Wednesday, February 08, 2017

Twistor lift and the reduction of field equations by SH to holomorphy

It has become clear that twistorialization has very nice physical consequences. But what is the deep mathematical reason for twistorialization? Understanding this might allow to gain new insights about construction of scattering amplitudes with space-time surface serving as analogs of twistor diatrams.

Penrose's original motivation for twistorilization was to reduce field equations for massless fields to holomorphy conditions for their lifts to the twistor bundle. Very roughly, one can say that the value of massless field in space-time is determined by the values of the twistor lift of the field over the twistor sphere and helicity of the massless modes reduces to cohomology and the values of conformal weights of the field mode so that the description applies to all spins.

I want to find the general solution of field equations associated with the Kähler action lifted to 6-D Kähler action. Also one would like to understand strong form of holography (SH). In TGD fields in space-time are are replaced with the imbedding of space-time as 4-surface to H. Twistor lift imbeds the twistor space of the space-time surface as 6-surface into the product of twistor spaces of M4 and CP2. Following Penrose, these imbeddings should be holomorphic in some sense.

Twistor lift T(H) means that M4 and CP2 are replaced with their 6-D twistor spaces.

1. If S2 for M4 has 2 time-like dimensions one has 3+3 dimensions, and one can speak about hyper-complex variants of holomorphic functions with time-like and space-like coordinate paired for all three hypercomplex coordinates. For the Minkowskian regions of the space-time surface X4 the situation is the same.

2. For T(CP2) Euclidian signature of twistor sphere guarantees this and one has 3 complex coordinates corresponding to those of S2 and CP2. One can also now also pair two real coordinates of S2 with two coordinates of CP2 to get two complex coordinates. For the Euclidian regions of the space-time surface the situation is the same.
Consider now what the general solution could look like. Let us continue to use the shorthand notations S21= S2(X4); S22= S2(CP2);S23= S2(M4).
1. Consider first solution of type (1,0) so that coordinates of S22 are constant. One has holomorphy in hypercomplex sense (light-like coordinate t-z and t+z correspond to hypercomplex coordinates).

1. The general map T(X4) to T(M4) should be holomorphic in hyper-complex sense. S21 is in turn identified with S23 by isometry realized in real coordinates. This could be also seen as holomorphy but with different imaginary unit. One has analytical continuation of the map S21→ S23 to a holomorphic map. Holomorphy might allows to achieve this rather uniquely. The continued coordinates of S21 correspond to the coordinates assignable with the integrable surface defined by E2(x) for local M2(x)× E2(x) decomposition of the local tangent space of X4. Similar condition holds true for T(M4). This leaves only M2(x) as dynamical degrees of freedom. Therefore one has only one holomorphic function defined by 1-D data at the surface determined by the integrable distribution of M2(x) remains. The 1-D data could correspond to the boundary of the string world sheet.

2. The general map T(X4) to T(CP2) cannot satisfy holomorphy in hyper-complex sense. One can however provide the integrable distribution of E2(x) with complex structure and map it holomorphically to CP2. The map is defined by 1-D data.

3. Altogether, 2-D data determine the map determining space-time surface. These two 1-D data correspond to 2-D data given at string world sheet: one would have SH.

2. What about solutions of type (0,1) making sense in Euclidian region of space-time? One has ordinary holomorphy in CP2 sector.

1. The simplest picture is a direct translation of that for Minkowskian regions. The map S21→ S22 is an isometry regarded as an identification of real coordinates but could be also regarded as holomorphy with different imaginary unit. The real coordinates can be analytically continued to complex coordinates on both sides, and their imaginary parts define coordinates for a distribution of transversal Euclidian spaces E22(x) on X4 side and E2(x) on M4 side. This leaves 1-D data.

2. What about the map to T(M4)? It is possible to map the integrable distribution E22(x) to the corresponding distribution for T(M4) holomorphically in the ordinary sense of the word. One has 1-D data. Altogether one has 2-D data and SH and partonic 2-surfaces could carry these data. One has SH again.

3. The above construction works also for the solutions of type (1,1), which might make sense in Euclidian regions of space-time. It is however essential that the spheres S22 and S23 have real coordinates.

SH thus would thus emerge automatically from the twistor lift and holomorphy in the proposed sense.
1. Two possible complex units appear in the process. This suggests a connection with quaternion analytic functions suggested as an alternative manner to solve the field equations. Space-time surface as associative (quaterionic) or co-associate (co-quaternionic) surface is a further solution ansatz.

Also the integrable decompositions M2(x)× E2(x) resp. E21(x)× E22(x) for Minkowskian resp. Euclidian space-time regions are highly suggestive and would correspond to a foliation by string wold sheets and partonic 2-surfaces. This expectation conforms with the number theoretically motivated conjectures.

2. The foliation gives good hopes that the action indeed reduces to an effective action consisting of an area term plus topological magnetic flux term for a suitably chosen stringy 2-surfaces and partonic 2-surfaces. One should understand whether one must choose the string world sheets to be Lagrangian surfaces for the Kähler form including also M4 term. Minimal surface condition could select the Lagrangian string world sheet, which should also carry vanishing classical W fields in order that spinors modes can be eigenstates of em charge.

The points representing intersections of string world sheets with partonic 2-surfaces defining punctures would represent positions of fermions at partonic 2-surfaces at the boundaries of CD and these positions should be able to vary. Should one allow also non-Lagrangian string world sheets or does the space-time surface depend on the choice of the punctures carrying fermion number (quantum classical correspondence)?

3. The alternative option is that any choice produces of the preferred 2-surfaces produces the same scattering amplitudes. Does this mean that the string world sheet area is a constant for the foliation - perhaps too strong a condition - or could the topological flux term compensate for the change of the area?

The selection of string world sheets and partonic 2-surfaces could indeed be also only a gauge choice. I have considered this option earlier and proposed that it reduces to a symmetry identifiable as U(1) gauge symmetry for Kähler function of WCW allowing addition to it of a real part of complex function of WCW complex coordinates to Kähler action. The additional term in the Kähler action would compensate for the change if string world sheet action in SH. For complex Kähler action it could mean the addition of the entire complex function.

For details see the new chapter Some Questions Related to the Twistor Lift of TGD of "Towards M-matrix" or the article Some questions related to the twistor lift of TGD.

For a summary of earlier postings see Latest progress in TGD.

At 11:28 AM,  Crow- said...

"The 1-D data could correspond to the boundary of the string world sheet."

This is interesting.. lets say, everything in this post is "true" or at least most of it... would it be possible to personally experience this 1 dimensionality in some way? I ask this, because, during one of my meditations before sleeping, I recalled relatively mundane events from earlier in the day but was seeing them unfold from a 3rd person perspective.. the "view" was monochromatic and a little bit fuzzy, but there was a very bright spark looking thing that zigged and zagged very quickly and was apparently tracing out a path that I understand now to be perhaps a worldline or something like that. The path the spark was travelling had both timelike and spacelike aspects to it

At 7:14 PM,  Matti Pitkänen said...

My view is that sensory experience does not directly reflect it. The data needed to construct space-time surfaces are 2-D. Same about quantum states. This does not mean that experience would be 2-D. Many-sheeted space-time also complicates things.

There is also the number theretic discretisation as intersection of real and p-adic surfaces consting of algebraic points. These data are needed at least for p-adic continuation and define cognitive representations. This breaks strongest form of holography.

This picture is outcome of long work to understand in more detail how TGD variants of twistors would allow to construct general solution of field equations. Holomorphy becomes the key notion and quite expectedly.

The ultimate goal is to understand the construction of scattering amplitudes in terms of twistors and classical 2-D data about space-time surfaces and connections with Witten's twistor theory are highly suggestive.

At 9:33 AM,  Crow- said...

Interesting, thank you for the reply. It's probably just the mathematical part of my brain adding stuff to the regular part of the sleep/wake cycle. I suppose the conclusion is that there is not much of a connection between physical reality and sensory perception as it has evolved? Did evolution have no need for this 2 dimensionality then? How do we get along at all? I understand the thing about spacetime surfaces being like zombies in TGD and there being building blocks and all that.. indeed multi-sheetedness seems to be there.

Another thing, I cant believe, that in this day and age, there are "serious scientists" still spouting the nonsense of this multiverse idea that there exists some other universe where almost everything is the same except one minor detail or the other. I heard some astronomer on a national radio program saying this the other day... I think they've lost their minds, and TGD makes more sense than what I gather from the "popular scientific consensus"

At 6:34 PM,  Matti Pitkänen said...

There is a very deep connection between sensory reality and quantum theory. Basically all sensory and cognitive percepts are quantum measurements of some commuting observables.

For instance, visual color can be related to measurements of - well- color charges. This is not a joke. The algebra of color charges is similar to algebra of colors and this motivated the term color as a joke. Joke that turned to be not at all a joke.

Effective 2-dimensionality is only an approximate notion in adelic physics: also discrete data is needed in p-adic sector and correponds to the extensions of rationals. There is also the interpretation of predictions of quantum theory and here 4-D space-time is absolutely necessary. Physics is much more than the calculations. Every quantum measurement is interpereted in terms of concepts related to classical space-time. The people talking about holography usually forget completely this aspect.

The 2-D in information theoretic sense reduces to holomorhy properties of the twistor spaces of space-time surfaces as surfaces in twistor space of M^4xCP2. The idea is very simple: analytic function is coded by data at 1-D curve. M^8-H duality says that space-time surfaces can be regarded as quaternionic surfaces in M^8 and here quaternicity of tangent spaces of space-time surface determines dynamics (not quite actually, they contain integrable distribution of commutative 2-planes). Quaternion analyticity comes in play and means again that 1-D data correpond to function in 4-D.

Popular science is its own fairy world, which does not have much to do with real science. I think that multiverse lives only in popular science anymore. Emergence of space-time is the newest fashion. These fads live few years in science and then continue decade or two as fads in popular science. A lot of bestsellers are written. Science industry needs this.