(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.
August 25, 2021
August 26, 2021
June 26, 2021
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 „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.
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, ... 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“.
September 15, 2021
May 22, 2021
June 15, 2021
July 31, 2019
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