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A. Einstein, "We can't solve problems by using the same kind of thinking we used when we created them".
This homepage provides solutions to the
following Millennium problems
(1) the Riemann Hypothesis
(2) the 3D-nonlinear, non-stationary Navier-Stokes equations problem
(3) the mass gap problem of the Yang-Mills equations.
A common distributional
Hilbert space framework enables
(4) a quantum gravity theory.
The proposed quantum
gravity theory is based on an only (energy related) Hamiltonian formalism, as the
corresponding (force related) Lagrange formalism is no longer defined due to the
reduced regularity assumptions to the domains of the concerned pseudo differential operators. It provides an answer to Derbyshine's question ((DeJ) p.
295): „ What on earth does the distribution of prime numbers
have to do with the behavior of subatomic particles?".
(1) The key ingredients of the Zeta function
theory are the Mellin transforms of the Gaussian function and the fractional
part function. To the author´s humble opinion the main handicap to prove the
RH is the not-vanishing constant Fourier term of both functions. The Hilbert
transform of any function has a vanishing constant Fourier term. Replacing the Gaussian function
and
the fractional part function by their corresponding Hilbert transforms enables an alternative
Zeta function theory based on two specific Kummer functions and the cotangens function. The imaginary part of the zeros of one of the Kummer functions play a key role defining alternatively proposed arithmetic functions to solve the binary Goldbach conjecture.
(2) The common distributional
Hilbert space framework goes along with reduced regularity assumptions for the domain of the momentum (or pressure) operator. In the context of the 3-D-NSE problem this enables energy norm estimates "closing" the Serrin gap, while at the same point in time overcoming current "blow-up" effect handicaps.
(3) The classical Yang-Mills theory is the generalization
of the Maxwell theory of electromagnetism where chromo-electromagnetic field
itself carries charges. As a classical field theory it has solutions which
travel at the speed of light so that its quantum version should describe
massless particles (gluons). However, the postulated phenomenon of color
confinement permits only bound states of gluons, forming massive particles.
This is the Yang-Mills mass gap. The variational representation of the time-harmonic Maxwell equations in the proposed "quantum state" Hilbert space framework H(-1/2) builds on truly fermions (with mass) & bosons (w/o mass) quantum states / energies, i.e. a Yang-Mills equations model extention is no longer required.
(4) The thermodynamic Hilbert (energy) space H(1) is
compactly embedded into the newly proposed Hilbert (energy) space
H(1/2). From a statistical point of view it means that the probability
to catch a quantum state/"elementary particle", which is able to collide
with another one, is zero. This compactly embeddedness enables a new
interpretation of the entropy phenomenon as the change process from thermodynamical (kinetic) energy to ether (ground state, "quantum potential", "Leibniz's living force") energy.
Mathematically
speaking the expanded new energy Hilbert space H(1/2) (where the
Heisenberg uncertainty inequality is valid) enables the Hamiltonian
formalism, only. Only for the standard energy Hilbert space H(1) (which
is a compactly embedded, separable Hilbert (sub-) space of H(1/2)) the
corresponding Lagrange formalism is defined due to a valid Legendre
transformation, because of appropriate regularity of the Hilbert space
H(1). In other words, Emmy Noether's theorem is valid only in the H(1)
framework. It means that if the Lagrange functional is an extremal, and
if under corresponding infinitesimal transformation the functional is
invariant to a certain definition, then a corresponding conservation law
holds true.
The proposed inflation model of A. Linde requires a very small amount of ("a priori" existing, which is a contradiction by itself) matter to generate an "initial vacuum", which then inflated / blowed up to the current universe (big bang). The newly proposed model assumes a mass-less initial vacuum state (w/o any "existing" space-time concept) generating first fermions at Planck time (going along with a space-time framework initiated at Planck time) by a „projection operator onto the observation/measure
space". Then, "caused" by the first generated fermions at Planck time, the Linde model can be applied.
______________________________________________________________
 Braun K., RH, YME, NSE, GUT solutions, overview |
______________________________________________________________
The journey
started in 2010 resulting in the paper
Braun
K., The three Millennium problem solutions (RH, NSE, YME) and a Hilbert
scale based quantum geometrodynamics, July 2019
The
journey is still ongoing: in order to support an appropriate change traceability this paper is split into
PART
I:
Braun
K., A Kummer function based alternative Zeta function theory to solve the Riemann
Hypothesis and the binary Goldbach conjecture
Braun K., RH solutions |
last update : July 31, 2019
PART
II:
Braun
K., 3D-NSE and YME mass gap solutions in a distributional Hilbert scale frame
enabling a quantum gravity theory
Braun K., 3D-NSE, YME, GUT solutions |
last update : July 31, 2019
With respect to the NSE topic we also refer to
http://www.navier-stokes-equations.com/
With respect to the ground state energy and quantum gravity topic we also refer to
https://quantum-gravitation.de/
