Riemann Hypothesis, Unified field theory, Euler-Mascheroni constant

www.unified-field-theory.de

A.Einstein, "For us believing in physicists, the distinction between past, present and future is only a stubbornly persistent illusion".

H. Weyl, "We are not surprised that a concrete chunk of nature, taken in its isolated phenomenal existence, challenges our analysis by its inexhaustibility and incompleteness; it is for the sake of completeness, as we have seen, that physics projects what is given onto the background of the possible", (WeH) p. 220.

The phenomenological world
In the context of his concept of an "inner time consciousness" Husserl differentiates between

(1) an "objective time of appearing objects"
(2) "asubjective or pre-empirical time of acts"
(3) "experiences and a pre-phenominalabsolute flow of the internal time consciousness".

The physical world
In the physical world the concept "time" is an "objective time of appearing objects". About 95% of the universe is about the phenomenon „vacuum“. The same proportion applies to the emptyness between a proton and an electron. The remaining 5% of universe’s vacuum consists roughly of 5% matter, of 25% dark matter, and of 70% dark energy. Nearly all (about 99%) of the 5% matter in the universe is in "plasma state". The presumed existence of „dark matter“ provides the baseline for a physical model of the galaxies‘ spiral shapes phenomenon. The presumed existence of „dark energy“ provides the baseline for a physical model of the cosmic microwave background phenomenon (CMP).

The mathematical world
The proposed Hilbert scale & MHD based unified field model is built on variational PDE in appropriate Krein spaces. The overall mathematical solutions are governed by conservation laws in an "extended" H(1/2) energy Hilbert space. The related physical PDE solution are approximated by variational PDE governed by its underlying "coarse grained" (i.e. compactly embedded) standard kinematical energy sub-Hilbert space H(1) of H(1/2).

The Krein space is based on orthogonal decompositions of Hilbert space, enabling the definition of a potential operator governing the "borderlines" between those sub-spaces. In the proposed model there are two potential operator types in line with corresponding type I and type II Hilbert space decompositions:

type 1: one kinematical energy Hilbert space is compactly embedded into an overall Hilbert space (including a ground state energy concept); this is the proposed framework for a 1-fermion (proton) & a 2-boson types model accompanied by orthogonal discrete and continuous eigen-functions resp. eigen-differential spectra

type 2: a kinematical energy Hilbert space is decomposed into two sub-spaces with identical numbers of basis functions; this is the proposed framework for a 2-plasma-fermions (+/- electrons) & a 1-boson type model accompanied by orthogonal discrete and continuous eigen-functions resp. eigen-differential spectra

The Krein space framework
A Krein space is a Hilbert space H, which can be represented as an orthogonal decomposition H=H(+)+H(-) accompanied by the conceptual elements of

i) a self-adjoint operator B defined on all of the Hilbert space inducing this decomposition

ii) the notions "positivity", "negativity", and "neutrality" of the elements of H.

Property i) is supposed to build wanted Hamiltonian operators, like a PDE system independent definition of a potential operator or the solution operator of the Berry-Keating conjecture.

The Krein space framework addresses the following modelling requirements to unify the quantum mechanics & quantum field theory and the general relativity theory

- in plasma physics there is a symmetry between the number of positively andnegatively charged interacting particles per considered volume element

- in quantum theory there is an asymmetry between the kinetics and dynamics of elementary particles and the ground state

- in the Maxwell theory there has to be a physical „electron“ (fermion) object (a priori) existing outside of the Maxwell equations framework in order to make its motion happen governed by electric and magnetic field operators accompaniedby a sophisticated „displacement current“; a similar situation is given by the „inflaton“ concept in the „big bang theory“

- conceptually, the Maxwell equations (accompanied by an "a priori" electron-fermion) are the role model for the SMEP, which is basically a "A x B x C collection" of three theories. Each of those theories starts with a set of fermions. If a theory (like Maxwell or Yang-Mills theory) is invariant under transformations by a symmetry group one obtains a conservation law and quantum numbers. Gauge symmetries are locally that act differently at each space-time point. They automatically determine the interaction between particles by introducing bosons that mediate the interaction. U(1) describes the electromagnetic interaction with one boson and one quantum number; SU(2) describes the weak force interaction with 3 bosons; SU(3) describes the strong force interaction with 8 gluon bosons

- there is the mathematical "mass gap" problem of the Yang-Mills equations, Physically speaking, this is about the difference in energy between vacuum energy and the next level lowest kinematical energy level

- it has been questioned whether the NSE really describes general flows. The difficulty with ideal fluids is that in such fluids there are no frictional forces- the proportional parts of massive particle matter, plasma particle matter, the „vaccum“ and the unknown rest of the universe

- in quantum theory the elements of a single geometric Hilbert space framework provide the model of the „state“ of an elementary particle. This „state“ is described with two attributes (location and momentum); from a mathematical perspective, those attributes relate to two different Hilbert spaces within a Hilbert scale

- in GRT the Einstein space is supposed to provide the framework for a geometric model of the phenomenon „gravity“. However it is a purely metric space accompanied by infinite numbers of locally euclidian space-time „frameworks“; mathematically speaking, the Einstein space has no geometric structure, at all

- there is still Einstein’s unfinished mission going beyond quantum mechanics to discover what is missing from a true theory of the atoms, (SmL), probably supported by Bohm's concepts of an "implicite order" and an "enfolding-unfolding universe", (BoD)

- there are many time-related reasons why a wide range of attempted approaches to quantum gravity failed to be satisfactory, (AnE), (BaJ1)

- a common modelling framework for the Hilbert-Polya conjecture and the Berry-Keating conjecture.

Braun, K., The mathematical and physical realities of an unified field model

                                                 August 25, 2022
                                           Sept. 8 updates: pp. 5, 8

Braun, K., A mathematical tool box for an unified field model

August 25, 2020

Braun K., A geometric Hilbert space based integrated quantum and gravity field model

Dec 2, 2020

homepage text: August 15, 2022

homepage text, A HS and MHD based UFT, August 15, 2022