An Unified Field Theory

                                                  Prologue

        The standard model of particle physics is not a well-defined theory

„A particularly worrying symptom of the current state of affairs in physics is the so-called discovery of the Higgs boson at CERN. … But what was actually discovered were a number of unexplained signals obtained by extensive filtering methods, raising many questions for everyone who takes a sober perspective. …

Nevertheless, these signals are pushed to serve as evidence for the long-theorized Higgs boson supporting the „standard model“ of particle physics, although this standard model is not even a well-defined theory. Such an interpretation speaks more of desperation to valididate the past six decades of research and to shore up a model that is wobbling precariously under the weight of all the bits and pieces glued onto it to make it work“, (UnA) Prologue.

                   The "After-Big-Bang" story is extremely unlikely

After the biggest black hole ever out of nowhere during "The First Three Minutes", (S. Weinberg), ...

„in order to produce an universe resembling the one in which we live, the Creator would have to aim for an absurdly tiny volume of the phase space of possible universes – about1/(10 exp(10 exp (123))) – of the entire volume, for the situation under consideration", (PeR), p. 444, Fig. 7.19

(PeR) Penrose R., The Emperor’s New Mind, Oxford University Press, Oxford, 1989

Braun, K., An unified field theory enabling a deductive structure of physics.pdf

SCOPE

The UFT is governed by two kinds of energy, the today’s mechanical energy (kinetic and potential energy) and (in line with Planck's dynamic type of physical law) a newly proposed dynamic energy.

In the calculus of variations the Dirichlet integral defines the inner product of the standard mechanical energy Hilbert space H(1). Krein (energy) spaces are proposed as mechanical x dynamic energy spaces. Those are indefinite inner product vector spaces with a symmeric (in the complex case: hermitian) bilinear form prescribed on it so that the corresponding quadratic form assumes both positive and negative values, (BoJ).

The physical scope of the UFT starts from an a priori dynamic vacuum energy field, which may create dynamic plasma energy fields; the latter one may create dynamic electromagnetic fields. The dynamic quanta pairs of the electromagnetic field may create three related types of atomic nuclei, in line with the three physical groups of atoms, the conductors, the semiconductors, and the non-conductors. This combined mechanical x dynamic quanta energy system is called the Dirac 2.0 energy field. The all-encompassing Krein energy space of this hierarchical structure of energy spaces is the dynamic vacuum energy space. The Dirac 2.0 energy field model is included in all other Krein spaces according to the above sequence. 

There is a consistent bottom up hierarchy of the above Krein-Hilbert (energy) spaces starting from an a priori dynamic vacuum energy field (the largest field) up to the classical mechanical energy field H(2); this is the standard domain of elliptic, parabolic, and hyperbolic PDE equations, which is compactly embedded into the mechanical (variational) energy Hilbert space H(1) accompanied by a related self-adjoint extension of the symmetric Laplace potential operator with domain H(2). The mechanical model "potential" operators of elliptic and parabolic PDE are the Laplace potential and the related heat equation operators; the related appropriate domains with regards to well defined convergent energy norms are the well-known Sobolev spaces, which correspond to the (variational) H(1) resp. the (classical) H(2) energy Hilbert spaces. It turned out that the mechanical model "potential" operator of hyperbolic PDE, the wave operator accompanied by the domain of the baseline Hilbert space of the proposed Krein spaces, is also well defined.  

Mathematically speaking, the compactly embeddingnesses enable corresponding well defined approximation methods of the solutions of PDE systems in all of the affected two (Krein-) Hilbert spaces, (NiJ) (NiJ1). Additionally, the smallest Dirac 2.0 energy (Krein-Hilbert space model can be approximated by a newly proposed purely extended Hilbert (mechanical + dynamic energy) space H(1/2); as the mechanical energy Hilbert space H(1) is compactly embedded into H(1/2), there is a closed complementary sub-space of H(1/2), (which is not a Hilbert space), which may be interpreted as dynamic energy field. In this case, the dynamic quanta energies (the quanta of the closed sub-space) of the considered mechanical particles (the quanta of the H(1) energy space) may be interpreted as alternative field intrinsic "potential" (better "dynamic") functions, (alternatively to the concept of physical case defined specific functions like the potential function V(r) in Schrödinger's harmonic quantum oscillator equation).

It turned out that the energy Hilbert space model H(1/2), which includes the standard mechanical energy Hilbert space H(1), solves the 3D-NSE problem. At the same time the (extended) Dirac 2.0 quanta energy field model solves the YME mass gap problem of the Clay Mathematics Institute.

While the anorganic chemistry is in scope of the UFT, the organic chemistry (and a field theory of light, see below) is out of scope. In the context of this UFT this especially excludes Schrödinger‘s related topics like „what is life?“, "the principle of objectivation", and "two ways of producing orderlines" and Schauberger's alternative world view accompanied by topics like "water, source of life", "different kinds of energy and physical movements", "the vortex as the (mechanical) key to create evolution", "the sun as a fertilizing entity", "gravity and levity", "entropy and ectropy", and "biologic vacuum", (BaA). To the author’s humble opinion beside the quoted references of E. Schrödinger the „Hidden Nature, The Startling Insights of Victor Schauberger“ (by Alick Bartholomew), "Viktor Schauberger, Das Wesen des Wassers" (by Jörg Schauberger) and "Lebendes Wasser" (by Olof Alexandersson) may be regarded as appropriate starting point of a top-down investigation approach from "water, the source of life" down to organic chemistry and biology governed by oxygen (while the anorganic chemistry is basically governed by hydrogen as the "carrier" of both carbone, i.e. more than just carbon and oxygen).

Further key words:

(AlO): living water; the secret of natural energy; birds are not flying, they are flown; a fish is not swimming, it is swum (S. 135)

(BaA): wrong and right form of motion; energies as creative process; water, source of life; Nature’s silent methods of producing energy, which are inherently more effective and powerful than mechanical techniques; oxygen as a lower form of solar energy; the sun is responsible for all life, the oxygen is responsible for organic growing and development; hydrogen as the "carrier" of both carbone (i.e. more than just "carbon") and oxygen, p. 51

(ScJ) organic vacuum; cycloid space curve; electrolysis does't work for distilled water.

CONCEPTUAL DESIGN ELEMENTS AND OPPORTUNITIES

Planck’s dynamic type of law & the dynamic energy concept in the UFT

(PlM) S. 90: „Immerhin erhellt aus der geschilderten Sachlage wohl hinreichend deutlich die überaus hohe Bedeutung, welche die Durchführung einer sorgfältigen und grundsatzlichen Trennung der beiden besprochenen Arten von Gesetzmaeßigkeit: der dynamischen, streng kausalen, und der lediglich statistischen, für das Verständnis des eigentlichen Wesens jeglicher naturwissenschaftlichen Erkenntnis besitzt“.

"The phase space of a physical system is the set of all possible physical states of the system when described by a given parametrization" (Wikipedia). Today, there is only one type of energy concept, the mechanical energy with its two forms, the kinetic (movement related) and the potential (change of position related) energy. The related statistical type of laws are formulated as PDE (e.g. in thermo-statistics) accompanied by an underlying (mechanical) self-adjoint potential operator accompanied by related Sobolev function domains (potential theory). In the proposed UFT there is an additional Hilbert-Krein space HK, which contains the full distributional Sobolev-Hilbert scale HS, providing the appropriate framework for a newly proposed dynamic energy concept. The full (polynomial-decay-mechanical) Sobolev-Hilbert scale HS is defined by a self-adjoint potential operator (potential theory for mechanical particles; elliptic & parabolic PDE); the extended (exponential-decay-dynamic) Hilbert-Krein space HK is defined by Krein-space specific self-adjoint potential operators accompanied by related inner products on all of the HK (potential theory for dynamic particles; hyperbolic PDE). The norms may be interpreted as dynamic potentials enabling the definition of dynamic quanta (i.e. dynamic quantum element types) on all of the Hilbert-Krein-space HK. A HK x HK orthogonal composition provides a two-component (purely) dynamic quanta system; a HS x HK orthogonal composition provides a two-component mechanical x dynamic quanta system. The latter systems support the below Dirac 2.0 quanta system; the first type of systems provides an appropriate framework for plasma dynamics and an enhanced Maxwell-Mie System; this Maxwell-Mie system is in line with the photophoresis discovery of F. Ehrenhaft.

Schrödinger’s permanent form of „ultimative particles“ and the UFT

(ScE3) p. 125, „(But) when you come to the ultimate particles constituting matter, there seems to be no point in thinking of them again consisting of some material. They are, as it were, pure shape, nothing but shape; what turns up again and again in successive observations is this shape, not an individual speck of material.“

In the proposed UFT Schrödinger's form of the „ultimate particles“ (quanta) is provided by the so-called kappa-quanta scheme. It is enabled by appropriately defined quanta type specific inner products, related norms and corresponding self-adjoint dynamic-potential operators as part of a Hilbert-Krein space HK; this Hilbert-Krein space HK contains all polynominal degree Hilbert scales, which are defined by the eigenpairs of self-adjoint (classical) mechanical-potential operators.

The dynamic plasma quanta pairs as condensed vacuum quanta pairs

Expressed colloquially, plasma is an ionized gas consisting of approximately equal numbers of positively and negatively charged particles. In the proposed UFT the (dynamic) plasma quanta pairs may be interpreted as condensed (dynamic) vacuum quanta pairs. Those plasma quanta pairs are characterized by an identical absolute amount of „charges“ close to 1/2, a property, which is in line with the Landau damping phenomenon. The composition of those pairs is called neutron accompanied by the quantum numbers one. The (purely mathematical) vacuum quanta pairs are characterized by nealy identical +/- „charges“ with quanta numbers only close to one; the related potential differences may be interpretated as „ground state energy“.

The dynamic electromagnetism quanta pairs as condensed vacuum and plasma quanta pairs

The dynamic electromagnetism quanta pairs are built by appropriate compositions of plasma and vacuum quanta. Similar as the (dynamic) plasma quanta pairs the (dynamic) electromagnetism quanta pairs are accompanied by two nearly identical +/- „charges“; however, in this case the quanta numbers are close to one.

The atomic Dirac 2.0 quanta built from the three possible combinations of the electromagnetic quanta

(FeE): „Dirac‘s theory of radiation is based on a very simple idea; instead of considering an atom and the radiation field with which it interacts as two distinct systems, he treats them as a single system whose energy is the sum of three terms: one representing the energy of the atom, a second representating the electromagnetic energy of the radiation field, and a small term representing the coupling energy of the atom and the radiation field."

In the theory of quantum mechanics each considered (Dirac) system is an element of a related Hilbert space. This mathematical concept is also applied to the QFT, the QCD, and the QED. The prize to be paid for this "force specific" modelling approach are "three independent "theories" for „strong interactions, weak interactions, and electromagnetic …  which are linked because they seem to have similar characteristics", R. Feynman, (GlJ) p. 433.

The Dirac 2.0 quanta scheme provides three atomic nuclei quanta, which are called positronium, electronium, and neutronium accompanied by correspondingly defined quanta numbers >2. They are built by the three possible combination of the two (dynamic) electromagnetic quanta, the electroton and the magneton. The three Dirac 2.0 nuclei basic types are in line with the three groups of anorganic atoms, the precious metals, the base metals, and the non-metals.

The simple H(1/2) energy Hilbert space approximation model of the atomic Dirac 2.0 quanta model

The (polynomial-decay) energy Hilbert space H(1/2), which contains the standard (variational (!)) mechanical energy Hilbert space H(1), provides the approximation mechanical x dynamic quanta energy model to the Dirac 2.0 model:

in the calculus of variations the ("weak" mechanical energy) Hilbert sub-space H(1) of (H(1/2) is governed by Fourier waves, while the related complementary closed ("weak" dynamic energy) sub-space of H(1/2) (which is not a Hilbert space (!)) is governed by Calderón wavelets.

Note: It turned out (see below) that the non-linear energy term of the 3D non-linear, non-stationary NSE system is bounded with respect to the H(1/2) energy norm as a simple consequence of the Sobolevskii inequality (BrK9).

The tool to build the orthogonal relationship between the mechanical and the dynamic (energy) worlds are the Riesz transforms (resp. the 1D Hilbert transform).

Dirac’s new basis for cosmology (DiP2)

(UnA2) p. 73: "Dirac’s first conjecture: In the hydrogen atom, nature’s simplest stable structure, the constituent proton and electron are held together by electric force. Yet their gravitational attraction, however small it may be, can also be calculated theoretically. Dirac noticed that the ratio of the two forces ... was an incredibly huge number (about 10 exp(39)) with almost 40 digits."

(UnA2) p. 74: "Dirac has discovered a second conjecture, connected to the first conjecture, which make a random coincidence extremely unlikely. After the first estimates of the total mass M(U) of the universe in the 1930s, Dirac divided M(U) by the mass of the proton m(p), thus arriving at the ball-park figure of the number of particles in the universe, 10 exp(78), the square of that other mysterious, but lent substance to the first obeservation. For decades, Dirac’s second observation has defied all attemps at explanation. In particular, it seems to jeopardize all established cosmological models. For normally the number of particles should be proportional to the volume, i.e. the third power of the linear dimensions of the cosmos, not just to the second. All the more remarkable, however, is that Dirac’s cosmology ultimately follows from Einstein’s pivotal idea of a variable speed of light back in 1911, if one consequently applies Dicke’s formulation of 1957", (see also (UnA1).

Schrödinger’s hour of glory "potential is simply energy per mass" (UnA1)

(UnA1) pp. 116, 118, UFT, pp. 146/147: „Dividing the visible mass of the universe (estimated 10 exp(53) kilogram) by the size R(U) of the universe (estimated 10 exp(26) meter) yields 10 exp(27) kilograms per meter, coinciding with the value derived from the ratio of the square of speed of light and the gravitational constant G. … There is a real gem of physical reasoning in a completely unknown article on cosmology published in 1925 by E. Schrödinger (ScE4).  … Whereas the relation above as such is only numerical, Schrödinger went a step further and realized that the concept of gravitational potential was concealed in this formula. Potential is simply energy per mass, for which Newton had derived an expression in his theory of gravitation with the distance r from the Sun (with mass M) in the denominator. … It looked plausible to him that the influence of the even more distant masses in the Milky Way had to be larger, even though it was impossible to perceive a force. .. With amazing intuition he suspected that all the potentials in the universe might just add up to the square of the speed of light. … In a way Schrödinger had thus anticipated the discovery of the size of the cosmos in the 1930s. He further insisted that Mach's principle had to be incorporated into the theory of relativity. In this respect, Schrödinger's intuition went beyond Einstein's.“

Mach's principle 2.0 (UnA1)

(UnA1) p. 156: „To round off the value of Dirac’s observation, however, one should mention that it is in complete harmony with Ernst Mach’s thoughts on gravity, though Dirac apparently never dealt with Mach. … The fact that Dirac considered the size and the mass of the universe, the two quantities that Mach also related to the origin of gravity, constitutes another piece in this fascinating puzzle.

However, Dirac’s observation goes beyond Mach’s principle. Imagine that the number of particles in the universe was billion times larger, while simulataneous their mass was a billion times smaller. This would change nothing about Mach’s principle (or ‚flatness‘) but it would alter Dirac’s observation. In other words, Dirac was the first to insinuate that the size and the mass of elementary particles had a meaning, and that it is no coincidence that they are as large and heavy as they are. Who thought soothe same? You’ve guessed it – Albert Einstein:

„The real laws of nature are much more restrictive than the ones we know. For instance would it not violate our known laws, if we found electrons of any size or iron of any specific weight. Nature however only realizes electrons of a particular size and iron of vry specific weight.“

A suggestion out of the Dirac conjectures (UnA)

(UnA) p. 225: "Dirac’s conjecture – that the number of particles in the universe (10 exp(80)) is related to its size (which is 10 exp(40)) proton radii- clashes with all established concepts …. It suggests that the quantum effects of gravity start at the size of an atomic nucleus and not at the far smaller, unobserved Planck’s length of 10 exp(-35) meters (that this is 20 powers of ten below the nucleus‘ radius is the result of Dirac’s hypothesis. (However, it doesn’t have any fundamental meaning.)"

Does a neutron feel the age of the universe? (UnA2)

(UnA2) p. 95: „From a philosophical point of view, the half-life of the neutron of about ten minutes is a basic quantity that calls for an explanation. … Claiming that the decay of the neutron can be deduced within the variable speed of light model would certainly be premature. In any case, however, the phenomenon of radioactivity will only be thorughly understood once th half-life of the neutron is calculated from first principles. …. However, if the mass ratio of  proton/electron does depend logarithmically on the age of the universe, then it follows that at the time of the „Big Flash“ the electron and proton were  of equal weight.  … The hydrogen atom would then be similar to an object now called positronium, consisting of an electron and its antiparticle positron that orbit each other. …. This would imply that the orbital speed of the electron in the hydrogen atom was equal to the speed of light…. This, in turn, suggests that the hydrogen atom – at that time an orbiting electron-positron pair – could simply be seen as a rotating light wave.“

Dirac, Unzicker and the UFT

Anticipating that gravity start at the size of an atomic nucleus puts the spot on the Big Bang story (creatio ex nihilo), which is basically the physical (!) "explanation" of the observed Cosmological Background Radiation (CMBR). Anticipating that nearly all of the universe is vacuum, and that nearly most of the space within the atoms is also vacuum, a physical (!) "creatio ex vacuum" concept sounds more reasonable than a (physical) "creatio ex nihilo" concept.

The assumption that the observed quantum effects of gravity start at the size of an atomic nucleus (governed by mechanical energy accompanied by the concept of physical time) is in line with the (one-component concept of the) proposed UFT.

The hydrogen atom, nature’s simplest stable structure, shows three states, the molecular, the semi-metallic, and the metallic states; the concept of a positronium, (UnA2) p. 95, is in line with the Dirac 2.0 quanta systems scheme; it also supports the concept of "a fluid sun, the coming revolution in astrophysics", (UnA4).

Expressed colloquially, the one-component quanta scheme of the proposed UFT may be interpretated as condensed dynamical energy accompanied by a related emerging potential difference to the underlying two-component quanta systems (creatio ex vacuum). The corresponding potential equalization (interitus in vacuum) is governed by the "least (mechanical) action principle" of Nature accompanied by the concept of "physical time".

Water is the combination of molecular hydrogen and oxygen and was the fundamental prerequisite to build organisms on Earth. The current theory „how the water came to our planet“ is that water reached Earth via meteorites and asteroids. This sounds very unlikely, as the given amount of water reached just our planet from outer solar system delivered in one or just a few units. At the same time the hydrogen particle is basically the only relevant atomic structure in the universe, whereas oxygen is a very Earth-specific atomic structure. In fact oxygen is the most prevalent element on Earth.

The theory of a liquid sun in combination with the (0) vacuum-, (1) plasma-, (2) electromagnetic-, and, (3) Dirac 2.0-quanta creation processes of the UFT may provide an alternative idea about the phenomenon of water. This idea may support Schrödinger‘s „order from order“ mechanism by which orderly events can be produced in the unfolding of life springs (ScE1) p. 80. Regarding the above quanta creation processes we note the key differentiator between the Earth and the other planets of our solar system is its electromagnetic field caused by the hot core of our planet.

Einstein's ether, Davidson's creative vacuum & dynamic quanta w/o motion

(EiA5): „Lorentz succeeded in reducing all electromagnetic happenings to Maxwell’s equations for free space.

As to the mechanical nature of the Lorentzian ether, it may be said of it, in a somewhat playful spirit, that immobility is the only mechanical property of which it has not been deprived by H. A. Lorentz. It may be added that the whole change in the conception of the ether which the special theory of relativity brought about, consisted in taking away from the ether its last mechanical quality, namely, its immobility.  …

Generalizing we must say this: -- There may be supposed to be extended physical objects to which the idea of motion cannot be applied. They may not be thought of as consisting of particles which allow themselves to be separately tracked through time."

(DaJ) p. 26: "In other words, what looks to us as solid is actually a dance of tiny particles in a vacuum. In fact, if all the space were compressed out of our planet earth, it would end up the size of a tennis ball, or some say, even a pea. The exact final dimension is irrelevant because firstly it is not likely to happen and secondly the point is made either way.

In summary, then, according to modern physics: physical material substance is mostly nothing or vacuum."

A Goethe-Newton field theory of light enabled by Fourier’s waves and Calderon’s wavelets?

Goethe's color theory is about a light field; the colors are a limited (light) field and the ray of light is the visible boundary of that limited field, (MüO) S. 9/10, (NuI) S. 85/86. Additionally to the refraction and the Reflexion "Forces" of Newton's optic, there is a third "force", which hasn't been discovered yet or not sufficiently described in more detail yet, (NuI) S. 69/70.

The counter argument of physicics to Goethe's comprehensive concept is (just) that the theory does not allow quantitative predictions, as there is no corresponding mathematical provided.

Newton's optic is about a ray theory of light, where the phenomenon of light is basically about rays of light consisting of smallest parts. Newton’s light/prism experiment is governed by the two principles, refraction & deflection.

The counter argument to Newton's optic is the fact that it demands that the rainbow spectral colors are indecomposable.

Newton's indecomposable (rainbow) spectral colors are in line with the concept of Fourier waves accompanied by the H(1) domain and a discrete spectrum. Goethe's comprehensive field model and his proposed concept of an "effective cloudy" (the third force) might be consistently modelled by wavelets with the related domain, i.e. the complementary closed ("weak" dynamic energy) sub-space of H(1/2) accompanied by a continuous spectrum.

In the calculus of variations the ("weak" mechanical energy) Hilbert sub-space H(1) of (H(1/2) is governed by Fourier waves, while the related complementary closed ("weak" dynamic energy) sub-space of H(1/2) (which is not a Hilbert space (!)) is governed by Calderón wavelets.

Note: Our sky is blue and the oxygen photosynthesis convert the energy of light in combination with green colored plants to molecular oxygen. According to Viktor Schauberger "the sun is responsible for all life, the oxygen (a lower form of solar energy) is responsible for organic growing and development; .. hydrogen (is considered) as the "carrier" of both, carbone (i.e. more than just "carbon") and oxygen. ... From a detached view, far outside the atmosphere, our planet, composed of carbones and fertilized by oxygen, is indeed floating in the hydrogen gas ocean of space", (BaA) p. 51.

Note: Regarding the growing of organic and the creation of life, the working assumptions (1) "the sun is responsible for all life, (2) the oxygen is responsible for organic growing & development, (3) there is a field theory of light, and (4) hydrogen is the gas ocean of space in the universe, (the "carrier" of both, carbone and oxygen) may lead to the following working hypothesis:  "let there be light".

Kant’s external resp. internal sense shaped space resp. time as a priori forms of knowledge

(RoC1) p. 159: „Inquiry into perception of the internal rather than external nature of time reoccurs frequently in Western philosophy. Kant discussed the nature of space and time in his Critique of Pure Reason, and interprets both space and time as a priori forms of knowledge, that is to say, things that relate not just to the objective world but also to the way in which a subject apprehends it. But he also observes that, whereas space is shaped our external sense, that is to say, by our way of ordering things that we see in the world outside of us, time is shaped by our internal sense, that is to say, by our way of ordering internal states within ourselves. Once again: the basis of the temporal structure of the world is to be sought in something that closely relates to our way of thinking and perceiving, to our consciousness. This remains true without having to get tangled up in Kantian transcendentalism.“

(BöG) S. 145, Die Zeit als Thema der Astronomie: „Kosmologisch gesehen sind die Gestirne um der Zeit willen da, und nicht umgekehrt. Sie sind, wie es im Timaios heißt, Werkzeuge der Zeit (42 d 5) oder Werkzeuge der Zeiten (41 e 5). Die Zeit ist der kosmologische Sinn der Gestirne, sie ist deshalb das eigentliche Thema der Astronomie.

UFT related Millennium Problems from the Clay Mathematics Institute

The proposed UFT solves the  (1) 3D non-linear, non-stationary Navier-Stokes Equation Problem, and the  (2) Yang-Mills & Mass Gap problem:

(1)  A well posed 3D-NSE system governed by a H(1/2) mechanical x dynamic energy Hilbert space

Navier-Stokes Equation - Clay Mathematics Institute

"This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations."

The (polynomial-decay) energy Hilbert space H(1/2), which contains the standard (variational (!)) mechanical energy Hilbert space H(1), provides the approximation mechanical x dynamic quanta energy model to the Dirac 2.0 model; in the calculus of variations the ("weak" mechanical energy) Hilbert sub-space H(1) is governed by Fourier waves, while the related closed ("weak" dynamic energy) sub-space of H(1/2) (which is not a Hilbert space (!)) is governed by Calderón wavelets. It turned out that the non-linear energy term of the 3D non-linear, non-stationary NSE system is bounded with respect to the H(1/2) energy norm as a simple consequence of the Sobolevskii inequality (BrK9). The tool to build the orthogonal relationship between the mechanical and the dynamic (energy) worlds are the Riesz transforms (resp. the 1D Hilbert transform).

Note: The second unknown variable of the NSE is the pressure function p; it can be represented as an appropriate function of Riesz transforms of the unknown velocity function u; the corresponding Neumann problem of the pressure field p may be modelled by the Prandtl operator accompanied by the domain H(1/2) enjoying appreciated properties, (LiI).

Note: Already this first approximation layer of a complementary mechanical x dynamic quanta energy field model provides an appropriate physical modelling framework for the alternatively proposed Schrödinger (-Calderon) quantum momentum operator, (BrK7), and the physical (mechanical (!)) "frictionless" boundary modelling problem of the NSE, (GaG).

In summary, the H(1/2) energy Hilbert space based proof of the 3D-NSE problem governing the flow of fluids such as water and air gives understanding. The combination with the understanding by an aligned Schrödinger (-Calderon) quantum momentum operator provides a kind of proof of concept (PoC) of this first approximation layer of the complementary mechanical x dynamic quanta energy field model (being followed by the second approximation layer, the Dirac 2.0 quanta dynamics model) as part of the proposed UFT.

(2)  The Dirac 2.0 quanta dynamics model making the YME system obsolete

The so-called standard model of elementary particles (SMEP) is about three de-coupled theories with similar characteristics; the underlying independent three groups of elementary particles are governed by Fourier waves with standard domain H(1). Accordingly the theory specific „wave dynamics“ requires similar but not identical wave equation models accompanied by related bounded energy norms. The considered exponential-decay Hilbert space of the UFT provides an appropriate domain framework for well defined hyperbolic wave operators. Accordingly, not the simplest energy Hilbert space model H(1/2) but the Dirac 2.0 modelling layer accompanied by well defined wave operators (i.e. including the necessary appropriately defined domains (!)) makes the YME system obsolete.

                                              Supporting data

Braun, K., A Krein space based quanta energy field model, supporting mathematics

Braun, K., Current physical and mathematical realities regarding an unified field theory

                                              

                               

Braun K., UFT related list of papers