A. The Gordian knot: „the principle of transfer causality“

The common denominator of all dynamic models in physics is "the principle of transfer causality". In SMEP this leads to the invention of two types of quantum elements, the fermions and the bosons. In the GRT this principle is addressed by the principle that „the boundary of the boundary of a manifold is zero, (BrK10) p. 15, (CiI) p. 49. 


Quote from M. Planck: „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 Gesetzmaßigkeit: der dynamischen, streng kausalen, und der lediglich statistischen, für das Verständnis des eigentlichen Wesens jeglicher naturwissenschaftlichen Erkenntnis besitzt; es sei mir daher gestattet, diesem Gegenstande und diesem Gegensatze heute einige Ausführungen zu widmen“, (PlM) S. 90.

Quote from A. Einstein: "Recapitulating, we may say that according to the general theory of relativity space is endowed with physcial quantities, in this sense, therefore, there exists an ether. According to the general theory of relativity without ether is unthinkable;  for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it", (EiA).

Quote from J. A. Wheeler et al.: „According to an idea of extreme simplicity of the laws at the foundations of physics, what one of us has called „the principle of austerity“ or „laws without law at the basis of physics“ in geometrodynamics it is possible to derive the dynamical equations for matter and fields from an extremely simple but central identity of algebraic topology: the principle that the boundary of the boundary of a manifold is zero“, (CiI) p. 49.

Quote from C. W. Weizäcker: „Das Seiende der Physik ist, so scheint es, die Materie“, (WeC) S. 344

Quotes from D. Bohm: „It is important to emphasize that mathematics and physics are not being regarded here as separate but mutually related structures (so that, for example, one could be said to apply mathematics to physics as paint is applied to wood). Rather, it is being suggested that mathematics and physics are two be considered as aspects of a single undivided whole“, (BoD) p. 199.

„We begin with the mathematical description of explicate order. Now, explicate orders arises primarily as a certain aspect of sense perception and of experience with the content of such sense perception. It may be added that, in physics, explicate order generally reveals itself in the sensible observable results of functioning of an instrument.      What is common to the functioning of instruments generally used in physical research is that the sensible perceptible content is ultimately describable in terms of a Euclidian system of order and measure, i.e., one that can adequately be understood in terms of ordinary Euclidian geometry. We shall therefore begin with a discussion of Euclidian systems of order and measure", (BoD) p. 200.

„We now discuss the mathematical description of implicate order. Implicate order is generally to be described not in terms of simple geometric transformations, such as translations, rotations, and dilations, but rather in terms of a different kind of operation. In the interests of clarity, we shall therefore reserve the word transformation to describe a simple geometric change within a given explicate order. What happens in the broader context of implicate order we shall call a metamorphosis“, BoD) p. 202.

The next step is to discuss the mathematization of the language for the description of implicate order. …This approach is indeed used in a great deal of modern mathematics, especially in number rheory. Thus, one can start with what are called undefinable symbols. The meaning of such a symbol is never directly relevant. Rather, only relationships and operations in which these symbols take part are relevant,“ (BoD) p. 202.

Quote from B. Russell: „Hume had proved that the law of causality is not analytic, and had inferred that we could not be certain of its truth. Kant accepted the view that it is synthetic, but nevertheless maintained that it is known a priori. He maintained that arithmetic and geometry are synthetic, but likewise a priori. He thus led to formulate his problem in these terms:     How are synthetic judgements a priori possible?     The answer to this question, with its consequences, constitutes the main theme of The Critique of Pure Reason“, (RuB) p. 680.

Quotes from A. N. Whitehead: „It cannot be too clearly understood that some chief motions of European thought were framed under the influence of a missapprehension, only partially corrected by the scientific progress of the last century. This mistake consists in the confusion of mere potentiality with actuality. Continuity concerns what is potential; whereas actuality is incurably atomic“, (WhA) p. 61.

„This account of „presentational immediacy“ presupposes two metaphysical assumptions:

(i) That the actual world, in so far as it is a community of entities which are settled, actual, and already become, condititions and limits the potentiality for creativeness beyond itself, (WhA) p. 65.

(ii) The second metaphysical assumption is that the real potentialities relative to all standpoints are coordinated as diverse determinations of one extensive continuum. This extensive continuum is one relational complex in which all potential objectivations find their niche. It underlies the whole world, past, present, and future“, (WhA) p. 66.

References

(BoD) Bohm D., Wholeness and the Implicate Order, Routledge & Kegan Paul, London, 1980

(CiI) Ciufolini I., Wheeler J. A., Gravitation and Inertia, Princeton University Press, Princeton, New Jersey, 1995

(EiA) Einstein A., Ether and the theory of relativity, An Address delivered on May 5th, 1920, in the University of Leyden

(PlM) Planck M., Dynamische und Statistische Gesetzmässigkeit, In: Roos, H., Hermann, A. (eds) Vorträge Reden Erinnerungen, Springer, Berlin, Heidelberg, (2001) 87-102

(RuB) Russell B., History of western philosophy, Routledge, London, 1995

(WeC) Weizsäcker C. F., Die Einheit der Natur, Carl Hanser Verlag, München, 1971

(WhA) Whitehead A. N., Process and Reality, An Essay In Cosmology, The Free Press, New York, 1985

  
B. The overall conceptual new paradigm

1. A quanta system intrinsic dynamic energy type

The current dynamic laws in theoretical physics (e.g., in thermo-statistics or in the quantum field theories) are based on statistical regularities. Planck proposed a distinction between dynamic (i.e. strictly causal) and statistical regularities, (PlM). The overall conceptual new paradigm is that the fundamental dynamic laws of the proposed unified field theory are built on two types of energy concepts, the current mechanical (kinetical and potential) energy type, and a new dynamic (potential) energy type.

The definition of the new dynamic energy type follows the same design principle as the definition of the kinetic-mechanical energy type. The latter one is based on the Friedrichs extension of the classical Laplacian (potential) operator with domain H(2). Its Friedrichs extension is a self-adjoint, positive definite (i.e. Hermitian) operator (its inverse operator is compact) accompanied by an extended domain H(1), (BrK0) pp. 21-22. The quanta system specific new dynamic energy potential operators of building block 1 below are defined by the same mathematical concept: this is the Krein space intrinsic so-called J-operator, (AzT), (BoJ), or potential operator, (VaM).

The new paradigm is based on two building blocks. Building block 1 is based on a priori (implicate) "meta-physical "ground state & perfect plasma dynamic systems" modelled in a Hilbert-Krein space based framework. Building block 2 is based on a purely Hilbert scale based approximation Hilbert space framework of the quanta scheme type specific Krein spaces. In the framework of building block 2 the dynamic energy potential operators of building block 1 are approximated by the Schrödinger 2.0 operator; its crucial differentiator to Schrödinger's original operator is "just" the combination with the Riesz transform operator, which gives the Calderón-Zygmund integrodifferential operator, (BrK0) pp. 27-28, 36, (BrK6), (EsG) pp. 39, 44). This Riesz operator enables an appropriate domain, which is complementary to mechanical energy Hilbert space H(1), (BrK6). While the mathematical tool to analyze functions with domain H(1) are Fourier waves, the related tool for Calderon-Zygmund integrodifferential operators are Calderon wavelets, (MeY).

2. A quanta system intrinsic least action princple

A physical system just governed by the conservation law of energy is insufficient to formulate any physical law. The ultimate cause of any dynamics lies in related concepts as "potentials" and "potential differences".

3. The new math. tool set for new paradigm groups 1-3

For more details we refer to the below sections. By purpose the order of the considered paradigm groups is reversed. The deductive structure of the proposed Krein space based quanta system scheme is based on a priori dynamic ground state (vacuum) & perfect plasma systems. The related quanta pairs of those systems are defined by quanta number sequences < 1. Those systems are considered as meta-physical (i.e., not-mechanical) energetical systems. The quanta of the physical systems are defined by quanta numbers >1, called as Dirac quanta systems, (the neutron system is defined by quantum number 1). There is one 2-component Dirac quanta system (called perfect (plasma medium type) electromagnetic quanta system), and there are three 1-component atomic nucleus quanta systems. The 2-component quanta systems are governed by the complex Lorentz group SU(2)xSU(2). The 1-component Dirac quanta systems are governed by SU(2), the hidden symmetry group of the Coulomb problem, (BrK14) p. 14. The 1-component Dirac systems can be approximated by a purely Hilbert space quanta system in a Minkowski space continuum, which is called dynamic fluid/gas particle system.

... re paradigm group 3

The a priori dynamic ground state (vacuum) & perfect plasma quanta systems are supposed to provide a model of a "darkness field", which is characterized by purely dynamic quanta types. The perfect electromagnetic quanta system is supposed to provide a related "brightness field". This field may allow an alternative interpretation of the CMBR and the Landau damping phenomena, (RoP2). In combination with the 1-component Dirac quanta systems it may also support the concept of "liquid metallic hydrogen as a solar building block", (RoP). The Dirac quanta systems may be interpreted as "curdled mass" quanta systems from the purely dynamic quanta system accompanied by a corresponding (gravitational) potential. This concept may provide an appropriate model to explain the inertial mass of bodies through a kind of interaction with all masses of the universe (i.e. Mach's principle). Accordingly, the Hilbert space based (approximation) dynamic fluid/gas particle concept is supposed to enable a SRT-Minkowski space accompanied by Einstein's lost key (i.e. "a variable speed of light", (BrK0) pp. 53, 152, 155, (UnA1)), and Christodoulou's and Klainerman's "global stability of nonlinear Minkowski space", (BrK10) p. 111/112, (BrK14) pp. 25-26, (ChD), (KlS1), (NaS). The (gravitational) potential may be simply interpreted as energy per mass, (UnA1) pp. 78, 118. In terms of the proposed energy types the inter-action potential between the total "dynamic energy" and the total "mechanical energy" in the universe provides a mathematical model for a gravitational potential in line with Mach's principle. The integrated Dirac quanta system enables a combination of Mach's principle with Dirac's "large number hypotheses" providing a mathematical modelling framework for Unzicker's propagated "Mach 2.0 principle", (BrK10) pp. 155-157, (UnA1).

... re paradigm group 2

The Dirac quanta systems accompanied with intrinsic (implicate in the sense of D. Bohm) dynamic anti-quanta replace the current "force" typ specific SMEP conceptual design. The Hilbert space based approximation layer H(1/2) enables the definition of the Schrödinger 2.0 dynamic Hermitian potential operator with complementary domain to the Laplacian self-adjoint potential operator, i.e., the sub-space (H1,ortho) of H(1/2), (BrK0) pp. 11, 18, 19, 27-28, 36, 38, (BrK6), (BrK9), (BrK10).  

... re paradigm group 1

The Hilbert space based approximation layer H(1/2) composed by the standard variational mechanical Hilbert space H(1) and the closed sub-space (H1,ortho) provides the appropriate domain for a well defined Prandtl/Plemelj double layer potential operator, (BrK0) pp. 13, 26, 64, (BrK7), (BrK9), (BrK10), (BrK14) p. 6. It turned out that its range, the distributional Hilbert space H(-1/2), is the appropriate variational framework to define a well-defined 3D non-stationary, non-linear NSE system, (BrK0), (BrK11). It also turned out that the underlying extended distributional Hilbert space of the Krein space systems allows variational coerciveness conditions for hyperbolic PDE operators, (BrK10) pp. 11, 25, 27.


Note: "The Newtonian (single layer) integral equation for the gravitational potential gives the fundamental solution of the inverted Poisson equation. The singular integral is taken over the entire volume where the mass is distributed. Its singular term in the kernel becomes infinite when the field point coincides with the source point", Wikipedia. In modern terms spoken, Newton's adapted original law (stated in terms of a system of two particle/bodies) lead to the very first invented whatever colored hole in astrophysics. Plemelj's concept of a double layer potential integral over a closed surface enables a well defined potential definition for continuously distributed matter particles on a closed surface, (BrK10) p. 6, (PlJ); quote (PlJ) p. 5: "Auch physikalisch gibt es für die Annahme, das Newtonsche Potential müsse im Unendlichen verschwinden, keinen hinreichenden Grund"; ...quote (PlJ) p. 12: "Die größere Tragweite des neuen Begriffes ist nicht schwer zu erkennen, denn jetzt ist man unabhängig sowohl von der Existenz der Normale als auch der Ableitungen am Rande."

Note: The Schwarzschild radius may be seen as the modern version of Newton's original law: "The Schwarzschild radius is a parameter (which can be calculated from the mass M of a considered body, the speed of light, and the Newtonian constant of gravitation) in the Schwarzschild solution to Einstein’s field equations. It corresponds to the radius of a sphere in flat space that has the same surface area as that of the event horizon of a Schwarzschild black hole of a given mass. It is a characteristic quantity that may be associated with any quantity of mass", Wikipedia.

Note: The mathematical model of Dirac’s point charge particle is the Dirac distribution "function" d(x). It is an element of the (distributional) Sobolev space H(-n/2-e), e>0, where n denotes the space dimension. Accordingly, the corresponding point charge particle of the 3D-Coulomb and 3D-Newton potential functions are elements of H(-3/2-e), e>0. Therefore, the above elements of the variational H(-1/2) Hilbert space representation of the 3D-NSE system may be interpreted as "Dirac 2.0 point charge particles". Mathematically spoken, the Dirac 2.0 point charge particles are the "dual" elements of their related variational mechanical energetical particles, which are the elements of H(1/2). The Noetherian Prandtl operator provides a well defined mapping between H(1/2) and H(-1/2) ((distributional) functions with closed surface domain), solving the exterior Neumann problem, (BrK0) p. 26, (LiI).

Note: If the concept of London dispersion forces is replaced by the concept of a dynamic potential difference "force" ... then the current physical law for an ideal gas in thermodynamics in the form P*V = k*N*T (k denotes the Boltzmann constant) becomes obsolet. As the pressure P becomes an explicate potential difference between two dynamic quanta systems and as the concept of kinetic energy is no longer required the volume V and the temperature T become obsolet, as well.

Note: The concepts of a meta-physical "darkness" quanta field and a physical "brightness" quanta field is supposed to enable a missing "theory of light". Such a theory would be in line with Bohm's conception of "wholeness, explicate and implicate order", (BoD1). The formation of stars might be interpreted as a kind of condension processes of Dirac quanta systems accompanied by a symmetry break down form SU(2)xSU(2) to SU(2). Such a process might provide an appropriate modelling framework to explain a liquid metallic hydrogen sun, (RoP), (UnA4). Accordingly, ...

 - ... the solar wind might be interpreted as a cold plasma quanta flow with supersonic quanta speed in a space-time continuum governed by the restricted Lorentz group

- ... the energy output of stars (i.e. their luminosity) is no longer governed by the Stefan-Boltzmann law but by a field theory of light accompanied by the concepts of "brightness" and "darkness".

4. Related notes

Note: „Einstein asserted that Mach had sought to explain the inertial mass of bodies through a kind of interaction with all masses of the universe“, (BaJ).

Note: The distinction between energetical dynamic and mechanical quanta systems is in line with Mie's conception an electromagnetic theory as "ether theory", where dynamic laws govern "the state of the ether and its dynamical evolution, and conceiving of elementary mechanical particles as stable "energy knots" in the ether (i.e., the discrete eigenpair spectrum of the mechanical potential operator) rather than independent entities", (SmC).

Note: The proposed mathematical formalism in a Krein space framework deals with indefinite metrics, which is in line with Heisenberg's mathematical formalism for a unified field theory, (HeW) vi.

Note: The Lamb shift phenomenon says that the energy values of an electron in the hydrogen potential field shows slighly different values than the related discrete energy knots n. The fine structure constant of the hydrogen atom is a mathematical correction term in the Sommerfeld energy formula derived from the „hydrogen“ Dirac equation governed by the Coulomb potential operator based on the „physical relevant“ stationary solutions of this equation. It is derived from Dirac‘s spin-orbit momentum operator by a mathematical „trick“, which may be called „two-component-separation of the angular momentum“. Beside the restriction to only physical relevant solutions of the Dirac equation there is also a mathematical approximation error, which is caused by a finite power series approximation of this separation (phys., a shrinking of the set of quantum numbers), (BrK14).

Note: The Ginzburg-Landau theory in a magnetic field accompanied with the concept of surfaces of superconductors. In (GiJ) Ginzburg has presented the notion of Krein spaces that is an extension of Hilbert spaces for studying in quantum mechanics, (BrK14) p. 21.

The BCS theory of superconductors is accompanied by the concepts of Cooper pairs and a mean-field-Hamilton operator. In the proposed model the Cooper pairs may be interpreted as alternating (maximal) pairs. The counterpart of the mean-field-Hamilton operator might become the plasma potential operator, (BrK14) p. 21.


C. The deductive structure of the quanta system hierarchy

There is a priori energetical dynamic (ground state and perfect plasma) world characterized by a 2-component quanta systems with quanta numbers sequences < 1.

From the a priori energetical dynamic quanta system there may be built two composed 1-component quanta systems with quanta number sequences > 1, called electroton and magneton, and a related 2-component (electroton - magneton) plasma type quanta energy field, called "perfect electromagnetic system".

From the a priori energetical dynamic quanta system there may be also built further composed 1-component quanta systems with quanta numbers sequences > 2, called Dirac 2.0 systems. The total of all quanta systems with with quanta number sequences > 1 are called Dirac quanta systems. 

The 2-component systems are governed by the complex Lorentz group SU(2)xSU(2), while the Dirac quanta systems are governed by the real (restricted) Lorentz group SU(2).

The Dirac systems can be approximated by a purely H(1/2) Hilbert space based energetical system in a Minkowski space framework. The equalization (entropy) processes of potential differences from this mechanical energy system back to "symmetric" dynamic energy systems become a model for observed particle decay processes. More generally spoken, the interface between the approximating H(1/2) Hilbert space energetical system and the "underlying" Dirac quanta systems provides an appropriate modelling framework for "Nature constants", like the CMBR, the Landau damping, the size of Bohr’s magneton, the spin hypothesis, the beta-decay phenomenon, and Ehrenhaft’s photophoresis phenomenon.


Note: The quantum system with quantum number one is called the neutron.

Note: The new paradigm is about "symmeties" of 2-component quanta systems, while Heisenberg's mathematical formalism for a unified field theory deals with the degeneracy of the ground state, (HeW) vi.

Note: The Planck action constant is the proportionality factor between the energy of a photon and its frequency. It was introduced to model the observed spectral distribution of black body radiations. Planck denoted it by the letter "h" to anticipated his interpretation that this constant is just an auxiliary constant, see also (RoP1).

Note: The constant speed of the photon is calculated from the permittivity and the permeability constants of the purely mechanical 1-component "vacuum" Maxwell equations (accompanied by the concepts of space, time, velocity, and mechanical mass).

Note: Practically, all processes in the quantum world are time reversal invariant. 

Note: In thermodynamics (e.g. in the entropy process) and in the relativity theory (with the related paradigm of an "expanding universe") there is a direction of time.

Note: One of the main mathematical tools in relativity theory (beside differentiable manifolds) are hyperbolic partial differential equations (PDE) accompanied by Sobolev space domain. The most prominent hyperbolic PDE models are wave and radiation initial-boundary value problems, which remain unchanged when time is reversed. 

Note: In opposite to parabolic (with direction of time) and elliptic (time independent) PDE, which both enjoy appropriate shift theorems in a Sobolev space framework, hyperbolic PDE are accompanied by the concepts of wave fronts and shocks, see e.g. (CoR) pp. 551 ff. We note that the baseline Hilbert space of the proposed Krein space framework (accompanied by a continuous parameter t>0) enables appropriate shift theorems for those type of PDEs, (BrK0) pp. 15/25.

Note: The standard variational domain of the mechanical (self-adjoint) potential operator is the H(1) energy space. Its complemetary sub-space of the newly proposed extended energy Hilbert space H(1/2) with respect to the H(1/2) norm (the energy invariant scalar function) provides the domain of a complementary dynamic potential operator. This operator is not self-adjoint, however, in the corresponding variational representation it may be interpreted as a compact disturbance of the mechanical potential operator enabled by the underlying coerciveness (Garding type) inequality, (BrK0) p. 26. Technically spoken, the extended energy Hilbert space H(1/2) is the new conceptual element of building block 2 below.

Note: The two current theories, the relativity theory and the quantum theory, describe two different universes. Colloquially spoken, the proposed UFT provides „superordinated laws establishing the constants of Nature“, which is one of the four conceivable explanations in (GaG3) S. 107/108.

D. Dynamic action types and related notes

Quote: „Das Infinitesimale ist (gar) keine Größe, sondern eine dynamische Konzeption. Es ist mehr ein Prozess, der eine Ausgangsgröße zum Verschwinden bringt, wenn sie ihre Schuldigkeit getan hat“, (FiE) S. 102; transl.: „The infinitesimal is not a stable value, but a dynamic conception. It is more a process, which makes an initial value disappear, when it is no longer required“.

1. Dynamic action types

a. Intra-action dynamics within the quanta systems

The intra-action dynamics within each quanta system is caused by the quanta intrinsic potential differences between the considered quantum "particle" and its related anti-quantum "particle". By design this potential difference is characterized by the quanta system intrinsic potential operator accompanied by appropriately defined quanta numbers sequences. This concept is in line with Bohm's concepts of wholeness, explicate and implicate order, (BoD1). 

b. Inter-action dynamics between the quanta systems

The inter-action dynamics between quanta systems is caused by the different balances of the system intrinsic potential differences within the considered quanta systems. In other words, it is caused by the different total dynamic energy of those systems.

c. Inter-action dynamics within the H(1/2) dynamic fluid system. 

The inter-action dynamics with the H(1/2) dynamic fluid system is governed by the intrinsic energy difference between the standard variational mechanical energy Hilbert space H(1) and its related complementary closed sub-space of H(1/2). It is supposed to replace Schrödinger's concept of case specific potential functions.


2. Related Notes

a. Intra-action dynamics

Note: The minimum volume of a substance on atomic level is determined by the nature of the electrons not so much by the (far larger) nucleus, (UnA4) p. 63.

Note: The Maxwell fields can carry energy from one place to another. It describes the electricity dynamics of an a priori existing charged elementary particle (electron) in an idealized semiconductor world governed by an electric and a magnetic field induced by the sum of a line current (in an electrical conductor world) and a so-called displacement current (a cross-section line reduced 1st order approximation of an electrical insulator world accompanied by the notions of „time“ and „distance“). Mathematically speaking, the energy tensor of the electromagnetic field is only known outside of such an a priori (!) electron (particle) energetical system.

Note: The Yang-Mills (gauge) theory is a generalization of Maxwell"s theory of electromagnetism and is fundamental to the SMEP, particularly for understanding the strong and weak nuclear forces, Wikipedia.

Note: Mie’s project in (MiG0,(MiG1),(MiG2) is „to derive electromagnetism, gravitation, and aspects of the emerging quantum theory from a single variational principle and a well-chosen Lagrangian, governing the state of the aether and its dynamical evolution, and conceiving of elementary particles as stable “knots” in the aether rather than independent entities“, (SmC).

Note (The Mie theory): „The aim of the trilogy on matter theory in (MiG), (MiG1), (MiG2) was to develop a unified theory able to account for the existence and properties of electrons (as well as atoms or molecules), explain recent observations of atomic spectra, and yield field equations for gravitation“, (SmC).

Note (The Mie theory, cohesive pressure keeping the electron together): "The theory of Maxwell and Lorentz cannot hold for the interior of the electron; therefore, from the point of view of the ordinary theory of electrons we must treat the electron as something given a priori, as a foreign body in the field. A more general theory of electromagnetics has been proposed by Mie, by which it seems possible to derive the matter from the field, (WeH1) p. 206.  ... In the static case the Mie equation states that the electric force is counterbalanced in the ether by an "electrical pressure". ..... this is the required cohesive pressure that keeps the electron together", (WeH1) p. 208.

Note (The Mie theory and a global nonlinear stability of the Minkowski space): „Mie aimed to derive electromagnetism, gravitation, and aspects of the emerging quantum theory from a single variational principle and a well-chosen Lagrangian. Mie’s main innovation was to consider nonlinear field equations to allow for stable particle-like solutions (now called solitons), and he clarified the use of variational principles in the context of special relativity“, (SmC). This is in line with Klainerman’s proof of a „global nonlinear stability of the Minkowski space, (ChD).

Note (The Mie theory, electromagnetic theory as “aether physics"): „Part of Mie’s project was to develop a relativistic theory of gravitation as a consequence of his generalized electromagnetic theory, and our second goal is to briefly assess this work, which reflects the conceptual resources available for developing a new account of gravitation by analogy with electro-magnetism. …. Mie characterized electromagnetic theory as “aether physics.” Mie emphasized the appeal of reducing physics to a simple set of equations governing the state of the aether and its dynamical evolution, and conceiving of elementary particles as stable “knots” in the aether rather than independent entities“, (SmC).   

Note (The Mie theory): „Die Grundannahme meiner Theorie ist, daß auch im Innern der Elektronen elektrische und magnetische Felder auftreten. Die Elektronen und demnach überhaupt die kleinsten Teilchen der Materie sind nach dieser Auffassung also mit dem Weltäther nicht wesensverschieden, sie sind nicht, wie man sich das vielleicht vor zwanzig Jahren dachte, Fremdkörper im Äther, sondern sie sind nur Stellen, wo der Äther einen ganz besonderen Zustand angenommen hat, den wir durch das Wort elektrischte Ladung bezeichnen.  ….    Man wird vielleicht denken, daß man mit der eben formulierten Grundannahme wenig anfangen könne. Sie führt aber immerhin zu einer allgemeinen Form für die Grundgleichungen der Ätherphysik, wenn man noch zwei weitere Annahmen hinzunimmt. Die erste ist, daß das Relativitätsprinzip allgemeine Gültigkeit haben soll, die zweite, daß die bisher bekannten Zustände des Äthers, nämlich elektrisches Feld, magnetisches Feld, elektrische Ladung, Ladungsstrom, vollständig ausreichen, um alle Erscheinungen in der materiellen Welt zu beschreiben“, (MiG).

Note: Mie’s nonlinear field equations allow for stable particle-like solutions using variational principles in the context of special relativity, (SmC). This is in line with Klainerman’s proof of a global nonlinear stability of the Minkowski space, (ChD). Technically speaking, the eigenpairs of the standard self-adjoint (mechanical!) Laplace operator with H(1)-domain become the model of Mie's (mechanical!) energy knots. The "complementary"  (dynamic) operator with the complementary domain in H(1/2) with respect to the H(1)-norm becomes the model of the "implicate" dynamic energy field, which is governed by the Schrödinger 2.0 operator. Technically speaking the Schrödinger 2.0 operator is "just" the Riesz transformed Schrödinger operator. For the appreciated properties of the Riesz transforms we refer to (BrK14) p. 33.

b. Inter-action dynamics

Note: The symmetry break down from the complex Lorentz group to the (real) restricted Lorentz group becomes a characteristic of the inter-action dynamics between purely dynamic 2-component quanta systems and 1-component mechanical quanta systems accompanied by the concept of mechanical energy and the Minkowski space-time continuum. SU(2) is isometric to the unit quaternions S(3). This puts the spot on the Courant conjecture, which would show that the four-dimensional Minkowski physical Minkowski space-time world of classical physics enjoys an essential distinction, (CoR) p. 763. 

Note: The three Dirac 2.0 systems are accompanied by a „symmetry break down“ from SU(2) x SU(2) to SU(2), the symmetry group of the Klein-Gordon equations. 

Note The group SU(2) is isometric to the unit quaternions S(3). The quaternions provide an appropriate field to address the „translation-rotation“ (linear and angular rotation) „permutation“ requirement. The perhaps primary application of quaternions is the quaternion rotation operator. This is a special quaternion triple-product (unit quaternions and rotating imaginary vector) competing with the conventional (Euler) matrix rotation operator, (KuJ). Its outstanding advantages compared to the Euler geometry are

- the axes of rotation and angles of rotation are independent from the                underlying coordinate system and directly readable
- there is no need to take care about the sequencing of the rotary axes.

Note: The math. proof of the CPT invariance phenomenon, the only fundamental law of nature requiring a „time arrow“, is enabled by the complex Lorentz transform (StR). In other words, as long as there are no decay processes of atomic nuclei in scope the laws of Nature allow a "reverse of time".

Note: The transitions from the 2-component quanta systems to the 1-component Dirac 2.0 quanta systems avoid Dirac's spin hypothesis.   

Note: "Plasma „matter“ is basically characterized by the following two requirements:

-   there is an interaction between two oppositely charged particle types 
-   the numbers of those two particle types may be arbitrarily small or large,
    but they need to be almost the same", (CaF) p. 1.

„The Landau damping phenomenon is a characteristic of collisionless plasmas, but it may also have applications in other fields. For instance, in the kinetic treatment of galaxy formation, stars can be considered as atoms of a plasma interacting via gravitational rather than electromagnetic forces. Instabilities of the gas of stars can cause spiral arms to form, but this process is limited by Landau damping“, (ChF) p. 245, see also (ChF) p. 402.

"Most of the visible matter in the universe exists as plasma, whereas lightning and the aurora are the only natural manifestations of the plasma state on Earth", (DeR) p. 1.

"The sun, like most stars, is composed of plasma; in its core, the kinetic energy of the atomic nuclei, dissociated from the electrons, is so great that they can overcome their mutual electrical repulsion and fuse together, releasing energy", (DeR) p. 1.

"The solar wind consists of a diffuse plasma that streams outwards from the sun and fills interplanetary space. Its density and velocity near the Earth fluctuate in time; ... The Earth's magnetic field is sufficiently strong to deflect the solar wind", (DeR) p. 82, see also (ShF) p. 372 ff.

"The kinetic description of galaxies has many similarities with that of plasmas. Because collisions between stars in galaxies are very rare, the evolution of the distribution of stars in phase space can be described by a continuity equation which has the form (5.5). Each star interacts with the rest of the galaxy through the local gravitational potential", (DeR) p. 122.

The dynamic 2-component "perfect plasma" system is in line with the baseline requirement for plasma matter associated with an "empty space potential/pressure" providing an appropriate explanation of the Landau damping phenomenon.

Note: The inter-action dynamics between the "perfect plasma" system and the mechanical 2-component „electromagnetic“ system (both governed by the complex Lorentz group) supports Robitaille's "Liquid Metallic Hydrogen Model of the Sun and the Solar Atmoshere", (RoP), (UnA4).

Note: There are only two superfluids which can be studied in laboratory. These are the two isotopes of helium. Unlike all other substances they are unique because they remain in the liquid state even down to absolute zero in temperature, (AnJ) p. 21.

Note: Sommerfeld’s fine structure constant is „just“ mathematically required to ensure convergent power series representations of the solutions of Dirac equation.

Note: 

(1) The CMBR (currently interpreted as the "echo of the early universe", (LaM)) is an essential element of theoretical and observational cosmology and one of the foundation stones of the big bang models; to the author's humble opinion, those models are extremely unrealistic because they are based on an a priori required mathematical singularity which caused for whatever reason the biggest explosion ever, (PeR) p. 444.

(2) There are currently two different (!) underlying physical "force" types for the Landau damping phenomenon depending from the considered mathematical linear (Coulomb potential based "hot plasma") or nonlinear (Landau collision operator based "cold plasma") model, (BrK14) p. 18.

(3) The cosmic microwave background radiation (CMBR) and the Landau damping phenomena may be interpreted as characteristic (echo) phenomena of the EMT electroton-magneton quanta creation process from the GSM and PPM, see also (BrK14) p. 26.

Note: The mechanical 2-component „electromagnetic“ system and the non-mechanical 2-component "perfect plasma" system enable consistent explanations of the Landau damping phenomenon and the related CMBR, (LaM), and Ehrenhaft's photophoresis phenomena, (EhF). It may also enable a missing theory of light anticipating „Einstein’s lost key“, (UnA1), Dirac's large number hypothesis, (UnA1) p. 150, (UnA2) p. 85, and Dicke’s related "theory of a variable speed of light", (UnA1) p. 129, accompanied by a mechanical global nonlinear stability of the Minkowski space, (ChD).

Note (Nature constants): The UFT indicates a new role of Nature constants. They may provide physical characterizations of the borderlines within the hierarchical quanta system structure of the above five dynamic quanta systems. The obvious characteristic borderline constant between ANT and PDT is Planck's quantum of action. In this context we refer to Robitaille’s „blackbody radiation and the loss of universality: implications for Planck’s formulation and Boltzman’s constant“, (RoP3). The observed duration for the beta-decay (about 15 min) might become another Nature constant with respect to the borderline between EMT and ANT. The magnetic moment interpretation of an electroton might become another characteristic constant. Basically Unzicker's approach investigating constants of nature and questioning their origin is reversed, (UnA2) p. 3. In other words, Planck's quantum of action become the most rough "approximation" constant within the deductive structure as its formula contains the generic term "temperature" for "energy". It also contains the speed of light, which can be calculated from the two electromagnetic Nature constants, the vacuum permittivity and the vacuum permeability resp. the Bohr magneton, i.e. the size of atomic magnetic moments, (BlS) p. 4.

c. Inter-action dynamics within the H(1/2) dynamic fluid system   

Note: In (RoP2) it is shown that hydrogen bonds within water should be able to produce thermal spectra in the far infrared and microwave regions of the electromagnetic spectrum. This simple analysis reveals that the oceans have a physical mechanism at their disposal, which is capable of generating the microwave background.

Note: The pressure p in the NSE (which may be interpreted as a "potential") can be expressed in terms of the velocity u by the formula p = R(u x u), where R denotes the Riesz operator and u x u denotes a 3x3 matrix.

Note: The  H(1/2) Hilbert space plays also a key role in the Teichmüller theory and the universal period mapping via quantum calculus accompanied by  a canonical complex structure for H(1/2), (NaS). Also, the degree or a winding number of maps of the unit circle into itself corresponds to a related H(1/2) -norm enabling the statement „one cannot her the winding number“, (BoJ).

Note (Einstein's lost key, (UnA1)): All known tests of the GRT can be explained with the concept of a variable speed of light, (DeH), (UnA1) p. 142. Additionally, there is a „nonlinear stability of the Minkowski space“, (ChD). Approximation theory of a nonlinear operator equation in Hilbert scales is enabled by an appropriate decomposition of the nonlinear operator N=L+R into a lineralized operator L and a remaining nonlinear operator R. In this context "nonlinear energy stability" is ensured if the nonlinear variational equation representation fulfills the Garding inequality with respect to the underlying „energy norm“ induced by the linearized term L. In this case the remaining nonlinear operator R may be interpreted as a compact disturbance of the linear operator, (BrK0) pp. 11, 26, (BrK13).

Note (Mechanical mass-energy equivalence): Einstein's famous formula  E = m*c*c  may be interpreted as approximation formula, where the energy terms on both sides of the equation are interpreted as norms of the underlying weak variational representation in an appropriately defined Hilbert-Krein space framework. In other words, the Hilbert-Krein space framework (accompanied by the concept of indefinite norms) avoids the problem of infinite negative eigenvalues. This problem occurs in Dirac's relativistic invariant wave equation for an one-electron system, which allows electrons to traverse very high potential thresholds with a certain probability, e.g. (HeW1) S. 76.


                     

Braun K., 3D-NSE, YME, GUT solutions, July 2019

 

Braun K., A distributional Hilbert space framework to prove the Landau damping phenomenon