(1) The Planck action constant

The Planck action constant is independent from any weak or strong gravitation field. It therefore somehow mirrors the fundamental difference of physical macro and micro world, (DeH).

(2) The conservation of energy

The conservation principles of energy, linear momentum, angular momentum, and electric charge are amoung the most fundamental principles of physics. … The notion „conservation“ as in „conservation of energy“ is not the same as „invariant“. They are related, …, but they are not synonymous. The momentum or energy of a system of particles may be conserved but not necessarily invariant, (NeD) pp. 1, 4. A second fundamental principle in physics is the least action principle; here physical action requires a potential difference or a pressure. There is no action just due to the presence of energy or a potential.

(3) Invariant quantities   

The invariant quantities in energy conservation laws are governed by functionals. The simplest model of functionals is a constant. In a Hilbert space framework such a functional is provided by its norm.

(4) The Noether theorem   

The probably most fundamental mathematical theorem in physics is E. Noether‘s theorem. It effects a huge class of conservation laws governing symmetries of space, time, and „internal“ variables. Noether’s theorem relates conservation to invariance, and thus to symmetry. This theorem provides the mathematical foundation of the whole quantum mechanics. However, the conservation of electric charge emerges from a more abstract symmetry called „gauge invariance“.

(5) An indefinite metric in a Hilbert space   

An indefinite metric in a Hilbert space is one of the unconventional features of Heisenberg's "Introduction to the Unified Field Theory of Elementary Particles", (HeW).

(6) Manifolds   

The terminology of "multiple extended quantities" was introduced by B. Riemann, synonymly to a "continuous manifold". The history of manifolds is the attempt to build a mathematical structure to model the whole (the continuum) and the particular (the part) to put its combination  into relationship to describe motion, action etc. It is conceptually based on two essential attributes: "continuity" and "multiple extension". Based on the concept of a manifold developing a mathematical framework which symbolically explores the "relationship between the part and the whole" for the case of the continuum lead to the concepts of of co-variant derivatives, affine connexion and Lie algebra.

(7) First appearance of manifolds in mathematical physics

Of course there are several semantical links of the manifold concept to physics, which could be pursued even in the 19th century. Riemann had already started to discuss such links on at least two levels. The final part and culmination of his Habilitations talk gave a sketch how in a subtle interplay between mathematical arguments and the evaluation of physical/empirical insights he proposed to come to a refined understanding of physical space. The essential bridge was an improved understanding of the microstructure of matter and its binding forces that should be, according to Riemann, as directly translated into differential geometric structures on manifolds as possible. But he also left the possibility open for further consideration that perhaps some time even a discrete structure of matter has to be taken into account, as it might very well be that the concepts of rigid body and light ray lose their meaning in the small. Still, so Riemann argued by reference to astronomical measurements, the acceptance of a Euclidean space structure was well adapted to the physical knowledge of the time, (ScE6).

(8) Hermann Weyl’s „Purely Infinitesimal Geometry“

"A general concept, the whole (here the continuum), has to be presupposed in order to give meaning to an individual determination, the particular, or the part (here the point). On the other hand, the whole, the general concept (the continuum) is constituted in a process of common generation by particulars (the parts).   

This conceptual figure came close to those procedures that had been called „impredicative“ by Russell and Poincaré and that had been blamed as being responsible for contradictions of the type of Russell’s antinomy“, (ScE6).

(9) (Plemelj’s) double layer potential

J. Plemelj proposed an alternative definition of a normal derivative, based on Stieltjes integral. It requires less regularity assumptions than standard definition; the "achieved" "regularity reduction"is in the same size as a reduction from a C(1) to a C(0) regularity. Plemelj's concept is proposed to be applied to ensure physical model requirements modeled by normal derivatives within a distributional Hilbert space framework, (PlJ).

(10) The problem of an ill-posed problemNSE system

"In the NSE with the unknowns velocity v(x,t) and the pressure p(x,t) fields the field p(x,t) can be formally obtained - by operating with "div" on both sides of the NSE - as a solution of a Neumann problem. From this it is clear that to describe the values of the pressure at the bounding walls or at the initial time independently of v, could incompatible with the NSE and, therefore, could render the problem ill-posed", (GaG).

(11) Exterior space problems in physics

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

(12) The Neumann PDE system: an exterior space problem

"In the Navier-Stokes Equation (NSE) system the field p(x,t) can be formally obtained - by operating with "div" on both sides of the NSE equation - as the solution of an exterior space Neumann problem. From this it is clear that to describe the values of the pressure at the bounding walls or at the initial time independently of v(x,t), could incompatible with the NSE system and, therefore, could render the problem ill-posed", (GaG).

(13) The d’Alembert “paradox”

The d’Alembert “paradox” is about the failure of the Euler equation as a model for fluid-solid interaction as in incompressible fluids there are no frictional forces.    

(14) Current matter and force particles of the universe 

„All known particles in the universe can be divided into two groups: particles of spin ½, which make up the matter in the universe, and particles of spin 0, 1, and 2, which give rise to forces between matter particles. The matter particles obey what is called Pauli’s exclusion principle“, (HaS1) pp. 69/70. 
„The neutron is observed to change spontaneously into a proton, an electron, and a neutrino. This process is so rare that we can often ignore it. Neutron decay takes on the average some thousand seconds for a free neutron, whereas within a nucleus the characteristic time between nucleon-nucleon collisions is only 10 exp(-21) second“, (BeH) p. 29.       

„The word „neutrino“ has been used to represent any assumed product of decay which has half-integer spin, no charge, and neglegible mass. Whether these particles are all identical with neutrinos of  beta-decay is so far conjecture“, (BeH) p. 32.

„Charge neutrality is one of the fundamental property of plasma, the fourth state of matter: it is about the shielding of the electric (Coulomb) potential applied to the plasma. When a probe is inserted into a plasma and positive (negative) potential is applied, the probe attracts (repulses) electrons and the plasma tends to shield the electric disturbance“, (MiK).

The conceptual element in superconducting metal models is an electron-phonon interaction governed by strongly electrostatic Coulomb repulsions accompanied by an electron-phonon coupling parameter. The related Bardeen-Cooper-Schrieffer (BCS) theory is governed by a Coulomb pseudopotential, (AnJ).

(15) Dirac‘s single atom-radiation system     

Dirac’s single atom-radiation system deals with energy, which 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“, (FeE).          

(16) Ether, perfect plasma and perfect electrodynamic fields

Einstein considered Maxwell’s equations in combination with the requirement that light moves with a certain velocity as laws of physics. 

Lorentz envisioned „various kinds of ether pressures, objects would be squeezed and therefore shortened", (SuL) pp. 60-62. 

Ehrenhaft interpreted his discovery called photophoresis, as „light induces electric and magnetic charges (poles) upon the particles if they are illuminated by concentrated light preponderantly shorter wave lengths“, (EhF) p. 242. 

The existence of quantum fluctuations dynamics in a „world“ without a time arrow and without entropy has been verified by the Casimir and the Lamb shift effects.  
        
(17) SMEP and gauge invariance

In SMEP (Standard Model of Elementary Particles) symmetry plays a key role. Conceptually, the SMEP starts with a set of fermions (e.g. the electron in quantum electrodynamics). If a theory is invariant under transformations by a symmetry group one obtains a conservation law and quantum numbers. Gauge symmetries are local symmetries that act differently at each space-time point. They automatically determine the interaction between particles by introducing bosons that mediate the interaction. The U(1) symmetry, i.e. the complex unit circle numbers, (where probability of the wave function is conserved) describes the electromagnetic interaction with one boson (photon) and one quantum number (charge Q). The group SU(2) of complex, unitary (2x2) matrices with determinant 1 describes the weak force interaction with 3 bosons W(+), W(-), Z, while the group SU(3) of complex, unitary  matrices describes the strong force interaction with 8 gluon bosons. This means that each of the observed Nature „force“ phenomena are related to a specific gauge group. The SMEP does not provide any explanation where the related elementary „particles“ were built during the inflation phase of the current big bang story and why their mass have their specific values.

For gauge symmetries the fundamental equations are symmetric, but e.g., the ground state wave function breaks the symmetry. When a gauge symmetry is broken the gauge bosons are able to acquire an effective mass, even though gauge symmetry does not allow a boson mass in the fundamental equations. 

(18) The Higgs effect (or mechanism)

The Higgs effect (or mechanism) builds on an extended from global to local U(1)  transformations symmetry group of the underlying Lagrangian. It explains the mass of the gauge W- and Z- (weak interaction) bosons of the weak “nuclear-force”. The Higgs boson is supposed to be a heavy elementary particle (with non-zero rest mass of about 125 GeV with spin 0). The Higgs field is supposed to fill the whole universe interacting with each particle, which “moves” through it by a kind of frictional resistance, i.e. which has kinetic energy. Therefore, the Higgs effect requires a Higgs field with not vanishing amplitudes in the ground state.

„The Higgs mechanism helps in two ways. First, gauge fields can acquire mass by the symmetry breaking. Second, the undesirable Goldstone bosons (which arise in the symmetry-breaking process) can be usually gauged away”, (BlD) 10.3. This degeneracy of the physical vacuum (as a result of a “spontaneous” breakdown of symmetry) permits … nonzero vacuum expectation values (or “vacuons”), (HiP).

(19) The standard model is not the final theory

„Theoretical physicists are convinced that the standard model is not the final theory. There are a number of phenomena which find no explanation in the context of the standard model and must be added in an ad hoc manner. For example, the Higgs mechanism, th mysterious field which gives mass to all other particles, does not follow in any sense from the standard model. The apparent asymmetry between matter and anti-matter is not explained by the standard model. Neutrino masses do not naturally arise in the context of the standard model. There is clearly physics, a deeper theory, beyond the standard model“, (SaR) p. 82.

(20) The London dispersion forces

(Wikipedia): The term "van der Waals force" is used to describe any dipole-dipole interactions in atom/molecules. Since hydrogen bonds involve interactions between permanent dipoles, they can be considered as a type of van der Waals force (and would fall under the category of Keesom interactions). The strength of van der Waals forces varies, with dipole-dipole interactions typically being stronger than covalent or ionic bonds. "London dispersion forces" are a specific type of intermolecular force present in nonpolar molecules. They are the weakest of all molecular forces. Molecules like noble gases (e.g., He, Ar), diatomic molecules (e.g., H2), and nonpolar organic molecules experience these forces.

(21) A sun basically consisting of liquid metallic hydrogen

A gaseous structure of the sun do not allow "white light" blackbody rays passing from its interior through the observed surface of the sun to Earth. At the same time there is no direct observation of the presence of classical concepts pressure & density in the atmosphere of the sun, which may explain a phase change from gaseous hydrogen to liquid metallic hydrogen governed by London dispersion forces ((Una4) p. 73. 

The fusion process from hydrogen to helium is the energy production process of the sun. The hydrogen molecules shows only two electrons, however there is an enormous bounding energy of the electron in a hydrogen molecule, (RoP), (UnA4) p. 64.

There is no direct observations neither of the pressure nor of the density in the entire atmosphere of the sun, (UnA4) p. 59.

Ironically, astronomers admit that liquid metallic hydrogen occurs in planets like Jupiter or Saturn, (UnA4) p. 61, Wikipedia

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. ... 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) pp. 63/66.

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.

(22) Oceans generating microwave background

"Though liquid water has a fleeting structure, it displays an astonishingly stable network of hydrogen bonds. Thus, even as a liquid, water possesses a local lattice with short range order. The presence of hydroxyl and hydrogen bonds within water, indicate that it can simultaneously maintain two separate energy systems. These can be viewed as two very different temperatures. The analysis presented uses results from vibrational spectroscopy, extracting the force constant for the hydrogen bonded dimer. By idealizing this species as a simple diatomic structure, 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", (RoP2)

(23) The extraordinarily special Big Bang story      

„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 – about 10exp(10)exp(123) – of the entire volume, for the situation under consideration“, (PeR) p. 444.    

 „The chaotic inflation state of the early universe does not match to the second law of thermodynamics as this law requires a permanent increase of the entropy of the universe over time, while the cosmos started with an incredible low probability, but also with an incredible high ordered state“, (PeR1) pp. 122 ff.    

 „It (the expanding universe) starts with high-entropy singularity which, it seems, could have been an initial state for our actual universe, and, indeed, would be a far more probable initial state (i.e. of much larger entropy) than the Big Bang that actually occurred. The black holes that congeal together in the final stage of our envisaged collapse would, when time-reversed to an expanding universe, provide us with the image of an initial singularity consisting of multiply bifurcating white holes! A white hole is the time-reverse of a black hole, and I have indicated the sort of situation that this provides us with in Fig. 2,45. It is the total absence of such white-hole singularities that singles out our Big Bang as being so extraordinarily special“, (PeR1) p. 125.

(24) Photophoresis: light beams causing particle motions     

„Particles illuminated by a beam of light or infrared radiation, having sufficient flux density, can move in various directions. This phenomenon has been mentioned by Thoré (1877), first described and later intensively studied by Ehrenhaft (1918) and co-workers/students. Since light is the cause for motion, the expression „photophoresis“ has become common. Preining (1966) gives a concise description of the important factors influencing photophoretic motion: „This motion can depend on the illumination, color, structure and shape of the light beam, on pressure and composition of the gas, on the particle’s size, shape and material, and on additional fields such as electric, magnetic, and so on“, (HoH).   

(25) Wavelets: a mathematical microscope   Wavelet analysis can be used as a mathematical microscope, looking at the details that are added if one goes from a scale "a" to a scale "a+da", where " da " is infinitesimally small. The mathematical microscope wavelet tool 'unfolds' a function over the one-dimensional space R into a function over the two-dimensional half-plane of "positions" and "details". This two-dimensional parameter space may also be called the position-scale half-plane.   The wavelet theory is established in the Fourier Hilbert space framework. In order to apply the Calderón inverse formula in a Hilbert scale framework it requires the so-called admissibility condition defining a wavelet. "A wavelet synthesis can be performed locally as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions. …. The wavelet transform unfolds any signal (e.g. in time) or any field (e.g., in three-dimensional space) into both space (or time) and scale (or time scale), and possible directions (for dimensions higher than one). … „Decomposing a vector field into orthogonal wavelets, scale-dependent distributions can be quantified at different length scales and in different directions and hence longitudinal or transverse contributions can be determined. In the case of an imposed magnetic field the constructions in the directions perpendicular or parallel to it can be dinstinguished“, (FaM1). „In practice the Fourier transform may be thought of as embedded into the wavelet transform, because it is, to first approximation, possible to compute the Fourier spectrum of a signal by summing its wavelet coefficients over all positions scale by scale. … In practice the wavelet should also be well-localized in both physical and Fourier spaces“, (FaM).