This homepage is dedicated to my mom, who died at April 9, 2020 in the age of 93 years. It considers multiple research areas. In retrospect, the proposed
solution concepts originate in some few simple ideas / basic conceptual
changes to current insufficient "solutions":
(A) A modified Zeta function theory is proposed to overcome current challenges
(a) to verify several Riemann Hypothesis (RH) criteria
(b) to prove the binary Goldbach conjecture
(a) the current two baseline functions to define
the Zeta functions, the Gaussian function and the (periodical) fractional part
function (resp. their corresponding Mellin transforms) are replaced by their
corresponding Hilbert transforms, which are the Dawson function (which is a
specific Kummer function) and the Fourier series representation of the log(sin)-function.
The convergence analysis is based on corresponding Hilbert space frameworks,
supporting especially Cardon’s „convolution operator representation“ and
Bagchi’s "Nyman-Beurling" RH criteria applied to the modified entire Zeta function. Thereby, the convolution operator representation goes along with convergent (Mellin transform) integrals, overcoming the corresponding challenge of an only formally valid self-adjoint invariant operator representation of the standard entire Zeta function ((EdH) 10.3). A corresponding effect is valid with respect to the log(sin) based periodical Hilbert space framework in the context of the Bagchi-"Nyman-Beurling" RH criteria
(b) the (periodical) Hilbert space
framework based on the Fourier series representation of the log(sin)-L(2)-function enables the definition of new
arithmetical functions going along with an alternative to the standard "Hardy-Littlewood circle
method" (with its underlying "open disk" domain), which is based on the „boundary of theunit
circle“ domain. The properties of the zeros of a specific
Kummer function (alternatively to the (Dawson
function related) exp(ix)-function) enable the definition of pairs of not-identical arithmetical functions to
analyse prime number pairs (p,q) of binary number theoretical problems. The proposed distributional Hilbert space framework also supports the verification of the Snirelmann density criterion to prove the Goldbach conjecture.
The Hilbert-Polya conjecture (which is
about the existence of a proper self-adjoint integral operator going
along with the concept of convolution operators, (CaD)) needs to
the mathematical problem of the not vanishing constant Fourier term
Jacobian theta function. Every Hilbert transformed function has a
vanishing constant Fourier term. With respect to (B) below, especially regarding a proof of the "Landau damping phenomenon", we note that a vanishing constant Fourier term of a L(2) function is a sufficient conditions to be a wavelet function.
changes to the previous (July 22, 2020) version: pp. 1,2,4,5
(B) A quantum gravity model requires
some goodbyes from current postulates of quantum mechanics/dynamics models and
Einstein’s field model as per definition both theories are not compatible:
(KaM) p. 12: „Because general relativity and quantum mechanics can be derived
from a small set of postulates, one or more of these postulates must be wrong.
The key must be to drop one of our commonsense assumptions about Nature (with
respect to the underlying physical models, which are (1)
continuity, (2) causality, (3) unitarity, (4) locality, (5) point particles),
on which we have constructued general relativity and quantum mechanics.“
The approach of this homepage is about challenging the
postulates of both theories with respect to the underlying mathematical postulated
The main gap of Dirac‘s quantum theory of radiation is the small
term representing the coupling energy of the atom and the radiation field.
The main gap of the Einstein field equations is, that it does
not fulfill Leibniz's requirement, that "there is no space, where no
matter exists"; the GRT field equations provide also solutions for a
vaccuum, i.e. the concept of "space-time" does not vanishes in
a matter-free universe.
The key ingredients of the proposed quantum gravity theory is about
differential forms equipped with an inner product of a distributional Hilbert
space. The (nonlinear) stability of the underlying Minkowski space framework
requires initial data sets with finite energy and linear and angular momentum
The variational representation of
the Maxwell equations in the proposed quantum element/energy Hilbert
space framwork (H(-1/2),H(1/2) conserves the two H(1)-based progressive (1-parameter (space or
time variable)) electric and magnetic waves concept while also allowing additional
standing (stationary) H(1,ortho)based (2-parameter) wavelets. The vaccuum
solution of the first ones conserves the linkage to the classical wave
equations for the electric and magnetic field (while this transformation still
requires additional, physical not relevant regularity requirements to the underlying
solution), while the second ones provides additional information regarding the
elementary particle dynamics.
Regarding the 3D NSE problem the
newly proposed "fluid element" Hilbert space H(-1/2) with
corresponding extended energy („momentum“, "velocity") space H(1/2) leads
to Ricci ODE estimates of order 1/2 enabling a corresponding bounded Sobolevskii
(energy inequality) estimate. Regarding the second unknown term of the NSE, the pressure, we note that "pressure" corresponds to "energy density" (Nm/volume ~ N/area).
The proposed quantum gravity
model overcomes the challenges to
prove well defined 3D-non-linear, non-stationary Navier-Stokes
Equations (the Serrin gap challenge) and Yang-Mills equations (the mass gap challenge).
The proposed quantum gravity model in a nutshell The
newly proposed energy Hilbert space H(1/2) (alternatively to the standard energy Hilbert space H(1)) is decomposed into a "kinematical"
energy / "kinematical" action Hilbert space H(1) and its complementary
"zero-point" energy & "zero-kinematical" action Hilbert space
H(1,ortho); mathematically speaking this is about a decomposition of the
Hilbert space H(1/2) into a "granular", compactly (dense) embedded
Hilbert space H(1) of H(1/2) and its complementary closed sub-space
H(1,ortho). Conceptually this decomposition corresponds to the
"decomposition" of the field of real numbers R into rational (countable)
numbers Q and irrational (non countable) numbers.
classical PDE model solutions are considered as approximation solutions
to the underlying weak variational formulation in the proposed Hilbert
space framework and not the other way around, e.g. (BrK6), (VeW). The
weak variational models are governed by a common energy model concept
(BrK), (BrK1), while the related "forces" phenomena become part of the specific corresponding classical PDE model, only.
The distributional (quantum state)
Hilbert space framework resp. its underlying norm (i.e. with its
underling "length measurements") is governed by the sum of the standard
(quantum mechanics / statistics) L(2)-Hilbert space norm and an "exponentical decay" (entropy measurements, (BrK1) note 2, (BrK6)) norm, which is weaker than any distributional "polynomial decay"
With additionally assumed regularity to the solutions of
the proposed weak PDE representations (which is without any quantum
theoretical physical meaning) the corresponding approximation solutions
of the related classical PDE are well defined (VeW), i.e. the
scalability from the "very small" quantum level to the "very large"
classical level is ensured. Therefore, for example the physical
concept of "force" (based on the Lagrange formalism) or the mathematical concept of "continuity" (due to the Sobolev embedding theorem) keeps being valid "on the classical PDE level" and the famous Einstein
equation E = m*c*c becomes an approximation formula, only, restricted to the
kinematical energy space H(1).
We note that the Fourier analysis
based applied spectral analysis methods (e.g. cosmoligical distance measurement or the Doppler effect in combination with the Hubble diagram leading to the interpretations of moving apart galaxies from each other galaxies with superluminal velocity in an expanding universe) is only defined in the „granular“ kinematical Hilbert
space framework H(1), i.e. the proposed quantum gravity
model allows an re-interpretation of the observed cosmological background
At the same point in time H. Weyl's requirement concerning a truly infinitesimal geometry are fulfilled as well, because ...
(WeH0): "… atruly infinitesimal geometry (wahrhafte Nahegeometrie) …
should know a transfer principle for
length measurements between infinitely close points only ..."
(WeH0) Weyl H., Gravitation und Elektrizität,
Sitzungsberichte Akademie der Wissenschaften Berlin, 1918, 465-48. https://arxiv.org/
The proposed model is
only about truly bosons w/o mass, modelled as elements of the H(1)-complementary
sub-space of the overall energy Hilbert space H(1/2). Therefore, the main gap
of Dirac‘s quantum theory of radiation, i.e. the small term representing the
coupling energy of the atom and the radiation field, becomes part of the H(1)-complementary
(truly bosons) sub-space of the overall energy Hilbert space H(1/2). It allows to revisit Einstein's thoughts on
ETHER AND THE THEORY OF RELATIVITY An Address delivered on May 5th, 1920, in the University of Leyden
in the context of the
space-time theory and the kinematics of the special theory of
relativity modelled on the Maxwell-Lorentz theory of the electromagnetic
Einstein’s field equations are
hyperbolic and allow so called „time bomb solutions“ which spreads along
bi-characteristic or characteristic hyper surfaces. Actual quantum theories are
talking about „inflations“, which blew up the germ of the universe in the very
first state. The inflation field due to these concepts are not smooth, but
containing fluctuation quanta. The action of those fluctuations create traces
into a large area of space. The existence of quantum fluctuations (in a „world“ without a time arrow and without entropy) has been verified by the Casimir and the Lamb shift effects.
The standard „big bang“ theory assumes that
the creation of the first mass particle (fermion) was the „birthday“ of the universe. This event
was caused by an „inflation“ energy field triggered by a „disturbance“, called
fluctuations. In the proposed quantum gravity model the „birthday“ of the
„granular“, compactly embedded fermion-energy Hilbert (sub-) space H(1) of
H(1/2) (coming along with the (kinematical) notions "space", "time", "action", etc.) is interpreted as first disturbance of the purely (pre-universe) boson
energy field H(1,ortho) with not existing entropy. The latter one can be
interpreted as the (in sync with the Casimir effect) not empty quantum vaccuum;
its oscillation is the cosmic background radiation, which contains all features
of dynamic energies.
With the „birthday“ of the fermions the correspondingly
adapted variational representation of the wave equation is then governed by the
purely kinematical (fermions) energy Hilbert space H(1), while its underlying
initial values are purely (undistorbed) vacuum (CBR, bosons) energy data from
H(1,ortho). As a consequence, the wave equation becomes time-asymmetric and the
second law of (kinematical) thermodynamics (the entropy phenomenon coming along
with the notions „mass“, „time“, „space“ etc.) can be interpreted (and derived
from this wave equation) as „action“ principle of the ground state energy to
damp and finally eliminate (remedy the deficiency) of any kinematical energy „disturbance“.
(C) Schopenhauer's"theory of explaining" (which he called "about the fourfold root of sufficient reason")
is about the different categories explaining the (his four) different
root causes & actions of the world's representations, answering the
"why?" question, based on the concept "something is, because something else has been before";
in today's world this would go along with the scope of all theoretical
physics & neuroscience phenomena/representations, but not including
the only suspected cause of a "big bang" "event".
Schopenhauer's "(the) world as will and representation" (written about 200 years ago) also addresses the "what?" question, which he answered with the concept of "will",
which is a kind of "vital principle" or "living energy" (or "living
force" according to Leibniz) affecting both, ("dead") matter and
In the context of this homepage this concept "will" might be
interpreted as analogy to the enlarged scope of the mathematical ("dark
energy", Einstein's vacuum "ether" energy) model as proposed in this homepage.
The linkage to the proposed quantum gravity model might be described with two quotes from Gyatso G. K., Modern Buddhism, The Path of Compassion and Wisdom,
Tharpa Publications UK, US, Canada, Australia, Asia, 2011
p. 113: „All phenomena that appear to my mind are the nature
of my mind. My mind is the nature of emptiness“
p. 120: „Emptiness is the true nature of all (mind produced)
phenomena (like clouds, mountains, planets, bodies, minds)“
(D) Officially accepted solutions of the considered research areas would be honored by several prizes. For hopefully understandable
reasons none of
the papers of this homepage are appropriately designed to go there. Therefore, after a 10 years long journey accompanied by four
main ingredients "fun, fun, fun and learning", it looks like a good point in time to share resp. enable more
fun to the readers‘ side, who showed their interest by more than
1 GB downloads per day (on average) during the last years. From (KoJ) p. 148 we quote:
„find a skillful motivation.
Then do the math and enjoy the creativity of the mind“
and, with the words of master Yoda: "may the Force be with you", ...:) .
For this purpose this page providing the MS-Word
based source documents of some key papers.
A small, closed building area to start with could be to go for "a truly proof of the observed non-linear
Landau damping phenomenon based on a variational representation of the
Disclaimer: None of the papers of this homepage have been
reviewed by other people; therefore there must be typos, but also errors for sure.
Nevertheless the fun part should prevail and if someone will become famous at
the end, it would be nice if there could be a reference found to this homepage somewhere.