflation with trans-Planckian excursions for the inflaton, which might even be forced on us experimentally by the discovery of primordial gravitational waves, seem to be pushing up against the limits of what quantum gravity seems to want to allow.

    And Lubos pretend not to know of this!
    Sorry Kea. You told me to look at Lubos.

    I have also another cit. from his blog:

    You may slice your spacetime by lightlike slices and all physical questions depend on the way how objects get from one slice to the next one (think about the light cone gauge). The physical systems on the opposite sides of a null slice may be entangled or related but they're also slightly different, because of the evolution in the other lightlike direction. All the dynamics is hidden in this dependence on the side, so dynamics can't be just about the entanglement.

    What is this if not The Big Book?

  6. Ulla, I did not seriously tell you to look at Lubos' rants. And Matti, you seem completely oblivious to the mountains of work done by first rate mathematicians over the last two decades to understand that black box. Now of course I agree with you that their ideas about the Physics are woefully incorrect, but I think it is fair to say that the categorical viewpoint (which is ALL ABOUT understanding the deeper structures behind diagrams) has some solid motivation.

  7. Dear Kea,

    I must say that your comment is completely out of hat. Why not to try to really read what I am saying and comment the *contents* rather than throwing random insults- they do note generate respect in any one.

    I am perfectly aware of mountains of work done with Feynman diagrams by first rate mathematicians- anyone claiming to work seriously with quantum theory is. The problem is that this work has been futile in the sense that the Feynman diagrams are still essentially the same old Feynman diagrams that we had at the times of Feynman. Individual loop diagrams diverge and you do not get unitarity and coupling constant evolution without loops. The situation is paradoxical and frustrating.

    When you have a paradox then the correct question is what goes wrong at the level of ontology. What is the too strong or erratic assumption? In the recent case the question is how to open the black box of loop diagram. A clever manner to sum them does not open it. You can apply twistors techniques. You can invent all kinds of clever and sophisticated algebraic anguages to talk about infinities as if they were not infinities -just as we talk about death- but the basic problem does not disappear. You can also probably cancel diverges in some SUSYs but not in realistic theories. This activity is extremely demanding technically but it has nothing to do with the opening of the black box. One must do what Columbus did.

    Bad ontology implies bad mathematics. Path integral is not mathematically existent notion. In TGD functional integral replaces it, is completely well-defined mathematically , and emerges naturally when you accept physics as infinite-D geometry program. This the first step.

    The string models where the first attempt giving hopes of the opening of the black box. After two decades it will be possible to say publicly that the tragedy of string models was the misinterpretation of trouser diagrams in terms of particle reactions. The natural interpretation is in terms of particles travelling via several routes simultaneously. I have strong temptation to believe that the leading figures have already realized this tragic error which takes mountains of literature by first rate mathematicians to the drain since it is not about physics. May be the realization of the tragedy might have something to do with the mysterious silence surrounding string theory nowadays. Even afficionados learned about the last super string conference when it had already begun!

    It is easy to characterize what went wrong with strings. Feynman diagrams are singular as 1-manifolds and smooth as 0-manifolds. Generalized Feynman diagrams are singular as 4-manifolds and smooth as 3-manifolds (same for light-like 3-surfaces and partonic 2-surface in dimensions D=3 and 2). Vertices must be smooth: for string diagrams they are not.

    Of course, string like objects are something extremely natural also in TGD as one learns by reading the article about how magnetic confinement leads to stringy flux tube like objects of length of order weak length scale producing as side product weak confinement as counterpart of ordinary massivation and also color confinement. One prediction is that this stringy structure of particles begins to make it visible at LHC energies.

  8. I would want you to tell me what you mean with an emergent space, Kea. 'The very meaning of space' doesn't sound so convicting.

    Suppose twistors are very important there.I have looked your links through.

    Sorry if I embarrass you.


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