A Krein space based matter field theory is
The mathematical model is based on the concept of a hermitian energy operator represented as the sum of two self-adjoint kinematic and potential operators.
The (self-adjoint) kinetic energy operator is defined by the considered physical problem, which is in line with the variational potential operator theory. Its related eigen-pair solutions define a kinematical energy sub-Hilbert space of an overall energy Hilbert space framework. The embedding of this kinematical Hilbert space into an overall energy Hilbert space is compact. Therefore, the physical interpretation of the discrete eigenvalues of the kinematical energy Hilbert space is in line with the concept of "discrete energy knots" as proposed in the Mie theory, (which is about extended Maxwell equations), accompanied by Mie's proposed physical concept of an "electric pressure".
The (self-adjoint) potential energy operator is independently defined from the considered physical problem. It is an intrinsic part of three connected Krein spaces with related potential energy norms on all of those Krein spaces. The three norms are built on two elementary particles, electrinos and positrinos, enabling three groups of composed quantum elements, the vacuum,
the plasma, and the electromagnetic quantum potential elements.
Each group is governed by one of the three Krein spaces and all Krein spaces are related to an energy Hilbert space H(1/2). The concept of Mie's "electric pressure" may be interpreted as a "potential energy difference" governed by the Mie-Krein space norm. The related potential energy norm of a H(1/2)
potential energy Hilbert space
- makes the Coulomb potential obsolete - makes the SMEP obsolete - makes the Yang-Mills equations obsolete - solves the 3D-NSE problem (www.navier-stokes-equations.com).
The proposed plasma quantum potential model and the Riemann Hypothesis The link of the proposed plasma quantum potential
model and the Riemann Hypothesis is provided by
the Berry-Keating (Hilbert-Polya) conjecture. This is about the existence of an appropriately defined self-adjoint operator governing the considered energy levels.
Conceptually speaking, this is when "Number Theory Meets Quantum Mechanics", (DeJ). Or more specifically, this is when number theory meets plasma quantum dynamics providing and appropriate mathematical model for the (physical) Montgomery-Odlyzko law, (see below).
Technically speaking, this is when the proposed Hermitian plasma quantum potential
operator meets a Gaussian Unitary Ensemple (GUE, i.e., the joint probability density function of Gaussian unitary matrices elements and the joint probability density function of the eigenvalues, (MeM)). "Their eigenvalues provide an excellent fit for the energy levels of the behavior of certain quantum dynamics
systems, where their spacing turned out to be not spaced at random", (DeJ).
We note that in a certain sense Gaussian-random
Hermitian matrices governing the behavior of quantum dynamics
systems somehow also relates to the Landau
The Montgomery conjecture never was
proved, not even on the assumption that the RH is true. The claim is that
interpreting the Gaussian-random Hermitian matrices as approximation operators
to the proposed plasma quantum potential operator (interpreted as the Hermitian
Berry-Keating operator) will prove the Montgomery conjecture and the RH.
We also note that the distributional Hilbert space formulation of the Bagchi-Nyman-Beurling RH criterion, where the zeta function may be interpreted as a H(-1) distributional function (BaB), provides an appropriate framework to the proposed H(1/2) quantum potential energy Hilbert space equipped with an inner product induced by the Friedrichs extension of a symmetric and positive definite operator with a H(-1) domain.
The Montgomery-Odlyzko law in a nutshell (DeJ) p. 292: "The distribution of the spacings between successive non-trivial zeros of the Riemann zeta function (suitable normalized) is statistically identical with the distribution of eigenvalue spacing in a Gaussian Unitary Ensemble (i.e. a collection of Gaussian unitary operators that share some common statistical properties)".
(DeJ):p. 280 ff.: … The eigenvalues (of Gaussian-random Hermitian matrices)…
are struggling to keep their distance from each other. … The statistical
properties of spacings between long non-uniform string of numbers are encapsulated
in a creature called „pair correlations function“ and a certain ratio
associated with this function is called its „form factor“. … The form factor for the pair correlation of
random Hermitian matrices is the conjectured distribution function for the
differences between the non-trivial zeros of Riemann’s zeta function. … The following
points look pretty plausible on the basis of related comparing figures of „the
eigenvalues of a 269-by-269-random matrix“,(p. 285) & p. 289: „The first 269 values of „t“,
where ½+it is a non trivial zero of the zeta function“ (p. 289):
1. neither the zeta zeros nor the eigenvalues look
much like randomly scattered points 2. they resemble each other 3. in particular, they both show the repulsion (energy
level) effect, trying to get as far as possible from each other, like a long
standing line of antisocial people".