M⁸-H duality, preferred extremals, criticality, and Mandelbrot fractals

M⁸-H duality (see this) represents an intriguing connection between number theory and TGD but the mathematics involved is extremely abstract and difficult so that I can only represent conjectures. In the following the basic duality is used to formulate a general conjecture for the construction of preferred extremals by iterative procedure. What is remarkable and extremely surprising is that the iteration gives rise to the analogs of Mandelbrot fractals and space-time surfaces can be seen as fractals defined as fixed sets of iteration. The analogy with Mandelbrot set can be also seen as a geometric correlate for quantum criticality.

M⁸-H duality

M⁸-H duality states the following. Consider a distribution of two planes M²(x) integrating to a 2-surface N² with the property that a fixed 1-plane M¹ defining time axis globally is contained in each M²(x) and therefore in N². M¹ defines real axis of octonionic plane M⁸ and M²(x) a local hyper-complex plane. Quaternionic subspaces with this property can be parameterized by points of CP₂. Define quaternionic surfaces in M⁸ as 4-surfaces, whose tangent plane is quaternionic at each point x and contains the local hyper-complex plane M²(x) and is therefore labelled by a point s(x)∈ CP₂. One can write these surfaces as union over 2-D surfaces associated with points of N²:

X⁴= ∪_{x∈ N²} X²(x)⊂ E⁶ .

These surfaces can be mapped to surfaces of M⁴× CP₂ via the correspondence (m(x),e(x))→ (m,s(T(X⁴(x)). Also the image surface contains at given point x the preferred plane M²(x) ⊃ M¹. One can also write these surfaces as union over 2-D surfaces associated with points of N²:

X⁴= ∪_{x∈ N²} X²(x)⊂ E²× CP₂ .

One can also ask what are the conditions under which one can map surfaces X⁴= ∪_{x∈ N²} X²⊂ E²× CP₂ to 4-surfaces in M⁸. The map would be given by (m,s)→ (m,T⁴(s) and the surface would be of the form as already described. The surface X⁴ must be such that the distribution of 4-D tangent planes defined in M⁸ is integrable and this gives complicated integrability conditions. One might hope that the conditions might hold true for preferred extremals satisfying some additional conditions.

One must make clear that these conditions do not allow most general possible surface. The point is that for preferred extremals with Euclidian signature of metric the M⁴ projection is 3-dimensional and involves light like projection. Here the fact that light-like line L⊂ M² spans M² in the sense that the complement of its orthogonal complement in M⁸ is M². Therefore one could consider also more general solution ansatz for which one has

X⁴= ∪_{x∈ Lx⊂ N²} X³(x)⊂ E²× CP₂ .

One can also consider co-quaternionic surfaces as surfaces for which tangent space is in the dual of a quaternionic subspace containing the preferred M²(x).

The integrability conditions

The integrability conditions are associated with the expression of tangent vectors of T(X⁴) as a linear combination of coordinate gradients ∇ m^k, where m^k denote the coordinates of M⁸. Consider the 4 tangent vectors e_i) for the quaternionic tangent plane (containing M²(x)) regarded as vectors of M⁸. e_i) have components e_i)^k, i=1,..,4, k=1,...,8. One must be able to express e_i) as linear combinations of coordinate gradients ∇ m^k:

e_i)^k= e_i)^α∂_αm^k .

Here x^α and e^k denote coordinates for X⁴ and M⁸. By forming inner products of of e_i) one finds that matrix e_i)^α represents the components of vierbein at X⁴. One can invert this matrix to get eⁱ⁾_α satisfying

eⁱ⁾_αe_i)^β=δ_α^β

and

eⁱ⁾_αe_j)^α=δⁱ_j.

One can solve the coordinate gradients ∇ m^k from above equation to get

∂_αm^k = eⁱ⁾_αe_i)^k== E_α^k .

The integrability conditions follow from the gradient property and state

D_αE^k_β= D_βE^k_α .

One obtains 8× 6=48 conditions in the general case. The slicing to a union of two-surfaces labeled by M²(x) reduces the number of conditions since the number of coordinates m^k reduces from 8 to 6 and one has 36 integrability conditions but still them is much larger than the number of free variables- essentially the six transversal coordinates m^k.

For co-quaternionic surfaces one can formulate integrability conditions now as conditions for the existence of integrable distribution of orthogonal complements for tangent planes and it seems that the conditions are formally similar.

How to solve the integrability conditions and field equations for preferred extremals?

The basic idea has been that the integrability condition characterize preferred extremals so that they can be said to be quaternionic in a well-defined sense. Could one imagine solving the integrability conditions by some simple ansatz utilizing the core idea of M⁸-H duality? What comes in mind is that M⁸ represents tangent space of M⁴× CP₂ so that one can assign to any point (m,s) of 4-surface X⁴⊂ M⁴× CP₂ a tangent plane T⁴(x) in its tangent space M⁸ identifiable as subspace of complexified octonions in the proposed manner. Assume that s∈ CP₂ corresponds to a fixed tangent plane containing M²_x, and that all planes M²_x are mapped to the same standard fixed hyper-octonionic plane M²⊂ M⁸, which does not depend on x. This guarantees that s corresponds to a unique quaternionic tangent plane for given M²(x).

Consider the map Tοs. The map takes the tangent plane T⁴ at point (m,e)∈ M⁴× E⁴ and maps it to (m,s₁=s(T⁴))∈ M⁴× CP₂. The obvious identification of quaternionic tangent plane at (m,s₁) would be as T⁴. One would have Tοs=Id. One could do this for all points of the quaternion surface X⁴⊂ E⁴ and hope of getting smooth 4-surface X⁴⊂ H as a result. This is the case if the integrability conditions at various points (m,s(T⁴)(x))∈ H are satisfied. One could equally well start from a quaternionic surface of H and end up with integrability conditions in M⁸ discussed above. The geometric meaning would be that the quaternionic surface in H is image of quaternionic surface in M⁸ under this map.

Could one somehow generalize this construction so that one could iterate the map Tοs to get Tοs=Id at the limit? If so, quaternionic space-time surfaces would be obtained as limits of iteration for rather arbitrary space-time surface in either M⁸ or H. One can also consider limit cycles, even limiting manifolds with finite-dimension which would give quaternionic surfaces. This would give a connection with chaos theory.

1. One could try to proceed by discretizing the situation in M⁸ and H. One does not fix quaternionic surface at either side but just considers for a fixed m₂∈ M²(x) a discrete collection X {T⁴_i⊃ M²(x) of quaternionic planes in M⁸. The points e_2,i⊂ E²⊂ M2× E²=M⁴ are not fixed. One can also assume that the points s_i=s(T⁴_i) of CP₂ defined by the collection of planes form in a good approximation a cubic lattice in CP₂ but this is not absolutely essential. Complex Eguchi-Hanson coordinates ξⁱ are natural choice for the coordinates of CP₂. Assume also that the distances between the nearest CP₂ points are below some upper limit.
2. Consider now the iteration. One can map the collection X to H by mapping it to the set s(X) of pairs ((m₂,s_i). Next one must select some candidates for the points e_2,i∈ E²⊂ M⁴ somehow. One can define a piece-wise linear surface in M⁴× CP₂ consisting of 4-planes defined by the nearest neighbors of given point (m₂,e_2,i,s_i). The coordinates e_2,i for E²⊂ M⁴ can be chosen rather freely. The collection (e_2,i,_i) defines a piece-wise linear surface in H consisting of four-cubes in the simplest case. One can hope that for certain choices of e_2,i the four-cubes are quaternionic and that there is some further criterion allowing to choose the points e_2,i uniquely. The tangent planes contain by construction M²(x) so that the product of remaining two spanning tangent space vectors (e₃,e₄) must give an element of M² in order to achieve quaternionicity. Another natural condition would be that the resulting tangent planes are not only quaternionic but also as near as possible to the planes T⁴_i. These conditions allow to find e_2,i giving rise to geometrically determined quaternionic tangent planes as near as possible to those determined by sⁱ.
3. What to do next? Should one replace the quaternionic planes T⁴_i with geometrically determined quaternionic planes as near as possible to them and map them to points s_i slightly different from the original one and repeat the procedure? This would not add new points to the approximation, and this is an unsatisfactory feature.
4. Second possibility is based on the addition of the quaternionic tangent planes obtained in this manner to the original collection of quaternionic planes. Therefore the number of points in discretization increases and the added points of CP₂ are as near as possible to existing ones. One can again determine the points e_2,i in such a manner that the resulting geometrically determined quaternionic tangent planes are as near as possible to the original ones. This guarantees that the algorithm converges.
5. The iteration can be stopped when desired accuracy is achieved: in other words the geometrically determined quaternionic tangent planes are near enough to those determined by the points s_i. Also limit cycles are possible and would be assignable to the transversal coordinates e_2i varying periodically during iteration. One can quite well allow this kind of cycles, and they would mean that e₂ coordinate as a function of CP₂ coordinates characterizing the tangent plane is many-valued. This is certainly very probable for solutions representable locally as graphs M⁴→ CP₂. In this case the tangent planes associated with distant points in E² would be strongly correlated which must have non-trivial physical implications. The iteration makes sense also p-adically and it might be that in some cases only p-adic iteration converges for some value of p.

One might hope that the self-referentiality condition sοT=Id for the CP₂ projection of (m,s) or its fractal generalization could solve the complicated integrability conditions for the map T. The image of the space-time surface in tangent space M⁸ in turn could be interpreted as a description of space-time surface using coordinates defined by the local tangent space M⁸. Also the analogy for the duality between position and momentum suggests itself.

Is there any hope that this kind of construction could make sense? Or could one demonstrate that it fails? If s would fix completely the tangent plane it would be probably easy to kill the conjecture but this is not the case. Same s corresponds for different planes M²_x to different point tangent plane. Presumably they are related by a local G₂ or SO(7) rotation. Note that the construction can be formulated without any reference to the representation of the imbedding space gamma matrices in terms of octonions. Complexified octonions are enough in the tangent space of M⁸.

Connection with Mandelbrot fractal and fractals as fixed sets for iteration

The occurrence of iteration in the construction of preferred extremals suggests a deep connection with the standard construction of 2-D fractals by iteration - about which Mandelbrot fractal is the canonical example. X²(x) (or X³(x)) could be identified as a union of orbits for the iteration of sοT. The appearance of the iteration map in the construction of solutions of field equation would answer positively to a long standing question whether the extremely beautiful mathematics of 2-D fractals could have some application at the level of fundamental physics according to TGD.

X² (or X³) would be completely analogous to Mandelbrot set in the sense that it would be boundary separating points in two different basis of attraction. In the case of Mandelbrot set iteration would take points at the other side of boundary to origin on the other side and to infinity. The points of Mandelbrot set are permuted by the iteration. In the recent case sοT maps X² (or X³) to itself. This map need not be diffeomorphism or even continuous map. The criticality of X² (or X³) could be seen as a geometric correlate for quantum criticality.

In fact, iteration plays a very general role in the construction of fractals. Very general fractals can be defined as fixed sets of iteration and simple rules for iteration produce impressive representations for fractals appearing in Nature. Therefore it would be highly satisfactory if space-time surfaces would be in well-defined sense fixed sets of iteration. This would be also numerically beautiful aspect since fixed sets of iteration can be obtained as infinite limit of iteration for almost arbitrary initial set.

What is intriguing and challenging is that there are several very attractive approaches to the construction of preferred extremals and the challenge of unifying them still remains.

For details and background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the same title.