Saturday, September 24, 2022

Riemann zeta and the number theoretical vision of TGD

There are strong indications that Riemann zeta (see this) has a deep role in physics, in particular in the physics of critical systems. TGD Universe is quantum critical. For what quantum criticaly would mean at the space-time level (see this). This raises the question whether also Riemann zeta could have a deep role in TGD.

First some background relating to the number theoretic view of TGD.

  1. In TGD, space-time regions are characterized by polynomials P with rational coefficients. Galois confinement defines a universal mechanism for the formation of bound states. Momenta for virtual fermions have components, which are algebraic integers in an extension of rationals defined by a polynomial P characterizing space-time region. For the physical many fermion states, the total momentum as the sum of fermion momenta has components, which are integers using the unit defined by the size of the causal diamond (CD).

    This defines a universal number theoretical mechanism for the formation of bound states. The condition is very strong but for rational coefficients it can be satisfied since the sum of all roots is always a rational number as the coefficient of the first order term.

  2. Galois confinement implies that the sum of the mass squared values, which are in general complex algebraic numbers in E, is also an integer. Since the mass squared values correspond to conformal weights as also in string models, one has conformal confinement: states are conformal singlets. This condition replaces the masslessness condition of gauge theories (see this).

    Riemann zeta is not a polynomial but has infinite number of root. How could one end up with Riemann zeta in TGD? One can also consider the replacement of the rational polynomials with analytic functions with rational coefficients or even more general functions.

    1. For real analytic functions roots come as pairs but building many-fermion states for which the sum of roots would be a real integer, is very difficult and in general impossible.
    2. Riemann zeta and the hierarchy of its generalizations to extensions of rationals (Dedekind zeta functions) is however a complete exception! If the roots are at the critical line as the generalization of Riemann hypothesis assumes, the sum of the root and its conjugate is equal to 1 and it is easy to construct many fermion states as 2N fermion states, such that they have integer value conformal weight.

      One can wonder whether one could see Riemann zeta as an analog of a polynomial such that the roots as zeros are algebraic numbers. This is however not necessary. Could zeta and its analogies allow it to build a very large number of Galois singlets and they would form a hierarchy corresponding to extensions of rationals. Could they represent a kind of second abstraction level after rational polynomials?

    See the article About TGD counterparts of twistor amplitudes: part II or the chapter About TGD counterparts of twistor amplitudes.

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

    Articles related to TGD.

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