Reducing non-planar diagrams to planar ones by a generalization of algorithm for calculating knot invariants?

I have been listening some lectures in Strings 2001. The lectures related to progress in the calculation of gauge theory and super-gravity amplitudes are really electrifying: one really feels the sparking enthusiasm of the speakers. Besides twistor revolution there is also other amazing progress taking place in QFT side.

At this morning I started to listen the talk of Henrik Johansson about Lie algebra structures in YM and gravitational amplitudes. I have already earlier written about the finding that there is a symmetry between kinematical numerators of the amplitudes involving polarizations and momenta on one hand and color factors on the other hand, and that one can in a well defined sense express gravitational scattering amplitudes in terms of squares of YM amplitudes. This holds true for on mass shell amplitudes. The reduction of the gravitational amplitudes to squares of YM amplitudes would be incredible simplification: Even 3-graviton off mass shell vertex contains about 100 terms! As a matter fact, gravitation is a gauge theory too with gauge group replaced with Poincare group so that it would not be totally surprising that this kind of duality between Poincare and gauge group kinematics would hold true.

This duality is not however the topic of this posting. As Johansson was explaining the Jacobi identity for the kinematical Lie algebra I got Eureka experience. What the kinematic Jacobi identity states is following:

The numerator for four-point amplitude with twisted legs in s-channel is expressible as a difference of planar s- and t-channel amplitudes.

If you did not get the association to twistor program already from this sentence, recall that the basic problem of twistor approach are non-planar diagrams. For them one cannot order the loop momenta in such a manner that the ordering would be universal and depend only on the number of loops as it is for planar diagrams without crossings. Hence one is not able to combine all diagrams to single integrand and this is related to the tricks one is forced to apply to make the loop integrals finite: same identification of loop momenta for all diagrams is not possible if one wants finiteness.

What one needs for a generalizaton of twistor approach to apply to non-planar diagrams is a universal identification of the loop momenta by cancelling all crossings: the amplitude itself need not be equal to the difference of the amplitudes obtained by reconnecting in two manners but could be something more general. This operation would be performed for internal lines only. For external lines it tells that the amplitudes changes possible sign when external lines are permuted. For braid statistics a more phase factor would result. The duality of old-fashioned string models says that the difference of s- and t-channel amplitudes vanishes so that one can say that amplitudes with twisted legs vanish. Also at large N (number of colors) limit of N =4 SUSY these differences vanish and YM theory behaves like string theory and planar twistor approach should give exact answers at this limit. In TGD framework the effective replacement of gauge group with infinite-dimensional symplectic group could have the same effect. But what about finite values of N in super YM theories?

Could one generalize the twistor approach so that one could calculate all amplitudes by recursion- not only the planar ones?

Alert reader has of course answered already but I try to explain for non-specialists (with me included). If one has worked with braids and knots, one realizes that the expression for the amplitude as difference of planar amplitudes is analogous to what you get in elementary un-knotting operation for braids annihilating one crossing in the knot diagram! In the process you form the difference of two possible reconnections at the crossing point. If you interpret the process as time evolution, it corresponds to two vertices in which interiors of strings touch each other and reconnect in a new manner. In the construction of Jones polynomial as a knot invariant the repeated application of these un-twisting operations eventually leads to un-knot and you get as an outcome the knot invariant. Also non-planar Feynman diagram is like a knot diagram and the outcome of similar procedure should consists of only planar amplitudes.

For Feynman diagrams one cannot distinguish between upper and lower crossings of the lines. This could be interpreted by saying that both crossings give the same contribution. This is the case if untwisting gives the difference of numerators in both color and momentum degrees of freedom so that the signs cancel and the integrals of both contributions are identical despite the fact that the propagator denominators are not identical. The most general outcome would be a term proportional to the sum of the four planar contributions and one could perhaps treat the situation using twistorial methods. Proportionality coefficient could depend on dimensionless Lorentz scalars constructed from the incoming momenta of the sub-diagram with crossing and dictated to high degree by conformal invariance. Professional could probably demonstrate in five minutes that the conjecture cannot hold true.

Especially, if you have written N times "Quantum TGD as almost topological QFT ..." you get at the large N limit the vibe in your spine. Because the combinatorics of an almost topological QFT must be that of a topological QFT and because braids are basic building brick of TGD amplitudes, it should be possible to reduce all non-planar amplitudes -both those of TGD and those of N=4 SUSY and even other gauge theories - by a repeated un-twisting to planar amplitudes. A generalization of the basic algorithm of knot theory would become part of twistorial Feynman diagrammatics and could perhaps also be used to define the integrand including also the loops with crossings!

The rules for calculating the twistor amplitudes would be simple.

1. You - or your knot theoretical friend- must first patiently unknot the Feynman diagrams involved by eliminating all twists using the basic formula allowing to express twisted sub-amplitude with a difference of un-twisted sub-amplitudes. You might even dream that he gives you explicit formulas for the outcome to get rid of your continual requests for help.

2. At the end of the day you get just planar diagrams and you can apply the general recursive formulas of Nima and others working for all numbers of external particles and all numbers of loops to get the integrand, which you should be able to integrate.

3. Unfortunately you are not! But you can knock the door of Goncharov and ask whether he could kindly perform the integral using his magic Symbolic Integration Machine about which Anastasia Volovich tells in her talk "Symblifying N=4 SUSY Scattering Amplitudes".

Is this idea just a passing daydream? Or morning dream- my hungry cat forced me to wake up at 3 a'clock so that I might be hallucinating in half-sleeping state. A specialist could immediately tell where this crazy idea of Europe's (if not World's) worst Feynman diagrammatician fails.