Thursday, February 03, 2022

Arnold's conjecture , generalization of Floer's theory, and TGD

There is a highly interesting popular article in Quanta Magazine with title "Mathematicians transcend the geometric theory of motion" (see this). The article explains the work of mathematicians Abouzaid and Blomberg represents a generalization of Floer homology, which, using popular terms, allows to "count holes" in the infinite-D spaces. The loop space is replaced with a more general space.
  1. The starting point is the conjecture by Arnold related to the Hamiltonian systems. These systems are defined in phase space whose points are pairs of position and momentum of the particle. This notion is extremely general in classical physics. The question of Arnold was whether there exist closed orbits, kind of islands of order, in the middle of oceans of chaos consisting of non-closed chaotic orbits. His conjecture was that there indeed exists and that homology theory allows us to understand them. These closed orbits would be minimal representatives for the homology equivalence classes of curves. These orbits are also critical.
  2. A 2-D example helps to understand the idea. How to understand the homology of torus? Morse theory is the tool.

    Consider the embedding of torus to 3-space. The height-coordinate H defines a function at torus. It has 4 critical points. H has maximum resp. minimum at the top resp. bottom of the torus. H has saddle points at the top and bottom of the "hole" of the torus. These correspond to two touching circles: the topology of the intersection changes. The situation is topologically critical and the criticality tells about the appearance of the "hole" in torus. The extrema code for the homology. Outside these points the topology of the intersection is a circle or two disjoint circles.

  3. One can deform the torus and also add handles to it to get topologies with a higher genus and reader is encouraged to see how the height function now codes for the homology and the appearance of "holes".
  4. This situation is finite-D and too simple to apply in the case of the space of orbits of a Hamiltonian system. Now the point of torus is replaced with a single orbit in phase space. This space is infinite-dimensional and the Morse theory does not generalize as such. The work of Abouzaid and Blomberg changes the situation.
I do not understand the technical aspects involved with the finding but it might have direct relevance for TGD.
  1. In the TGD Universe, space-time is a 4-surface in H=M4× CP2, in a loose sense an orbit of 3-surface. General Coordinate Invariance (GCI) requires that the dynamics associates to a given 3-surface a highly unique 4-surface at which the 4-D general coordinate transformations act. This 4-surface is a preferred extremal of the action principle determing space-time surfaces in H and analogous to Bohr orbit. GCI gives Bohr orbitology as an exact part of quantum theory and also holography.

    These preferred extremals as 4-surfaces are analogous to the closed orbits in Hamiltonian systems about which Arnold speculated. In the TGD Universe, only these preferred extremals would be realized and would make TGD an integrable theory. The theorem of Abouzaid and Blomberg allows to prove Arnold's conjecture in homologies based on cyclic groups Zp. Maybe it could also have use also in the TGD framework.

  2. WCW generalizes the loop space considered in Floer's approach. Very loosely, loop or string is replaced by a 3-D surface, which by holography induced is more or less equivalent with 4-surface. In TGD just these minimal representatives for homology as counterparts of closed orbits would matter.
  3. Symplectic structure and Hamiltonian are central notions also in TGD. Symplectic (or rather, contact) transformations assignable to the product δ M4+× CP2 of the light-cone boundary and CP2 act as the isometries of the infinite-D "world of classical worlds" (WCW) consisting of these preferred extremals, or more or less equivalently, corresponding 3-surfaces. Hamiltonian flows as 1-parameter subgroups of isometries of WCW are symplectic flows in WCW with symplectic structure and also Käehler structure.
  4. The space-time surfaces are 4-D minimal surfaces in H with singularities analogous to frames of soap films. Minimal surfaces are known to define representatives for homological equivalence classes of surfaces. This has inspired the conjecture that TGD could be seen as a topological/homological quantum theory in the sense that space-time surfaces served as unique representatives or their homological classes.
  5. There is also a completely new element involved. TGD can be seen also as number theoretic quantum theory. M8-H duality can be seen as a duality of a geometric vision in which space-times are 4-surfaces in H an of a number theoretic vision in which one consideres 4-surfaces in octonionic complexified M8 determined by polynomials with dynamics reducing to the condition that the normal space of 4-surface is associative (quaternionic). M8 is analogous to momentum space so that a generalization of momentum-position duality of wave mechanics is in question.
A generalization of Floer's theory allowing to generalize Arnold's conjecture is needed and the approach of Abouzaid and Blomberg might make such a generalization possible.
  1. The preferred extremals would correspond to the critical points of an analog of Morse function in the infinite-D context. In TGD the Kähler function K defining the Kahler geometry of WCW is the unique candidate for the analog of Morse function.

    The space-time surfaces for which the exponent exp(-K) of the Kähler function is stationary (so that the vacuum functional is maximum) would define preferred extremals. Also other space-time surfaces could be allowed and it seems that the continuity of WCW requires this. However the maxima or perhaps extrema would provide an excellent approximation and number theoretic vision would give an explicit realization for this approximation.

  2. A stronger condition would be that only the maxima are allowed. Since WCW Kähler geometry has an infinite number of zero modes, which do not appear in the line elements as coordinate differentials but only as parameters of the metric tensor, one expects an infinite number of maxima and this might be enough.
  3. These maxima or possibly also more general preferred extrema would correspond by M8-H duality to roots of polynomials P in the complexified octonionic M8 so that a connection with number theory emerges. M8-H duality strongly strongly suggests that exp(-K) is equal to the image of the discriminant D of P under canonical identification I: ∑ xnpn→ ∑ xnp-n mapping p-adic numbers to reals. The prime p would correspond to the largest ramified prime dividing D (see this and this).
  4. The number theoretic vision could apply only to the maxima/extrema of exp(-K) and give rise to what I call a hierarchy of p-adic physics as correlates of cognition. Everything would be discrete and one could speak of a generalization of computationalism allowing also the hierarchy of extensions of rationals instead of only rationals as in Turing's approach. The real-number based physics would also include the non-maxima via a perturbation theory involving a functional integral around the maxima. Here Kähler geometry allows to get rid of ill-defined metric and Gaussian determinants.
For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

No comments: