Monday, June 10, 2024

Holography=holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M8-H duality

Holography = holomorphy vision generalizes the realization of quantum criticality in terms of conformal invariance. Holography = holomorphy vision provides a general explicit solution to the field equations determining space-time surfaces as minimal surfaces X4⊂ H=M4× CP2. For the first option the space-time surfaces are roots of two generalized analytic functions P1,P2 defined in H . For the second option single analytic generalized analytic function defines X4 as its root and as the base space of 6-D twistor twistor-surface X6 in the twistor bundle T(H)=T(M4)× TCP2) identified as a zero section.

By holography, the space-time surfaces correspond to not completely deterministic orbits of particles as 3-surfaces and are thus analogous to Bohr orbits. This implies zero energy ontology (ZEO) and to the view of quantum TGD as wave mechanics in the space of these Bohr orbits located inside a causal diamond (CD), which form a causal hierarchy. Also the consruction of vertices for particle reactions has evolved dramatically during the last year and one can assign the vertices to partonic 2-surfaces.

M8-H duality is a second key principle of TGD. M8-H duality can be seen a number theoretic analog for momentum-position duality and brings in mind Langlands duality. M8 can be identified as octonions when the number-theoretic Minkowski norm is defined as Re(o2). The quaternionic normal space N(y) of y∈ Y4⊂ M8 having a 2-D commutative complex sub-space is mapped to a point of CP2. Y4 has Euclidian signature with respect to Re(o2). The points y∈ Y4 are lifted by a multiplication with a co-quaternionic unit to points of the quaternionic normal space N(y) and mapped to M4⊂ H inversion.

This article discusses the relationship of the holography = holomorphy vision with the number theoretic vision predicting a hierarchy heff=nh0 of effective Planck constants such that n corresponds to the dimension for an extension rationals (or extension F of rationals). How could this hierarchy follow from the recent view of M8-H duality? Both realizations of holography = holomorphy vision assume that the polynomials involved have coefficients in an extension F of rationals Partonic 2-surfaces would represent a stronger form of quantum criticality than the generalized holomorphy: one could say islands of algebraic extensions F from the ocean of complex numbers are selected. For the P option, the fermionic lines would be roots of P and dP/dz inducing an extension of F in the twistor sphere. Adelic physics would emerge at quantum criticality and scattering amplitudes would become number-theoretically universal. In particular, the hierarchy of Planck constants and the identification of p-adic primes as ramified primes would emerge as a prediction.

Also a generalization of the theory of analytic functions to the 4-D situation is suggestive. The poles of cuts of analytic functions would correspond to the 2-D partonic surfaces as vertices at which holomorphy fails and 2-D string worlds sheets could correspond to the cuts. This provides a general view of the breaking of the generalized conformal symmetries and their super counterparts as a necessary condition for the non-triviality of the scattering amplitudes.

See the artice Holography = holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M8-H duality 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.

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