Albert Einstein, "we can't solve problems by using the same kind of thinking we used when we created them",
Wolfgang E. Pauli, "all things reach the one who knows how to wait".
This homepage addresses the following three Millenium problems (resp. links to corresponding homepages):
A. The Riemann Hypothesis
The proposed framework also provides an answer to Derbyshine's question, ("Prime Obsession")
The answer, in a nutshell:
"identifiying "fluids" or "sub-atomic particles" not with real numbers (scalar field, I. Newton), but with hyper-real numbers (G. W. Leibniz) enables a truly infinitesimal (geometric) distributional Hilbert space framework (H. Weyl) which corresponds to the Teichmüller theory, the Bounded Mean Oscillation (BMO) and the Harmonic Analysis theory. The distributional Hilbert scale framework enables the full power of spectral theory, while still keeping the standard L(2)=H(0)-Hilbert space as test space to "measure" particles' locations. At the same time, the Ritz-Galerkin (energy or operator norm minimization) method and its counterpart, the methods of Trefftz/Noble to solve PDE by complementary variational principles (A. M. Arthurs, K. Friedrichs, L. B. Rall, P. D. Robinson, W. Velte) w/o anticipating boundary values) enables an alternative "quantization" method of PDE models (P. Ehrenfest), e.g. being applied to the Wheeler-de-Witt operator.
Regarding the proposed alternative quantization approach we also refer to the Berry-Keating conjecture. This is about an unknown quantization H of the classical Hamiltonian H=xp, that the Riemann zeros coincide with the spectrum of the operator 1/2+iH. This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principle and the natural numbers as spectrum of the harmonic quantum oscillator. The Hamiltonian needs to be self-adjoint so that the quantization can be a realization of the Hilbert-Polya conjecture.
B. The Navier-Stokes Equations
The Navier-Stokes equations describe the motion of fluids. The Navier–Stokes existence and smoothness problem for the three-dimensional NSE, given some initial conditions, is to prove that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass.
In the context of the below we note that the unit of measure of the pressure is Newton/area, which is equivalent to Newton*length/volume, i.e. "pressure" corresponds to "energy density".
In order to prepare the relationship to the below, which is concerned with quantum field theory and its relationship to a proposed quantum gravity model, we recall from the Maxwell equations the following:
the Maxwell equations are about 4 PDE constituting a complete description of the behavior of electric and magnetic fields. They sum up the experimential results of Coulomb, Ampere, and Faraday. Maxwell fixed them up so that they made mathematical sense by introducting the concept of displacement current (one could also call it a mathematical artefact/trick). With the inclusion of the displacement current the Maxwell equations treat electric and magnetic fields on equal footing, i.e. electric fields can induce magnetic fields and vice versa. The displacement current is defined in terms of the rate of change electric displacement field. It has the same unit as electric current, and it is a source of the magnetic field just as actual current is. However it is not an electric current moving charges, but a time-varying electric field.
The displacement current is a crucial additition that completed Maxwell's equations and is necessary to explain many phenomena, most particularly the existence of electro-magnetic waves (i.e. there is no need to differentiate between electric waves and magnetic waves and corresponding forces/energies !!!). In the light of the above we claim that in the same framework as for the proposed NSE solution the displacement current can be interpreted as a kind of "vacuum action" (related to time), in contrast to the "physical performance" of electric current to move charges.
C. The Yang-Mills Equations
Topic B. above is implicitly addressing the Serin gap problem; the proposed fractional Hilbert space framework can also be applied to address the mass gap problem of the Yang-Mills Equations (YME).
The YME are concerned with quantum field theory. The challenge is about an appropriate mathematical model to govern the "mass gap" (i.e. to end up with finite energy norms), which is the difference in energy between the vacuum and the next lowest energy field.
In some problem statements of the YME there are basically two assumptions made (which are not clearly defined):
1. the energy of the vacuum energy is zero
2. all energy states can the thought of as particles in plane-waves.
As a consequence the mass gap is the mass of the lightest particle.
Our challenge of proposition 1 is about the measure of the vacuum energy, which gives the value "zero". While the energy norm in the standard H(1) Hilbert space might be zero, the value of the quantum state with respect to the energy norm of the sub-space H(1/2) still can be >0.
Our challenge of Proposition 2 is going the same way: a particle with mass can be measured (condensed energy), i.e. it is an element of the test space H(0), while there still can be "waves" in the closed complementary space H(-1/2)-H(0), where the test space is "just" compactly embedded. Those "waves" might be interpreted as all kinds of today's massless "particles" (neutrinos and photons) with related "dark energy".
As a consequence there is no mass gap, but there is an additional vacuum energy governed by the Heisenberg uncertainty principle.
In summary, the common denominator of the topics above is about a mathematical (variational) framework for an integrated "4 Nature "forces"" gravity & quantum field model with (standard) test space L(2)=H(0), a quantum state space H(-1/2) and the related energy space H(1/2). The embeddings of the Hilbert spaces H(1/2) into H(0) into H(-1/2) is compact.
The proposed mathematical framework above is supposed to provide a truly infinitesimal geometry (H. Weyl), enabling the concept of Riemann that force is a pseudo force only, which results from distortions of the geometrical structure. The baseline is a common Hilbert space framework (providing the mathematical concept of a geometrical structure, while Riemann's manifold concept provides only a metric space and related affine connections) replacing "force type" specific gauge fields and its combinations to build an integrated model.
A physical interpretation could be about "rotating differentials" ("quantum fluctuations"), which corresponds mathematically to Leibniz's concept of monads. Its mathematical counterpart are the ideal points (or hyper-real numbers). This leads to non-standard analysis, whereby the number field has same cardinality than the real numbers. It is "just" the Archimedian principle which is no longer valid. This looks like a cheap prize to be paid, especially as hyper-real numbers might provide at least a proper mathematical language for the "Big Bang" initial value "function" and its related Einstein-Hilbert action functional.
Looking on hyper-real numbers from the "real" number perspective one must admit to classify the term "real" is a contraction in itself, if it is understood as real. Already the existence of an irrational number (not only the existence of a transcendental numbers) and also the cardinality of the irrational numbers, which is very much different from the rational numbers) is ensured by an axiom, "only" (the Cauchy convergence criterion), i.e. the "empty space" between two rational numbers is filled with infinite irrational numbers with same cardinaility as the field of real numbers itself, i.e. with multiple "universes". The difference of real numbers to hyper-real numbers is "just" the fact that there are additionally infinite small and large numbers "existing", ensured "just" by the missing Archimedean property. This principle is basically nothing else than the property that any finite distance can be measured by a given standard measure (i.e. for any real positive number r and a given standard measure of length a (e.g. a=1) there is an integer n that n*a>r). We emphasis that this is still given in the test space, but no longer valid in the distributional Hilbert space, where L(2) is a closed sub-space of.
We further emphasis that
- the differentable manifold framework of Einstein's field theory does not allow singularities, as required to model black whole and dark energy phenomena
- the hyper-real /ideal points /monads above map to the "proper and terminal indecomposable past-sets/ideal points in space-time (PIPs and TIPs)" in the context of the comological censorship and the existence of past and future time-space singularities (GeR).
- the idea to apply Non-standard Analysis (and its related non-classical motion) to explore a Quantum-relative Universe is not new (PoP).
- the model enables an alternative concept to current symmetry breaking and inflation model for the early universe by which the required energy to generate matter out of photons (w/o violating conservation laws) is released during the symmetry breaking process.
(GeR) Geroch R., Kronheimer E. H., Penrose R., Ideal points in spacetime, Proc. Roy. Soc., London, A347, 545-567, 1972
(PoP) Poluyan P. V., Non-Standard Analysis of Non-classical Motion; do the hyperreal numbers exist in the Quantum-relative universe?