Zeros of Riemann Zeta as conformal weights, braids,
Jones inclusions, and number theoretical universality of quantum TGD
Quantum TGD relies on a heuristic number theoretical vision lacking a rigorous justification and I have made considerable efforts to reduce this picture to as few basic unproven assumptions as possible. In the following I want briefly summarize some recent progress made in this respect.
1. Geometry of the world of classical worlds as the basic context
The number theoretic conjectures has been inspired by the construction of the geometry of the configuration space consisting of 3surfaces of M^{4}× CP_{2}, the "world of classical worlds". Hamiltonians defined at δM^{4}_{+/}× CP_{2} are basic elements of supercanonical algebra acting as isometries of the geometry of the "world of classical worlds". These Hamiltonians factorize naturally into products of functions of M^{4} radial coordinate r_{M} which corresponds to a lightlike direction of lightcone boundary δM^{4}_{+/} and functions of coordinates of r_{M} constant sphere and CP_{2} coordinates. The assumption has been that the functions in question are powers of form (r_{M}/r_{0})^{Δ} where Δ has a natural interpretation as a radial conformal conformal weight.
2. List of conjectures
Quite a thick cloud of conjectures surrounds the construction of configuration space geometry and of quantum TGD.
 Number theoretic universality of Riemann Zeta states that the factors 1/(1+p^{s}) appearing in its product representation are algebraic numbers for the zeros s=1/2+iy of Riemann zeta, and thus also for their linear combinations. Thus for any prime p, any zero s, and any padic number field, the number p^{iy} belongs to some finitedimensional algebraic extension of the padic number field in question.
 If the radial conformal weights are linear combinations of zeros of Zeta with integer coefficients, then for rational values of r_{M}/r_{0} the exponents (r_{M}/r_{0})^{Δ} are in some finitedimensional algebraic extension of the padic number field in question. This is crucial for the padicization of quantum TGD implying for instance that Smatrix elements are algebraic numbers.
 Quantum classical correspondence is realized in the sense that the radial conformal weights Δ are represented as (mapped to) points of CP_{2} much like momenta have classical representation as 3vectors. CP_{2} would play a role of heavenly sphere, so to say.
 The third hypothesis could be called braiding hypothesis.
 For a given parton surface X^{2} identified as intersection of Δ M^{4}_{+/}× CP_{2} and lightlike partonic orbit X^{3}_{l} the images of radial conformal weights have interpretation as a braid.
 The KacMoody type conformal algebra associated with X^{3}_{l} restricted to X^{2} acts on the radial conformal weights like on points of complex plane. Also the KacMoody algebra of X^{3}_{l} acts on the radial conformal weights in a nontrivial manner. There exists a unique braiding operation defined by the dynamics of X^{2} defined by X^{3}_{l} . This operation is highly relevant for the model of topological quantum computation and TGD based model of anyons and quantum Hall effect.
 These braids relate closely to the hierarchy of braids providing representation for a Jones inclusion of von Neumann algebra known as hyperfinite factor of type II_{1} and emerging naturally as the infinitedimensional Clifford algebra of the "world of the classical worlds".
 These braids define the finite sets of points which appear in the construction of universal Smatrix whose elements are algebraic numbers and thus can be interpreted as elements of any number field. This would mean that it is possible to construct Smatrix for say padictoreal transitions representing transformation of intention to action using same formulas as for ordinary Smatrix.
3. The unifying hypothesis
The most recent progress in TGD is based on the finding that these separate hypothesis can be unified to single assumption. The radial conformal weights Δ are not constants but functions of CP_{2} coordinate expressible as
Δ= ζ^{1}(ξ^{1}/ξ^{2}),
where ξ^{1} and ξ^{2} are the complex coordinates of CP_{2} transforming linearly under subgroup U(2) of SU(3). The choice of this coordinate system is not completely unique and relates to the choice of directions of color isospin and hyper charge. This choice has a correlates at spacetime and configuration space level in accordance with the idea that also quantum measurement theory has geometric correlates in TGD framework. This hypothesis obviously generalizes the earlier assumption which states that Δ is constant and a linear combination of zeros of Zeta.
A couple of comments are in order.
 The inverse of zeta has infinitely many branches in oneone correspondence with the zeros of zeta and the branch can change only for certain values of r_{M} such that imaginary part of Δ changes: this has very interesting physical implications.
 Accepting the universality of the zeros of Riemann Zeta, one also ends up naturally with the hypothesis that the points of the partonic 2surface appearing in the construction of the number theoretically universal Smatrix correspond to images ζ(s) of points s=∑ n_{k}s_{k} expressible as linear combinations of zeros of zeta with the additional condition that r_{M}/r_{0} is rational. In this manner one indeed obtains representation of allowed conformal weights on the "heavenly sphere" defined by CP_{2} and also other hypothesis follow naturally.
 In this framework braids are actually replaced by tangles for which the strand of braid can turn backwards.
For a detailed argument see the chapter Equivalence of Loop Diagrams with
Tree Diagrams and Cancellation of Infinities in Quantum
TGD.
