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The Riemann Hypothesis, the Goldbach conjecture(s), NSE, YME, quantum gravity & the zero state energy (field/matter) modelling challenge

A. Einstein: "We can't solve problems by using the same kind of thinking we used when we created them."   

This homepage addresses the Millenium problems (resp. links to corresponding homepages) of the areas:

A. The Riemann Hypothesis (RH) and the Goldbach conjecture
B. The 3D-Navier-Stokes equations (NSE) and the Serrin gap
C. The Yang-Mills equations (YME) and quantum gravity modelling

and related mathematical and theoretical physics research areas.

A. The Riemann Hypothesis

All nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Zeta function corresponds to eigenvalues of an unbounded self adjoint operator.

We provide a solution for the RH building on a new Kummer function based Zeta function theory, alternatively to the current Gauss-Weierstrass function based Zeta function theory. This priminarily enables a proof of the Hilbert-Polya conjecture (but also of other RH criteria like the Bagchi formulation of the Nyman-Beurling criterion or Polya criteria), whereby the imaginary parts of the zeros of the corresponding alternative Zeta function definition corresponds to eigenvalues of a bounded, self adjoint operator with (newly) distributional Hilbert space domain. At the same time this new theory enables an alternative Ei(x), resp. li(x)-function (built on a corresponding Kummer function), which is very "close" to the the "standard" exponential integral ("density") function. This enables also other proofs of the RH in the context of Riemann function convergence criteria.

The new theory is also proposed to leverage standard circle method, enabling proofs of the weak and strong Goldbach conjectures w/o using Weyl exponential sums (minor arcs) estimates.

B. The Navier-Stokes equations and the Serrin gap

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.

The Serrin gap occurs in case of space dimension n=3 as a consequence of the Sobolev embedding theorem with respect to the energy Hilbert space H(1) with the Dirichlet integral as its inner product.

We provide a global unique (weak, generalized Hopf) H(-1/2)-solution of the generalized 3D Navier-Stokes initial value problem. The global boundedness of a generalized energy inequality with respect to the energy Hilbert space H(1/2) is a consequence of the Sobolevskii estimate of the non-linear term (1959).

With respect to the below we mention that there are basically two kinds of required "energy" measures as part of the energy inequality which are defined by the H(1/2)- and the L(2)(H(1/2)) norms. The first one could be interpreted as the quantum energy measure while the second one could be interpreted as the (average) energy of this quantum energy on the "classical "phenomenon PDE level.

Building on the "H(-1/2)-solution concept" a corresponding Ritz-Galerkin approximation theory for non-linear parabolic PDE with non-regular inital value data is provided.

The Maxwell equations build a symmetric hyperbolic system. The Yang-Mills equations are a nonlinear generalization of the Maxwell equations which is semi-linear. The Euler and Einstein equations are quasi-linear, but not semi-linear. Even the simplest linear hyperbolic equation (the wave equation) shows different ('not optimal') properties of corresponding shift theorems. The latter ones indicate the appropriate definition of adequate weak energy inner products. Building on the "H(-1/2)-solution concept" this is addressed in


where we present an unusual 'optimal order' shift theorem for hyperbolic problem with a corresponding "unusual" energy norm. This indicates that classical energy measurements and corresponding norms are mathematical problem type (elliptic, parabolic, hyperbolic) dependent reflecting the corresponding physical model situation (and the corresponding observed phenomena) and not a physical model situation independent "truly" physical phenomenon which is valid across different model type situations. At the same time the common quantum energy across such mathematical model types (as proposed in this page, see C.) is measured by same quantum energy norm.

We further emphasis the elegant role of the alternatively proposed H(1/2) energy Hilbert space in universal Teichmüller theory.

C. The Yang-Mills equations and the mass gap

The classical Yang-Mills theory is a generalization of the Maxwell theory of electromagnetism where the 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 mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.

We provide a new mathematical "single force" fermion/boson (particle/energy) quantum element model alternatively to today's 3-forces SMEP model with its related Lagrange densities and the corresponding zoo of different fermions and bosons types. As a consequence the model covers also the 4th (gravity) "Nature force".

The new mathematical model is enabled by Plemelj's extended Dirichlet (energy inner product) integral definition: the standard Dirichlet integral is related to the Laplacian operator by the Green formula. It defines the energy inner product of the L(2) Hilbert space with corresponding domain H(1). Plemelj's extended Green formula ((PlJ) I, §4,5) require reduced regularity assumptions to the potential function V, i.e. grad(V) does not need to exist on the boundary. As a consequence Plemelj's extended Green formula lead to a Plemelj Dirichlet integral definition with domain H(1/2). We emphasis that this definition can be also applied to define a weak representation of the NSE which motivated the provided solution of B. The concept finally leads to PDE classes (elliptic, parabolic, hyperbolic) specific energy norm definitions, whereby the energy norms for elliptic and parabolic are conceptually similar. The energy norm for hyperbolic PDE needs to be conceptually different to anticipate e.g. wave fronts phenomena and the Huygens (space dimension depending) principle.

Additionally we propose an alternative quantum harmonic oscillator model based on an alternatively defined Schrödinger momentum operator. The proposed solution concept allows a generalization to space dimensions n>1 (in contraction to the today's Dirac function concept) enabled by the Riesz operators, which are the corresponding n>1 operators of the n=1 Hilbert transform operator.

D. Common denominator of A, B, C

The common denominator of the proposed solutions resp. solution concepts A, B, C is about the proposed distributional fractional Hilbert scale with related generalized waves/modes. Classical short distance solutions of PDE are approximations of corresponding weak (variation) short distance solutions and not the other way around. There is only one fractional energy Hilbert space. The different forms of "forces" are expressions of "energy/actions" per considered weak PDE. We don't discuss the impact on corresponding interpretations to quantum mechanics and the mind-body problem, but just want to refer in this context to the following books:

(EaS) Easwaran S. E. The Upanishads, Nilgiri Press, Tomales, California, 1987

(NaT) Nagel Th., Mind and Cosmos: Why The Materialist Neo-Darwinian Conception of Nature is Almost Certainly False, Oxford University Press, 2012

(ScA) Schopenhauer A., Die Welt als Wille und Vorstellung, erste Auflage, Leipzig, 1818

(ScE) Schrödinger E., The Interpretation of Quantum Mechanics, Ox Bow Press, Woodbridge, Connecticut, 1995

(ScE1) Schrödinger E., My View of the World, Ox Bow Press, Woodbridge, Connecticut, 1983

(WeH) Weyl H., Gravitation und Elektrizität, Sitzungsberichte Akademie der Wissenschaften Berlin, 1918, 465-480

(WeH1) Weyl H., Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, New Jersey, 1949, 2009

From (ScE1) we quote:

VIII, Consciousness and mneme: "Thus Schopenhauer's line of demarcation may be regarded as highly suitable, when he says that in inorganic being 'the essential and permanent element, the basis of identity and integrity, is the material, the matter, the inessential and mutable element being the form. In organic being the reverse is true; for its life, that is, its existence as an organic being, consists precisely in a constant change of matter while the form persists' ...".

IX, On becoming conscious: "consciousness is bound up with learning in organic substance; organic competence is unconscious. Still more briefly, and put in a form which is admittedly rather obscure and open to missunderstanding: Becoming is conscious, being unconscious." We note that in same section of the original german issue Schrödinger uses the term "Differential", which has been translated into "change" and "difference". If a world class theoretical physicist and mathematician like Schrödinger chooses the word "Differential" in his mother language he does not mean "Wechsel/Veränderung (change)" or "Differenz (difference)", that's for sure.

With respect to the mentioned Schoperhauer quote above we refer to Schopenhauer's concept of "will" in "The world as will and representation". His definition of "will" is not about the capability of decision making, but it describes a principle of life which is about the energy of life as it is part of any anorganic object (matter) or organic object (organism). By this concept the human body (organism) is the visible form of apperance (manifestation/ image) of the not visible acting principle of life. Schopenhauer and Schrödinger were very much impressed from the Upanishads (EaS). From (EaS) we mention A. Huxley's quote (p. 24): "all science is the reduction of multiplicities (=manifolds) to unities (=entities)" and quote further...p. 24 " Nothing is more characteristic of Indian thought. In the context of the terms "multiplicity vs. entity", "autonomy vs. variety" we quote from (EaS), epilog:

    "multiplicity/manifold arises from entity/singularity in a regulated order".

In our distributional Hilbert scale framework the "differential" corresponds to an "energy" element:

- with respect to the concept of causality between mind/body elements we quote from (ScA) §18, p.151: "Der Willenasakt und die Aktion des Leibes sind nicht zwei objektiv erkannte verschiedene Zustände, die das Band der Kausalität verknüpft,  stehen nicht im Verhältniß der Ursache und Wirkung; sondern sie sind Eines und das Selbe, nur auf zwei gänzlich verschiedene Weisen gegeben: einmal ganz unmittelbar und einmal in der Anschauung für den Verstand"

- with respect to the laws of transformation of most physical quantities which are intimately connected with that of the differentials and differentiable manifolds we quote from (WeH1) p.86: "While topology has succeeded fairly well in mastering continuity, we do not yet understand the inner meaning of the restriction to differentiable manifolds. Perhaps one day physics will be able to discard it ... Only in the infinitely small may we expect to encounter the elementary and uniform laws, hence the world must be comprehended through its behavior in the infinitely small."

Schopenhauer gives four different forms of causality as part of the "world as representation" ("On the fourfold root of the principle of sufficient reason").With the "world as will" principle Schopenhauer (inspired by the universal force and energy of the Upanischaden which they called "Brahma") does not mean neither a "conscious will" nor a "reasonable will" or a "chosen will". It is the purely opposite/counterpart of the "world as representation" principle with no common denominator or overlap. Therefore it is also not governed by the law of causality and has neither mind nor reason or space & time. As a consequence from the later two ones the will can not be split into individual wills.

In simple words: "causality" (and related "force" phenomena) is part of the "world as representation" principle (which is related to classical PDE with weak energy Hilbert space H(1)), while "energy" (which is related to singular integral (Pseudodifferential) equations with weak, fractional (energy) Hilbert space H(1/2)) is part of the "world as will" principle.

We give a trial to summaries the four forms of truths which build the baseline of the principle of sufficient reason:

- the logical truth (i.e. the formal logic; causality which is comprehends by mind with the three forms cause, appeal, motif)

- the empirical truth

- the transcendental truth

- the meta-logical truth (formal condition of thinking: principle of the identity, principle of contradictionary, principle of excluded third identity, principles of sufficient reason).

The common mathematical and philosophical concept

With respect to Schoperhauer's two world principles our central idea can be phrased in the following way:

Schopenhauer's "world as representation" principle (built on the four different forms of causality which are the root causes of the principle of reason) corresponds to the Lagrange world ("principle of reason") which is about classical PDE and related modelling of e.g. force phenomena. Its solutions are interpreted as approximations only to the Hamiltonian world ("principle of purpose"), which is about weak Pseudodifferential equations and related modelling of e.g. ("living force") mass/energy elements. By this the Hamiltonian world corresponds to Schopenhauer's "world of will" anticipating his concept of energy of life (which is part of any anorganic object (matter) or organic object (organism)). From a mathematical point of view both worlds are isomorh only in case the Legendre (contact) transform is defined. With respect to this we quote from (WeH):

" ...truly infinitesimal geometry ...should know a transfer principle for length measurements between infinitely close points only."

It is encouraging and at the same time still astonishing that classical (approximation) and related "truly" (fractional) energy Hilbert scale elements (a=1 and a=1/2) and its related weak (PDE resp. PDO) equations provide a model describing

- how the distribution of prime numbers are related to the behavior of subatomic particles and harmonic music based on same framework which is also applicable to the Maxwell & Einstein field equations

- how force phenomena versus energy entity concepts (including a length measurement concept between infinitely "close points") are related to the two "world principles" of the Western philosophical system of Schopenhauer and the Upanishads, an Indian collection of philosophical writings.

For more about this we refer to