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 3-surfaces of M4× CP2, the "world of classical worlds". Hamiltonians defined at δM4+/-× CP2 are basic elements of super-canonical algebra acting as isometries of the geometry of the "world of classical worlds". These Hamiltonians factorize naturally into products of functions of M4 radial coordinate rM which corresponds to a lightlike direction of lightcone boundary δM4+/- and functions of coordinates of rM constant sphere and CP2 coordinates. The assumption has been that the functions in question are powers of form (rM/r0)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+ps) 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 p-adic number field, the number piy belongs to some finite-dimensional algebraic extension of the p-adic number field in question.
- If the radial conformal weights are linear combinations of zeros of Zeta with integer coefficients, then for rational values of rM/r0 the exponents (rM/r0)Δ are in some finite-dimensional algebraic extension of the p-adic number field in question. This is crucial for the p-adicization of quantum TGD implying for instance that S-matrix 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 CP2 much like momenta have classical representation as 3-vectors. CP2 would play a role of heavenly sphere, so to say.
- The third hypothesis could be called braiding hypothesis.
- For a given parton surface X2 identified as intersection of Δ M4+/-× CP2 and lightlike partonic orbit X3l the images of radial conformal weights have interpretation as a braid.
- The Kac-Moody type conformal algebra associated with X3l restricted to X2 acts on the radial conformal weights like on points of complex plane. Also the Kac-Moody algebra of X3l acts on the radial conformal weights in a non-trivial manner. There exists a unique braiding operation defined by the dynamics of X2 defined by X3l . 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 II1 and emerging naturally as the infinite-dimensional Clifford algebra of the "world of the classical worlds".
- These braids define the finite sets of points which appear in the construction of universal S-matrix 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 S-matrix for say p-adic-to-real transitions representing transformation of intention to action using same formulas as for ordinary S-matrix.
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 CP2 coordinate expressible as
where ξ1 and ξ2 are the complex coordinates of CP2 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 space-time 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.
For a detailed argument see the chapter Equivalence of Loop Diagrams with
Tree Diagrams and Cancellation of Infinities in Quantum
- The inverse of zeta has infinitely many branches in one-one correspondence with the zeros of zeta and the branch can change only for certain values of rM 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 2-surface appearing in the construction of the number theoretically universal S-matrix correspond to images ζ(s) of points s=∑ nksk expressible as linear combinations of zeros of zeta with the additional condition that rM/r0 is rational. In this manner one indeed obtains representation of allowed conformal weights on the "heavenly sphere" defined by CP2 and also other hypothesis follow naturally.
- In this framework braids are actually replaced by tangles for which the strand of braid can turn backwards.