A. Einstein, "
(2) A non-harmonic Fourier series based circle method to solve additive number theory problemsThe specific common properties of the real negative zeros of the Digamma function and the imaginary part of the only complex valued zeros of a specific Kummer function allow the definition of corresponding weighted „retarding“ sequences fulfilling the Kadec condition. This enables the full power of non-harmonic Fourier series theory on the periodic L(2) Hilbert space with its relation to the Paley-Wiener space. In line with the proof of the RH those sequences allow a split of the Riemann density function J(x) into a sum of two number theoretical non-harmonic Fourier series, each of them governing one of two unit half-circles. Correspondingly, each pair of primes (p,q) of binary number theoretical problems can be governed by those two different number theoretical distribution functions. This overcomes current challenges caused by the dilution of the prime number sequence as x tends to infinite.
August 25, 2021
August 26, 2021 Further supporting data:
March 27, 2017
January 8, 2015
July 31, 2019
Dec 2, 2021
April 14, 2021
May 22, 2021
September 15, 2021 Supporting data:
Weyl H., "Space, Time, Matter", p. 206 ff.
The Emperor's New Mind“ one might say „look, the emperor is naked“.The common denominator with Heisenberg’s mathematical tool set for " a unified field theory of elementary particles", (HeW), is about a Hilbert space framework accompanied with an indefinite inner product resp. metric (norm), (HeW). The essential differentiators are1. there is only one fundamental (Hamiltonian based) conservation law accompanied with two underlying connected „symmetry“ groups, the two components of the complex Lorentz transform 2. the several possible invariants of other fundamental laws (resulting into corresponding observables, which hold unchanged over time during those processes, which are described by those laws) are modelled by an appropriately defined „self-adjoint“ operator, where the operator mapping describes the law, while the operator domain provides the required discrete and continuous spectra, where only discrete spectra become relevant for the (Lagrange formalism governed) physical world. From a physical perspective the proposed gravity and quantum field model is basically the variational representation of the Hamiltonian built from an enhanced Mie electrodynamics accompanied by the conception of an „electromagnetic pressure“. In this context we note the Novel Prize awards to W. Lamb & P. Kusch, (1955) for „ the discoveries concerning the fine structure of the hydrogen spectrum“ & „the precision determination of the magnetic moment of the electron'', i.e. there are „a so-called Lamb shift of the Schrödinger equation calculated energy levels“ and "a magnetic moment of an electron". The Hilbert space based model and point 1 overcomes the main difficulty of the GRT, which is basically caused by a missing truly geometric structure of the underling manifolds. Regarding the two connected group components of the complex Lorentz transform we note that in order to fulfill the required symmetry of the SRT the wave equation of a relativistic, force-free Dirac particle needs to be of order one with respect to the time and to the space variables . The corresponding Dirac matrix equations are determined by the „rest matrix“ R, the „velocity matrix“ V, the „spin matrix“ S, and the „pseudo-scalar matrix“ T, which links V=T*S. The matrices R and T, resp. the matrix S build two groups, where their related matrices are mutually interchangeable; on the other hand within each group they are anti-interchangeable, (MaW). The Hilbert space based model and point 2 overcomes the push back argument of Mie’s theory, which is about the selection of physical relevant solution (the physical world law) out of the infinite numbers of possible Mie solutions. In the context of a Hamiltonian formalism and the notion „spontaneous symmetry break down“ we recall from (BiJ) p. 48: „ When an exact symmetry of the laws governing a system is not manifest in the state of the system the symmetry is said to be spontaneously broken. Since the symmetry of the laws is not actually broken it would perhaps be better described as „hidden“, but the term „spontaneously broken symmetry“ has stuck.“ Devoted to hydrodynamics and turbulence R.Feynman observed, that „ we very possible already have the equation to a sufficient approximation of an equation for life, the equation of quantum mechanics, ... and ... we have the NSE for a detailed observation and the restruction of turbulent flow of an incompressible fluid“ (from this equation for life), (FrU) p. 1. Supporting data:
August 18, 2013
May 29, 2017
December 31, 2017
„ Plasma is that state of matter in which the atoms or molecules are found in an ionized state. The interactions of electrons and ions are determined by long-range electrical forces. The many forms of collective motion in a plasma are the result of coupling the charged-particle motion to the electromagnetic field. Therefore, the electromagnetic field which accompanies the particle motion is also a random nonreproducible quantity in a turbulent plasma. Measurements have shown that the fields excited in a plasma during the development of turbulence do in fact have a random nature.“, (TsV) p. 4." The turbulence of plasma differs from the hydrodynamic turbulence by the action of the magnetic field. A more relevant difference is due to the hydrodynamic interaction between the plasma particles, the interaction with the magnetic fields, and the interaction between the electromagnetic waves. ... All of them are the root cause of electromagnetic plasma turbulence. ... The case of interactions between quasi-stationary electromagnetic waves is called weak turbulence. ... The case of non-linear Landau damping (strong plasma turbulence) leads to the generation of virtual waves, which transfer their energy to the affected particles asymptotically with 1/t; the plasma is heated (turbulence heating) faster than this may happen by purely particles collisions", (CaF) p. 390 ff:.
- enables corresponding (weak variational) well-posed 3D non-linear, non-stationary Navier-Stokes equations (NSE) accompanied by a non-vanishing, bounded H(1/2)-energy norm non-linear term as a consequence of the lemma of P. E. Sobolevskii, (see Lemma 3.2 in (GiY), resp. the original proof of P. E. Sobolevskii, (SoP)); we note that the L(2)-based non-linear term (Bu,u) of the NSE vanishes, i.e. it provides no contribution to the energy ("stability") inequality - provides problem adequate Hilbert space norms for a mathematical proof of the non-linear Landau damping phenomenon. Mathematical speaking the non-linear Landau damping (the strong plasma turbulence case) is a specific behavior of linear waves in plasma governed by the non-linear term of the considered PDE system - provides a Hilbert space based variational plasma heating model governed by a mathematical Hamilton formalism enabling an approximating (statistical) physical Lagrange formalism governed by the Heisenberg uncertainty inequality, accompanied by approximation theory in Hilbert scales, and supported by related numerical approximation methods, (FEM, BEM) (ArF) pp. 639, 646: " In quantum field theory particle states correspond to „positive frequency“ solutions of the corresponding classical field theory, while anti-particle states correspond to „negative frequency“ solutions. Since PT turns positive frequency solutions into negative frequency solutions, PT in quantum field theory turns particles into anti-particles.Tentative conclusions: Whether a particle has positive or negative charge is determined by the temporal direction in which the four-momentum of particle points. … the CPT theorem should be called the PT-theorem. It holds for classical and quantum tensor fields theories, fails for classical spinor field theories, but it holds for quantum spinor fields. The fact that it holds for quantum field theories suggests that space-time has neither a temporal orientation nor a spatial handedness."In the context of the CPT symmetry and Lee-Yang’s law of parity conservation (Nobel prize 1957) we quote from (UnA2), " The dance of electrons and light":…. „ Long before the symmetry fashion took over, Richard Feynman became famous for his intriguing interpretation of the interactions of electrons, positrons, and light. The basic idea is fairly easy to grasp. Thanks to Heisenberg’s uncertainty principle, a traveling electron can borrow for a little time t an amount of energy E = h/t. Electrons may use this energy for juggling with photons. Like two people sitting on wheeled office chairs who are throwing heavy medicine balls to one another and rolling backward every time they pitch or catch the ball, two electrons that exchange photons knock each other back, too. Feynman managed to reformulate the laws of electrodynamics—two electrons feel a repulsive force—in these funny terms. The calculations based on this have led to predictions that have been precisely tested and are considered the best-measured results of all physics (The magnetic moment of an electron (its inherent magnetism) and the so-called Lamb shift in the spectral lines of a hydrogen atom). Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga were justifiably awarded the Nobel Prize for this in 1965. The big insight of the theory is that light and the most basic particles, electrons and positrons, show such a puzzling similarity. Yet nobody knows the reason for it."Regarding the approach to extend the concept of complex analyticity to quarternions by considering differential quotients we refer to (KrR). Gravity and the Mach hypothesisRegarding „gravity“ and the proposed quanta field model we note Mach’s knowledge, (UnA1) pp. 62,65, 66: „ the laws of dynamics could depend only on the motion of masses relatively to each other, ... andthe laws of nature are independent to accelerated motion. The Mach hypothesis (anticipating Einstein’s later comparison of inertial and gravitational mass known as the equivalence principle) is that distant celestial objects must be responsible for masses having gravitational properties. … The Mach principle has two different aspects. First, and qualitatively, just as the (Einstein) equivalence of principle, it says that inertia and gravitational mass are mystereriously connected. Secondly, Mach also claimed that inertia (i.e. the resistance to acceleration) must have its origin in the relative acceleration with respect to all other masses in the universe.This meant that the strength of gravity was also determined by every other celestial body – and suddenly we have a quantitiative statement“. Compactly embedded physical macro & micro "realities" into a mathematical "reality"The conceptual framework of the proposed quanta (dynamics) field model is about two compactly, densely embedded "physical worlds" into an overall "mathematical world"; whereby the two "physical worlds" describe the classical mechanics and the quantum mechanics "world". The two physical "worlds" are concerned with matter particle interactions. The corresponding mathematical models are classical resp. variational representation of 2nd order PDE, governed by the sum of a H(a)-coercive, linear operator with a=1/2 and a non-linear operator. The latter one can be split into a sum of a linearized, compact operator and a "remaining" operator, i.e. the two (physical "world") linear (observable) operators are governed by the Garding inequality. As "physical worlds" approximation solutions are at least elements of the compactly embedded Hilbert space H(1) the prerequisites of the Lax-Milgram theorem are fulfilled ensuring well defined classical or quantum mechanical "physical world" models, (AzA). From Cantor's cardinality perspective accompanied by Lebesgue's integral concept (which is the baseline tool for the probability & statistics theories) the "classical physical world" is a "zero-(sub-) set" of the "physical quanta mechanical world", which is a "zero-(sub-) set" of an overall "mathematical world". The domain inclusions of considered "observable" operators "promote" symmetric operators with a sub-space domain into self-adjoint operators with including domain.The mathematical model framework (compactly embedded Hilbert spaces H(2),H(1),H(1/2),L(2),H(-1/2)) enables - the following hierarchy of (variational) PDE models: the solution of a classical PDE of 2nd order is an H(2)-approximation of the underlying (kinematical) variational solution, which is an H(1)-approximation of the underlying (dynamical) variational H(1/2) solution - puts the spot on the philosophical world pictures of (NaT), (with the sub-title statement "...
From (HeM1) we
quote „Modern
physics is called mathematical because, in a remarkable way, it makes use of a
quite specific mathematics. But it can proceed mathematically in this way only
because, in a deeper sense, it is already itself mathematical.“
(73) „The rigor
of mathematical physical science is exactitude. Here all events, if they are to
enter at all into representation as events of nature, must be defined
beforehand as spatiotemporal magnitudes of motion. Such defining is
accomplished through measuring, with the help of number and calculation. But
mathematical research into nature is not exact because it calculates with
precision; rather it must calculate in this way because its adherence to its
object-sphere has the character of exactitude. The humanistic sciences, in
contrast, indeed all the sciences concerned with life, must necessarily be
inexact just in order to remain rigorous. A living thing can indeed also be
grasped as spatiotemporal magnitude of motion, but then it is no longer
apprehended as living. The inexactitude of the historical humanistic sciences
is not a deficiency, but is only the fulfillment of a demand essential to this
type of research. It is true, also, that the projecting and securing of the
object-sphere of the historical sciences is not only of another kind, but is
much more difficult of execution than is the achieving of rigor in the exact
sciences.“
June 15, 2021
July 31, 2019
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Streater R. F., Wightman A. S., " PCT, spin & statistics, and all that", p. 9 ff.
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