Quantum geometry and M8-H duality

Quantum geometry and quantum metric are very interesting notions (see this). What does one mean with a quantum metric?

1. In condensed matter physics many particle states are labelled by particle momenta. Berry phase is associated with U(1) connection in this momentum space. Quantum metric means extension to Kähler metric involving both the U(1) connection and metric. It defines a distance between quantum states labelled by momenta at 2-D Fermi surface.
2. Quantum metric in condensed matter sense is defined in momentum space, or rather, Fermi surface, rather than Hilbert space. Here I disagree with the claim of (see this). At Hilbert space level geometrization would replace the flat Kähler metric with a curved metric and replace global linear superposition with its local variant.
3. What is essential for this interpretation, is that the momentum space (2-D Fermi sphere) is endowed with a Kähler geometry. Both momentum space and position space are geometrized.

In the TGD framework, M⁸-H duality implies something analogous but not equivalent.

1. Space-times are 4-surfaces X⁴ in H=M⁴× CP₂: this hypothesis geometrizes standard model interactions and solves the energy problem of general relativity. Holography = holomorphy hypothesis leads to an exactly solvable classical theory.
2. M⁸ serves as the analog of 8-D momentum space for H=M⁴× CP₂ and Y⁴ generalizes the notion of momentum space. One can define for the points of M⁸ Minkowskian inner product xy as a real part of their octonionic product.
3. Space-time surfaces X⁴ in H=M⁴× CP₂ have as M⁸ duals 4-surfaces Y⁴ in M⁸ related by M⁸-H duality. Y⁴ generalizes the notion of the Fermi sphere and can be regarded as the 4-D space of 4-momenta representing the dispersion relation (see this and this).

   Associativity condition for the tangent space of Y⁴ defines the number theoretic dynamics of Y⁴ and local G₂ transformations allow us to construct a general solutions Y⁴ from very simple basic solutions Y⁴(f) determined by the roots f(0)=0 of analytic functions f(o) with real coefficients, which can be restricted to an extension of rationals. Polynomials and rational functions f appear as important special cases and form hierarchies since basic arithmetic operations and functional composition produce new solutions.

4. How to define the analog of quantum geometry?
   1. The values of functions f(o) can be regarded as octonions such that the imaginary part is proportional to the radial octonion unit and thus allows interpretation as an ordinary imaginary unit. For two tangent vectors of x, y of quaternionic Y⁴ the real part of xy defines Minkowskian inner product. The product xy is a quaternion and could be seen as a quaternionic analog of Kähler form. An analog of quaternion structure would be in question. Could this define the number theoretic version of quantum geometry?
   2. CP₂ allows a quaternion structure but does not allow hyper-Kähler structure. Hyper-Kähler structure with 4 covariantly constant quaternionic units defined by metric and 3 covariantly constant Kähler forms is not possible for CP₂. And define induced quaternionic structure in X⁴. Could one induce the metric and spinor curvature of X⁴ to Y⁴?

      The quaternionic tangent spaces of Y⁴ are labelled by the points of CP₂ and the corresponding CP₂ point can be taken as a local coordinate of M⁸. The metric of Y⁴ could be induced from that of X⁴ by M⁸-H duality. In M⁴⊂ M⁸ degrees of freedom the inversion map M⁴→ M⁴⊂ H motivated by Uncertainty Principle defines the M⁸-H duality.

      There are singularities at which the CP₂ point associated with Y⁴ is not unique. In the case of CP₂ type extremals the CP₂ points form a 3-D surface X³ and X³ points correspond to single point y in Y⁴: Y⁴ has a coordinate singularity at y, y blows up to X³ in H.

   Could these two approaches give equivalent quaternionic quantum metrics in Y⁴?

   See the article "Does M⁸-H duality reduce to local G₂ symmetry?" or the chapter with the same title.

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

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.