                     (1) A Digamma/Kummer functions based proof of the Riemann Hypothesis

Based on the negative real zeros of the Digamma function an alternative representation of the Riemann density function J(x) is provided where its critical (oscillating) sum is replaced by two non-oscillating sums, both enjoying the required asymptotics O(root of x) which proves the Riemann Hypothesis.

(2) A non-harmonic Fourier series based circle method to solve additive number theory problems

The 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.

(3) A Bessel function based proof that the Euler-Mascheroni constant is irrational Braun K., A Bessel function based proof that the Euler constant is irrational

June 26, 2021

(4) A Hilbert scale based integrated gravity & quantum field model

The gravity field theory and the quantum field theory are inconsistent from a physical and from a mathematical perspective.

Handicap 1: Lacking a common mathematical framework there is a large zoo of elementary particles. The root of the evil is already in place in Maxwell’s phenomenogical theory of electricity, as the theory cannot hold for the interior of the electron. From the point of view of ordinary theory of electrons one must treat the electron as something given a priori, as a foreign body in the field.

Handicap 2: The concealed motions of the electrons are not taken into account as motions of matter, consequently electricity is not supposed attached to matter in the Maxwell theory. The only way to explain how it is that a piece of matter carries a certain charge is to say this charge is that which simultaneously in the portion of space that is occupied by the matter at the moment under consideration. From this it comes that the charge is not, as in the theory of electrons, an invariant determined by the portion of matter, but is dependent on the way the world has been split up into space and time.

The Mie theory

A more general theory of electrodynamics has been proposed by Mie, by which it seems possible to derive the matter from the field. Mie’s theory resolves the problem of matter into a determination of the expression of the Hamiltonian function in terms of four quantities and the laws for the field may be summarised in a Hamilton’s principle.

In mechanics, a definite function of action corresponds to every given mechanical system and has to be deduced from the constitution of the system. Mie’s theory is only concerned with a single system, the world. This is were the real problem of matter takes its beginning: to determine the Mie „world-function of action“, belonging to the physical world.

The proposed gravity and quantum field model is basically an enhanced Mie electrodynamic overcoming the above difficulty which is basically caused by a missing truly geometric structure of the underling manifolds w/o any conceptual relationship to all possible mathematical solution of the Mie equations. Therefore, the enhancement is concerned with a replacement of the manifold framwork by a Hilbert space, where its inner product induces a corresponding norm and where an existing hermitian operator induces a corresponding energy norm, governing e.g. least action or energy minimization formalisms.

The proposed gravity and quantum field model

From a mathematical perspective the proposed gravity and quantum field model is about a variational representation of a Hamiltonian operator with defined domain in an appropriate Hilbert scale framework. The Heisenberg (matter particle) matrices mechanics and the Schrödinger (matter wave) PDE mechanics are equivalent with respect to their common related mapping descriptions of the corresponding Hilbert space based linear operators. However, a linear operator is only well defined in combination with an appropriately defined domain, which differs in case of the Lagrange and the Hamilton formalisms. Paraphrasing Roger Penrose‘s „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 are

1. 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 „pseudoscalar 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.

An accepted purely quanta field theory

- is based on extended Maxwell-Mie equations, where the (positively charged) proton and the (negatively charged) electron masses are energetically „balanced/generated“ by Mie‘s electromagnetic pressure concept, alternatively to the SMEP concept of "strong elementary particles interaction". The „beta decay“ process (also called „weak elementary particles interaction“) is when a single neutron decays into a proton, an electron, and an anti-neutrino. The proposed underlying Hilbert space decomposition H(+)+H(-)+H(~) provides a suitable framework for an integrated model of electromagnetic and "weak elementary particles interactions". In other words, the Maxwell-Mie equations make the Yang-Mills equations obsolete and the related Millennium problem (the YME mass gap problem) is solved

- enables corresponding (weak variational) well-posed 3D non-linear, non-stationary Navier-Stokes equations (NSE) with convergent energy norm estimates including a not vanishing non-linear term, which solves the related Millennium problem

- provides a Hilbert space based variational plasma heating model governed by a mathematical Hamilton formalism enabling approximating physical Lagrange formalisms governed by the Heisenberg uncertainty inequality, accompanied by approximation theory in Hilbert scales, and supported by related numerical approximation methods, (FEM, BEM)

- enables an (enhanced Mie equation based) enhanced SRT (replacing the GRT) where the Maxwell-Lorentz group with its underlying four disconnected components is replaced by the complex Lorentz group with its underlying two connected components.

The complex Lorentz group provides the central tool in the proof of the CPT theorem. It says that any Lorentz invariant quantum field theory must also be invariant under the combined operation of charge conjugation C, parity P, and time reversal T, even though none of those individual invariances need hold.

Physically speaking the CPT theorem says, that in quantum field theory there is a mathematically proven symmetry of the combined three physical (measurable) attributes of a quantum artefact (i.e. temporal orientation, spatial handedness, and matter–anti-matter transformation). In other words, the CPT theorem provides a mathematically proven physical law.

(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 „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, ... and

the 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“.

Further supporting data are provided in

z-lib.org

The mathematical reality. Why Space and Time are an Illusion by Alexander Unzicker

Einstein's Lost Key - How We Overlooked the Best Idea of the 20th Century by Alexander Unzicker

Bankrupting Physics How Today's Top Scientists are Gambling Away Their Credibility by Alexander Unzicker

The Higgs Fake. How Particle Physics Fooled the Nobel Committee by Alexander Unzicker

References

(ArF) Arntzenius F., The CPT Theorem, in Callender C., The Oxford Handbook of Philosophy of Time, Oxford University Press, Oxford, 2013

(BiJ) Binney J. J., Dowrick N. J., Fisher A. J., Newman M. E. J., The Theory of Critical Phenomena, Oxford Science Publications, Clarendon Press, Oxford, 1992

(BoD) Bohm D., Wholeness and the Implicate Order, Routledge & Kegan Paul, London, New York, 2005

(FrU) Frisch U., Turbulence, Cambridge University Press, Cambridge, 1995

(FoC) Foias C., Manley O., Rosa R., Teman R., Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001

(HeW) Heisenberg W., Introduction to the Unified Field Theory of Elementary Particles, Interscience, London, 1966

(HoE) Hopf E., Ergodentheorie, Springer-Verlag, Berlin, Heidelberg, New York, 1070

(MaW) Macke W., Quanten und Relativität, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1965

(SeE) Seneta E., Regularly Varying Functions, Springer-Verlag, Berlin, Heidelberg, New York, 1976

(UnA) Unzicker A., The mathematical reality, Why Space and Time are an Illusion

(UnA1) Unzicker A., Einstein's Lost Key - How We Overlooked the Best Idea of the 20th Century

(UnA2) Unzicker A., Bankrupting Physics How Today's Top Scientists are Gambling Away Their Credibility.   