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

The common denominator of all dynmic models in physics is the 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.


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 is based on the same mathematical concept as the definition of the kinetic energy type, i.e., it is based on the Laplacian (potential) operator: this is a self-adjoint, positive definite (i.e. Hermitian) operator accompanied by an appropriately defined domain.

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 Hilbert space based (approximation) dynamic fluid/gas particle concept is supposed to enable a SRT-Minkowski space accompanied the concepts of  "a variable speed of light, (BrK0) pp. 53, 152, 155, (UnA1),a global stability of non-linear Minkowski space, (BrK10) p. 111/112, (BrK14) pp. 25-26, (ChD), (KlS1), and the Teichmüller theory related to the H(1/2) Hilbert space, (NaS).

The distinction between purely dynamic energy and mechanical energy in the universe provides a mathematical model 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 an Schrödinger 2.0 dynamic Hermitian potential operator with (H1,ortho) sub-space domain, (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. 


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 new dynamic quanta systems

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.


C. The two building blocks

1. In a nutshell: the building blocks 1 & 2

The below building blocks 1 & 2 are supposed to enable a conceptual paradigm change. Each of two proposed building blocks is governed by the conservation law of total (mechanical and dynamic) energy. The mathematical framework for building block 1 is the Krein space. The mathematical framework for building block 2 is the Hilbert space.

The main conceptual change coming along with building block 1 is that the mechanical potential energy based Newton/Coulomb potential concept is replaced by a dynamic potential energy based indefinite norm based potential concept enabled by a system intrinsic potential energy operator.

The construction process for the building block 1 structure is a bottom-up approach from a priori (i.e. meta-physical) „ground state“ & „perfect plasma“ systems.

The proposed energetical dynamic and mechanical systems of building block 1 always deal with quanta types equipped with system intrinsic potential differences. Additionally, there are also potential differences between those systems. The intrinsic potential differences do have the role of a "cohesive Mie pressure". The explicate potential differences between the several quanta systems do have the role of "action causes". 

Building block 2 below provides an appropriate framework to solve the 3D-NSE problem in aligment with the intrinsic Neumann (pressure) problem, (BrK0) p. 26, (BrK11). It is also proposed to be applied to define an integrated mechnical & dynamic Landau equation model. Technically spoken, in case of the 3D-NSE problem this concept provides an dynamic fluid particle. In case of the Landau damping phenomenon it avoids the concepts of a Debye shield and it overcomes the current issue of two different incompatible mathematical Landau damping models (i.e., the Vlasov and the Landau equations) explaining the Landau damping phenomenon by two different underlying governing "forces", (BrK0) p. 63 ff.
 
The construction process for the building block 2 structure is a top-down approach from the (standard variational) domain of the kinetic Laplacian (potential) operator, which is the so-called H(1)-"energy" Hilbert space and a sub-Hilbert space of H(1/2). In this two-layer structure the corresponding (not self-adjoint) dynamic potential operator may be interpreted as a compact disturbance equipped with the domain H(1,ortho); this is the complementary sub-space of H(1) in the newly proposed dynamic H(1/2) Hilbert space.

The physical actions within building block 2 are governed by an implicate potential difference between the standard variational mechanical domain of the Laplacian (potential) operator and its complementary sub-space with respect to the overall H(1/2) Hilbert space norm.

2. Building block 1

   ... is an integrated hierarchical ordered Krein space based scheme of energetical quanta systems accompanied by

- a new dynamic energy type
- Krein space intrinsic concepts of potential and potential differences
- a Krein space intrinsic self-adjoint potential operator
- dynamic energy fields complementary to mechanical (kin./pot.) energy fields
- Krein-space based dynamic fields with related dynamic quanta systems
- 2-component (quanta pair) and 1-component (quanta) systems 
- a priori 2-component „ground state“ & „perfect plasma“ quanta pair systems
- a mechanical 2-component "perfect electromagnetic " quanta pair system
- quanta pair systems governed by the complex Lorentz group SU(2) x SU(2)
- 1-component quanta systems governed by the restricted Lorentz group SU(2)
- implicate intra-quanta system dynamics
- explicate inter-quanta systems dynamics
- dynamic energy generated by intra-quanta potential differences ("pressures")
- mechanical actions triggered by inter-quanta potential differences 
- a cohesive „Mie pressure“ of a generalized „ether physics“.

a. Underlying concepts

The quanta creation process from a priori 2-component „ground state“ & „perfect plasma“ quanta pairs is governed by a mathematical number theoretical "probability" principle. The underlying mathematical rule to build appropriate quanta numbers is based on the different "Schnirelmann densities" of odd (density = 1/2) and even (density = 0) integers. Colloquially spoken, the probability that electrinos or electrons may connect with positrinos or positrons is >0, while the probability of the reverse is 0. 

Quanta compositions from the baseline quanta pairs are acompanied by correspondingly recalculated quanta numbers. The dynamic quantum element types are characterized by quanta numbers < 1. The mechanical quantum element types are characterized by quanta numbers >1. The only quantum element with quanta number =1 is the neutron.

The equalization (entropy) processes of potential differences from mechanical energy systems back to purely "symmetric" dynamic energy systems become a model for related quanta annihilation (decay) processes.

Colloquially spoken, the 2-component systems provide a mathematical model of "the mysterious fabric of our reality", (GaG1), (UnA2). They may be interpreted in the sense of H. Weyl as the whole, which has to be presupposed in order to give meaning to the mechanical particulars, or as a kind of Higgs field.

The invariant quantities of the above three 2-component layers are governed by the two isomorphic normal subgroups of the group SO(4), which are directly related to the complex Lorentz group SU(2)xSU(2), (BrK0) p. 40.

The 1-component atomic dynamics system is governed by the restricted Lorentz group SU(2) (isomorphic to the unit quaternions) accompanied by the concept of a Maxwell-Mie-pressure enabling links to the SRT-Minkowski space and to "Einstein's lost key", (UnA1). In other words, the creation of the 1-component atomic system from the perfect 2-component electromagnetic system is accompanied by a symmetry breakdown from the complex Lorentz group (the main tool to prove the CPT theorem, StR)) down to the real restricted Lorentz group.

The 1-component atomic dynamics system is governed by the restricted Lorentz group SU(2) (isomorphic to the unit quaternions). In other words, the creation of the 1-component atomic system from the perfect 2-component electromagnetic system is accompanied by a symmetry breakdown from the complex Lorentz group (the main tool to prove the CPT theorem, StR)) down to the real restricted Lorentz group.

b. Opportunities

Note: The "perfect plasma" quanta pair system "explains" the decay of a neutron into an electron and a proton.

Note: The a priori 2-component dynamic „ground state“ & „perfect plasma“ quanta pair systems provide an appropriate modelling framework for the hypothesis of "vacuum flucuations" in line with the observed "Casimir effect" phenomenon.

Note: Plasma physics is about classical statistical fluid mechanics and classical fluid dynamics. The Landau damping phenomenon is a characteristic of collisionless plasma dynamics. It is a wave damping without energy dissipation by elementary particle collisions. Therefore, the a priori 2-component dynamic „perfect plasma“ quanta pair system provides an appropriate model enabling plasma dynamics without energy dissipation by elementary mechanical (!) particle collisions.

Note: The mechanical 2-component "perfect electromagnetic" quanta pair system provides an appropriate modelling framework for the CMBR phenomenon and Ehrenhaft's photophoresis phenomenon.

Note: The CMBR phenomenon and the physical mechanism of oceans generating microwave background, ((RoP2), can be interpreted as an (interaction) echo caused by system intrinsic potential differences of dynamic 2-component systems like the ground state, the perfect plasma, the perfect electromagnetic, and the hydroxyl-hydrogen systems.

Note: The mechanical electroton quanta system solves the "self-energy problem" of Dirac's electron system providing an explanation of the beta decay process.

Note: The atomic nucleus Dirac 2.0 systems are accompanied by three types of atomic nuclei, N(+), N(-), and N(0): 

"The Meissner effect shows that a bulk superconductor behaves as if the magnetic field inside the specimen vanishes. ....from Ohm’s law one may concluded that the flux through the metal cannot change on cooling through the transition. The Meissner effect suggests that perfect diamagnetism (external magnetic field and an induced intrinsic magnetic field) is an essential property of the superconducting state“, (KiC) pp. 262/263. 

What if, the three atomic nucleus types of the Dirac 2.0 systems enable a new concept to differentiate between superconductors (--> diamagnetism), insulators, and bulk conductors (--> external magnetic field and an induced intrinsic electric field; replacing the Maxwell displacement current) ?). This would provide as a new basis for a quantum theory of superconductivity replacing the BCS theory.

3. Building block 2

  ... is the energetical H(1/2) Hilbert space approximation system of the system scheme of building block 1 accompanied by dynamic fluid particles and

- a new "mass element" as distributional function of the H(-1/2) Hilbert space
- a well-posed NSE system aligned with Plemelj's enhanced Newton potential
- a properly defined Prandtl operator (incl. domain) for the Neumann problem
- a resolved d'Alembert "paradox", in fact the failure of the Euler equation (the    model of an ideal incompressible fluid) as a model for fluid-solid interaction
- a nonlinear dynamic potential operator interpreted as compact disturbance of     the mechanical Laplacian potential operator, (BrK0) p. 11
- mechanics / dynamics governed by Fourier waves / Calderon wavelets
- an alignment with the global nonlinear stability of the Minkowski space
- ...
- ... (BrK10).   

a. Underlying concepts 

Building block 2 provides a purely Hilbert space based modelling framework accompanied by the concept of a dynamic fluid particle. It may be interpreted as an approximation framework to the Krein space based quanta field scheme. Technically spoken, the H(1/2) Hilbert space is an extension of the standard variational mechanical energy Hilbert space H(1), which becomes a compactly embedded sub-Hilbert space of H(1/2) = H(1) x H(1,ortho). The complementary closed sub-space may be interpreted as dynamic fluid energy space.

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.

There is a mathematically correspondence between the norm (the invariant of a Hilbert space based energy system) of the proposed extended energy Hilbert space H(1/2) of building block 2 and the "wave energy" norm of the baseline energy Hilbert space of building block 1, (BrK0) p. 17.

Building block 2 provides an appropriate framework to solve the 3D-NSE problem in aligment with the intrinsic Neumann (pressure) problem, (BrK0) p. 26, (BrK11):

it turned out that the non-linear energy term of the 3D-NSE system is bounded with respect to the  H(1/2) energy norm as a simple consequence of the Sobolevskii inequality (BrK11), (GiY) lemma 3.2. At the same time, the dynamic fluid H(1/2) energy concept is in line with a well-defined Plemelj’s double layer potential function. The related Prandtl operator accompanied by a Hilbert scale domain H(r) (where  ½ smaller or equal than r smaller than 1) provides a unique solution of the underlying Neumann boundary value problem for the pressure p(x,t), (BrK7), (BrK11), (LiI) p. 95 ff., (PlJ).

The concept of a H(1/2) energy field based dynamic fluid element is accompanied by a well-posed exterior Neumann problem, a well defined related Prandtl operator, (BrK7), and Plemelj's double layer potential, where the mass density (du/ds)(s) of the single layer potential is replaced by the differential du(s), (PlJ).

The inner product of H(1/2) is isometric to an inner product in the form (Qx,Px), where Q resp. P denote Schrödinger‘s position & momentum operators. The combination with the Riesz transform operator provides the link to the considered operators in (BrK0), and is in line with the alternatively proposed Schrödinger operator in (BrK6) and related early thoughts on a new ground state energy model in (BrK8).

b. Opportunities

Note: The Prandtl operator enables a concept of a H(1/2) energy system intrinsic potential difference. It „solves“ the source of the d’Alembert “paradox”. This is about the failure of the Euler equation as a model for fluid-solid interaction, as in incompressible fluids there are no frictional forces. In other words, the Prandtl operator enables a dynamic fluid accompanied by frictional forces, which is applicable as mathematical tool for exterior space problems in physics.

Note: Building block 2 is also supposed to define an integrated mechanical & dynamic Landau equation model avoiding the concept of a Debye shield. An integrated mechanical & dynamic Landau system will overcome the current issue where two different Landau damping models (i.e., the Vlasov and the Landau equations) explain the Landau damping with two different underlying governing "forces", (BrK0) p. 63 ff.  

Note: The 1-component atomic dynamics system of building block 1 is governed by the restricted Lorentz group SU(2) (isomorphic to the unit quaternions). In building block 2 this concept may be adapted to Mie's original approach enabling links to the SRT-Minkowski space and to "Einstein's lost key", (UnA1).

Note: Exterior space problems in physics primarily revolve around the limitations and inconsistencies between general relativity and quantum mechanics, particularly when dealing with extreme conditions and the nature of space-time itself. Key areas include the nature of dark matter  and dark energy, the unification of gravity with other forces, and the behavior of space-time at the Big Bang and black holes. Additionally, the concept of quantum gravity  and the potential for emergent space-time are active areas of research, Wikipedia.

Note: There is no direct observations neither of the pressure nor of the density in the entire atmosphere of the sun, (UnA4) p. 59. … The energy production process of the sun is the fusion process from hydrogen to helium. The hydrogen molecules shows only two electrons, however there is an enormous bounding energy of the electron in a hydrogen molecule, (UnA4) p. 64. … Metals appear incompressible, but exposed to an exterior pressure they can reduce their volume. ... According to de Broglie's wave length formula for a electron such a volume reduction would lead to a reduction of the wave lengths of all electrons (i.e. basically the inverse of the mechanical momentum of those electrons) in the metal, (UnA4) p. 66. … There is an interesting phenomenon when water is subjected to massive pressure: water close by a nuclear explosion gets black and opaque, (UnA4) pp. 71/72.

Note: The Stefan-Boltzmann law states that "the total energy radiated per unit surface area of a black body (e.g., main sequence stars from the Herzsprung-Russell diagram) across all wavelengths per unit time is directly proportional to the fourth power of the black body's absolute temperature", Wikipedia.

Note: What if, 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"?

Note: "What if?" ... 

... the "perfect (electron-positron) plasma" and the "perfect (electroton-magneton) electromagnetics" quanta fields ...

(1) ... generate stars composed of mechanical 1-component atomic nucleii? ... Then, mechanical matter may be interpreted as condensed dynamic energy and the related process is governed by a symmetry breakdown from complex Lorentz group down to real restricted Lorentz group.

(2) ... accompanied by the concept of explicate and implicate potential differences governed by purely dynamic energy providing an appropriate modelling framework explains a liquid metallic hydrogen sun, (RoP), (UnA4)? In this case 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, and, as the pressure P becomes an explicate potential difference between two dynamic quanta systems and as the concept of kinetic energy is not required the volume V and the temperature T become obsolet, as well.

(3) ... are interpreted as as a kind of „darkness“ resp. „brightness" fields as a foundation for a theory of light? ... Then for instance, 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.

(4) ... 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"?

D. Dynamic action type related notes

1. Intra-action dynamics enabled by Krein space based quanta systems

2. Inter-action dynamics enabled by Krein space based quanta systems 

3. Inter-action dynamics within a Hilbert space based dynamic fluid system.  

1. Intra-action dynamics enabled by Krein space based quanta systems   

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.

2. Inter-action dynamics enabled by Krein space based quanta systems  

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.

3. Inter-action dynamics within a Hilbert space based 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