### About the definition of Hamilton-Jacobi structure

I have talked in previous postings a lot about Hamilton-Jacobi structure without bothering to write detailed definitions. In the following I discuss the notion in more detail. Thanks for Hamed who asked for more detailed explanation.

**Hermitian and hyper-Hermitian structures**

The starting point is the observation that besides the complex numbers forming a number field there are hyper-complex numbers. Imaginary unit i is replaced with e satisfying e^{2}=1. One obtains an algebra but not a number field since the norm is Minkowskian norm x^{2}-y^{2}, which vanishes at light-cone x=y so that light-like hypercomplex numbers x+/- e) do not have inverse. One has "almost" number field.

Hyper-complex numbers appear naturally in 2-D Minkowski space since the solutions of a massless field equation can be written as f=g(u=t-ex)+h(v=t+ex) whith e^{2}=1 realized by putting e=1. Therefore Wick rotation relates sums of holomorphic and antiholomorphic functions to sums of hyper-holomorphic and anti-hyper-holomorphic functions. Note that u and v are hyper-complex conjugates of each other.

Complex n-dimensional spaces allow Hermitian structure. This means that the metric has in complex coordinates (z_{1},....,z_{n}) the form in which the matrix elements of metric are nonvanishing only between z_{i} and complex conjugate of z_{j}. In 2-D case one obtains just ds^{2}=g_{zz*}dzdz*. Note that in this case metric is conformally flat since line element is proportional to the line element ds^{2}=dzdz* of plane. This form is always possible locally. For complex n-D case one obtains ds^{2}=g_{ij*}dz^{i}dz^{j*}. g_{ij*}=(g_{ji*})* guaranteing the reality of ds^{2}. In 2-D case this condition gives g_{zz*}= (g_{z*z})*.

How could one generalize this line element to hyper-complex n-dimensional case? In 2-D case Minkowski space M^{2} one has ds^{2}= g_{uv}dudv, g_{uv}=1. The obvious generalization would be the replacement ds^{2}=g_{uivj}du^{i}dv^{j}. Also now the analogs of reality conditions must hold with respect to u^{i}↔ v^{i}.

**Hamilton-Jacobi structure**

Consider next the path leading to Hamilton-Jacobi structure.

4-D Minkowski space M^{4}=M^{2}× E^{2} is Cartesian product of hyper-complex M^{2} with complex plane E^{2}, and one has ds^{2}= dudv+ dzdz* in standard Minkowski coordinates. One can also consider more general integrable decompositions of M^{4} for which the tangent space TM^{4}=M^{4} at each point is decomposed to M^{2}(x)× E^{2}(x). The physical analogy would be a position dependent decomposition of the degrees of freedom of massless particle to longitudinal ones (M^{2}(x): light-like momentum is in this plane) and transversal ones (E^{2}(x): polarization vector is in this plane). Cylindrical and spherical variants of Minkowski coordinates define two examples of this kind of coordinates (it is perhaps a good exercize to think what kind of decomposition of tangent space is in question in these examples). An interesting mathematical problem highly relevant for TGD is to identify all possible decompositions of this kind for empty Minkowski space.

The integrability of the decomposition means that the planes M^{2}(x) are tangent planes for 2-D surfaces of M^{4} analogous to Euclidian string world sheet. This gives slicing of M^{4} to Minkowskian string world sheets parametrized by euclidian string world sheets. The question is whether the sheets are stringy in a strong sense: that is minimal surfaces. This is not the case: for spherical coordinates the Euclidian string world sheets would be spheres which are not minimal surfaces. For cylindrical and spherical coordinates hower M^{2}(x) integrate to plane M^{2} which is minimal surface.

Integrability means in the case of M^{2}(x) the existence of light-like vector field J whose flow lines define a global coordinate. Its existence implies also the existence of its conjugate and together these vector fields give rise to M^{2}(x) at each point. This means that one has J= Ψ∇ Φ: Φ indeed defines the global coordinate along flow lines. In the case of M^{2} either the coordinate u or v would be the coordinate in question. This kind of flows are called Beltrami flows. Obviously the same holds for the transversal planes E^{2}.

One can generalize this metric to the case of general 4-D space with Minkowski signature of metric. At least the elements g_{uv} and g_{zz*} are non-vanishing and can depend on both u,v and z,z*. They must satisfy the reality conditions g_{zz*}= (g_{zz*})* and g_{uv}= (g_{vu})* where complex conjugation in the argument involves also u↔ v besides z↔ z*.

The question is whether the components g_{uz}, g_{vz}, and their complex conjugates are non-vanishing if they satisfy some conditions. They can. The direct generalization from complex 2-D space would be that one treats u and v as complex conjugates and therefore requires a direct generalization of the hermiticity condition

g_{uz}= (g_{vz*})*, g_{vz}= (g_{uz*})* .

This would give complete symmetry with the complex 2-D (4-D in real sense) spaces. This would allow the algebraic continuation of hermitian structures to Hamilton-Jacobi structures by just replacing i with e for some complex coordinates.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.

## 6 Comments:

Dear Matti,

I write what I understand to represent my misunderstandings from your writings and then ask my questions between them.

You decompose locally the tangent space of M4 into direct sum of M2+E2 . M2 has complex structure and E2 has hypercomplex structure.

But M4 is a flat space and the tangent space at each point of it, is like the other points. Therefore why do you decompose locally the tangent space and not globally? Maybe this is because the space-time surface in M4*Cp2 is not flat and this leads to the projection of the space-time surface to M4 is not flat, therefore one must decompose locally the tangent space of M4 at each point to M2+E2?

I understand a globally decomposition is this: S+=t+ez , s-=t-ez , w+= x+iy , w-=x-iy

Space-time surfaces have dual slicings to string world surfaces and partonic 2-surfaces. I have very dimly understanding about the two expressions. Now I want to understand them by the last decomposition of M4:

Projection of the space time surface into M2 and E2 spaces are correspond to partonic surface and string world surface respectively? Can be regard it a definition of the two expressions and start from this like a base and deduce properties of partonic surfaces and string world surfaces? If it can, is it possible for you to give me the physical intuition of them by starting from the base?

Dear Hamed,

thank you for questions. You probably meant to say that M^2 (with signature (1,-1)) has hypercomplex structure and E^2 (with signature (-1,-1) complex structure;-).

Consider first M^4.

a) Linear M^4 coordinates define one example of H-J structure. In this case choices of M^2 and E^2 do not vary spatially. Physically they correspond to the planes in which light-like momentum and polarization lie and thus to simplest plane waves for massless field. Lorentz transformations define non-equivalent structures of this kind and this is true quite generally: they define something analogous to a moduli space of complex structures.

b) I want to generalize: local momentum direction and polarization direction orthogonal to each other. If this decomposition is integrable one can assign to it slicings by 2-D surfaces with Minkowskian and Euclidian signatures respectively.

c) The two kinds of surfaces obtained from integrable distributions of respective 2-planes are well defined. The mathematical challenge is to deduce all possible decomposition in this manner. In the case of M^4 this should be possible: spherical and cylindrical variants of Minkowski coordinates provide 2 examples of these decompositions. In the case of X^4 situation is much more non-trivial since one must know the preferred extremals! Also non-flatness that you mentioned becomes crucial: consider only wormhole throats and also the possibility of both Minkowskian and Eucildian signature of induced metric.

It is important to distinguish between general 2-surfaces in this slicing and with what I have called string world sheets and partonic 2-surfaces. I have certainly not done this systematically;-). String world sheets and partonic 2-surfaces can be defined as special 2-surfaces in these slicings: they correspond to light-like orbit of wormhole throat (4-metric effectively 3-D) and string world sheets at which the signature of the effective metric defined by modified gammas changes signature from (1,1,-1,-1) to (-1-1,-1,-1) so that space-time (or rather its tangent space) becomes locally 2-D with respect to effective metric.

To be continued....

Continuation of the answer to Hamed....

The physical vision is more or less the following and inspired by AdS^5 duality and general coordinate invariance in strong form (GCI).

a) AdS^5 duality suggest that string world sheets are in the same role as string world sheets of 10-D space connecting branes in AdS^5 duality for N=4 SYM. What is important is that there is duality and two manners to calculate the amplitudes. What it could mean now?

My guess inspired by strong GCI is that string world sheet -partonic 2-surface duality and its generalization to arbitrary pair of surfaces in the slicing holds true. The functional integrals over the 2 kinds of 2-surfaces should give same result and functional integration over either kinds of 2-surfaces is enough. Note that the members of a given pair intersect at discrete set of points and these points defined braid ends carrying fermion number. Discretization and braid picture follows automatically.

b) Scattering amplitudes in twistorial approach can be calculated by using *any* pair in the slicing - or any half of it if a) is true. The possibility to choose any pair in the slicing means general coordinate invariance symmetry of metric of WCW and of the entire theory. For a general pair one obtains functional integral over deformations of space-time surface inducing deformations of 2-surfaces with only other kind 2-surface contributing to amplitude. This means the analog of QFT in 2-D spaces: something very stringy. Minkowskian or Euclidian strings.

c) When string world sheets and partonic 2-surfaces are used one obtains enormous simplification. The propagators for fermions and correlation functions for deformations reduce to 1-D instead of being 2-D: propagation takes place only along the light-like lines at which the string world sheets with Euclidian signature (inside CP_2 like regions) change to those with Minkowskian signature of induced metric. This would be enormous simplification.

These lines represent orbits of point like particles carrying fermion number at orbits of wormhole throats. Furthermore, the expansions coming from string world sheets and partonic 2-surfaces are identical automatically! Duality is trivially true. This would be a "proof" for the analog of AdS^5 duality for N=4 SYMs. One can indeed use integral over either string world sheets or partonic 2-sheets to deduce the amplitudes.

This kind of Feynman diagrammatics involving only braid strands is what I have indeed ended up earlier so that it seems that I can trust good intuition combined with a sloppy mathematics sometimes works;-).

In his Strings 2012 talk Nima tells that they have also realized that 2-surfaces must be there at space-time level as carriers of singularities and their next goal is to identify them. In TGD string world sheets and partonic 2-surfaces would be these singular surfaces. The singularities of Feynman amplitudes central for their approach would have singularity of the induced and effective metric as space-time counterpart.

Dear Hamed,

warning!

I decided to add to blog posting about Nima's talk something and ended up to talk about partonic 2-surfaces and string world sheets as generic 2-surfaces in the slicing of space-time surface.

The special surfaces at which effective metric is degenerate I call singular partonic 2-surfaces and string world sheets.

Sorry for this fuss. It is difficult to be consistent in terminology.

Dear Matti,

Thank you very much. i am in struggling with your answers.

Best wishes for you.

I added a little posting about the generalization of AdS^5 duality to TGD framework and the simplification of perturbative approach for singular string world sheets and partonic 2-surface: strings would reduce effectively to points.

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