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This homepage is dedicated to my mom, who died at
April 9, 2020 in the age of 93 years. In retrospect, the proposed
solution concepts of different problem areas (the Riemann Hypothesis
& the inconsistent quantum theory with Einstein's gravitation
theory) originate in some few simple common ideas / basic conceptual
changes to current insufficient "solution attemps".
This page is structured into
Part A: A Kummer function based Zeta function theory
Part B: A Hilbert space based quantum gravity model
Part C: Linkages between the quantum gravity model and philosophy.
(A) A Kummer function based Zeta function
theory
An alternative
Kummer function based Zeta function theory is proposed to enable
(a) the verification of
several Riemann Hypothesis (RH) criteria
(b) a truly circle method for the
analysis of binary number theory problems.
The Kummer function based Zeta function theory is basically about a
replacement of the integral exponential function Ei(x) by a corresponding
integral Kummer function. It enables the validation of several RH criteria,
especially the "Hilbert-Polya conjecture", the "Riemann error
function asymptotics" criterion and the „Beurling“ RH criterion. The
latter one provides the link to the fractional function and its related periodical
L(2) Hilbert space framework, (TiE).
Regarding
the tertiary Goldbach problem Vinogradov applied the Hardy-Littlewood
circle method (with its underlying domain "open unit disk") to derive
his famous (currently best known, but not sufficient) estimate. It is derived
from two estimate components based on a decomposition of the (Hardy-Littlewood) "nearly"-circle
into two parts, the „major arcs“ (also called „basic intervals“) and the „minor
arcs“ (also called „supplementary intervals“). The „major arcs“ estimates are sufficient to prove the Goldbach conjecture, unfortunately the „minor arc“ estimate is insufficient
to prove the Goldbach conjecture. The latter one is purely based on "Weyl sums"
estimates taking not any problem relevant information into account. However,
this estimate is optimal in the context of the Weyl sums theory. In other words,
the major/minor arcs decomposition is inappropriate to solve the tertiary and
the binary Goldbach conjecture.
The primary technical challenge regarding number theoretical problems is
the fact that only the set of odd integers has Snirelman density ½, while the
set of even integers has only Snirelman density zero (because the integer 1 is
not part of this set).
The additional challenge regarding binary number theoretical problems is
the fact that the problem connects two sets of prime numbers occuring with
different density (probability) during the counting process; regarding the
Goldbach conjecture this concerns the fact, that the number of primes in the
interval (2n-p) is less than the number of primes in the interval (1,p).
Therefore, two different „counting methods“ are required to count the numbers
of primes in the intervals (1,p) and (p,2n-p).
In order to overcome both technical challenges above a truly circle
method in a Hilbert space framework with underlying domain „boundary of the
unit circle“ is proposed. The nonharmonic Fourier series theory in a
distributional periodic Hilbert scale framework replaces the power series theory
with its underlying domain, the "open unit disk".
The proposed nonharmonic
Fourier series are built on the (non-integer) zeros of the considered Kummer function, (which
are only imaginary whereby for their real parts it holds >1/2) replacing the
role of the integers of exp(inx) for harmonic Fourier series. They are accompanied by the zeros of the digamma function (the Gaussian psi function). The set of both sequences are supposed to enable appropriate non-Z based lattices of functions with domain "negative real line & "positive" critical line". This domain is supposed to replace the full critical line in the context of the analysis of the Zeta function, in order to anticipate the full information of the set of zeros of the Zeta function (including the so-called trivial zeros), while omitting the redundant information provided by the critical zeros from the zeros from the "negative" part of the critical line.
With respect to
the analysis of the Goldbach conjecture it is about a replacement of the
concepts of trigonometric (Weyl) sums in a power series framework by Riesz
bases, which are "close" (in a certain sense) to the trigonometric
system exp(inx).
The nonharmonic
Fourier series concept of almost periodic functions is basically about the
change from integers n to appropriate sequence a(n). Such a change also makes
the difference between the Weyl method and the van der Corput method regarding
exponential sums with domains (n,n+N), (GrS), (MoH). Selberg‘s proof of the
large sieve inequality is based on the fact, that the characteristic functions
of an interval (n,n+N) can be estimated by the Beurling entire function of
exponential type 2*pi, applying its remarkable extremal property with respect
to the sgn(x) function, (GrS).
The Riesz based nonharmonic Fourier theory enables the split of number
theoretical functions into a sum of two functions dealing with odd and even
integers separately, while both domains do have Snirelman density ½. In case of an
analysis of the Goldbach conjecture it also enables the definition of two different
density functions, „counting“ the numbers of primes in the intervals (1,p)
resp. (p,2n-p).
The trigometric system exp(inx) is stable under sufficienctly small
perturbance, which leads to the Paley-Wiener criterion. Kadec's 1/4-theorem
provides the "small perturbance" criterion, which is fulfilled for
both sets of zeros, the considered Kummer function and the digamma function. A striking generalization of
"Kadec's 1/4-theorem", (YoR) p. 36, with respect to the below is
Avdonin's "Theorem "1/4 in the mean", (YoR) p. 178.
The Fourier transformed system of the trigonometric system forms an orthogonal
basis for the Paley-Wiener Hilbert space (PW space), providing an unique
expansion of every function in the PW-space with respect to the system of
sinc(z-n)-functions. Therefore, every PW-function f can be recaptured from its
values at the integers, which is achieved by the cardinal series representation
of that function f (YoR) p. 90.
When the integers n are replaced by a sequence a(n)) the correspondingly
transformed exponential system builds a related Riesz basis of the PW-space
with the reproducing sinc(z-a(n))-kernel functions system.
For the link of the nonharmonic Fourier series theory with its underlying
concepts of frames and Riesz bases to the wavelet theory and sampling theorems,
which is part of the solution concept of part B, we refer to (ChO), (HoM),
(ReH).
 Braun K., Looking back, part A, (A1)-(A3) |
April 10, 2021, change to previous version, p. 18
(B)
A single truly geometric gravity and quantum field theory
After having passed the two milestones, Einstein’s Special
Relativity Theory, (BoD1), and Pauli’s spin concept (accompanied by the spin
statistics CPT theorem, (StR)), the General Relativity Theory (GRT) and the
quantum theory of fields became two „dead end road“ theories towards a
common gravity and quantum field theory. The physical waymarking labels directing into
those dead end roads may be
read as
dead end road label (1):
"towards space-time regions with not constant
gravitational potentials governed by a globally constant speed of light",
(UnA)
dead end road label (2):
"towards Yang-Mills mass gap".
The waymarker labels of the royal road
towards a geometric gravity and quantum field theory may be
royal road label 1:
towards mathematical concepts of „potential“,
„potential operator“, and „potential barrior“ as intrinsic elements of a geometric
mathematical model beyond a metric space (*)
royal road label 2:
towards a Hilbert space based hyperboloid
manifold with hyperbolic and conical regions governed by a „half-odd-integer“
& „half-even integer“ spin concept
royal road label 3:
towards the Lorentz-invariant, CPT theorem supporting
weak Maxwell equations model of „proton potentials“ and „electron potentials“ as
intrinsic elements of a geometric mathematical model beyond a metric space
royal road label 4:
towards „the understanding of physical
units“, (UnA) p. 78, modelled as „potential barrior" constants, (*),(**), (***), (****), (*****)
(*) Einstein quote, (UnA) p. 78: „The principle of the
constancy of the speed of light only can be maintained by restricting to space-time
regions with a constant gravitational potential.“
(**) The Planck action constant may mark the "potential barrior" between the
measurarable action of an electron and the action of a proton, which "is acting"
beyond the Planck action constant barrior.
(***) The „potential barrior“ for the validity of the Mach
principle determines the fine structure constant and the mass ratio constant of
a proton and an electron: Dirac’s
large number hypothesis is about the fact that for a hydrogen atom with two masses, a proton
and an electron mass, the ratio of corresponding electric and gravitational
force, orbiting one another, coincides to the ratio of the size of a proton and the size of the
universe (as measured by Hubble), (UnA) p. 150. In the proposed geometric model
the hydrogen atom mass is governed by the Mach principle, while the Mach
principle is no longer valid for the electron mass, governed by the CPT spin statistics.
(****) The norm quadrat representation of the proposed "potential" definition indicates a representation of the fine structure constant in the form 256/137 ~ (pi*pi) - 8. In (GaB) there is an interesting approach (key words: "Margolus-Levitin theorem", "optimal packaged information in micro quantum world and macro universe") to „decrypt“ the fine
structure constant as the borderline multiplication factor between the range of
the total information volume size (calculated from the quantum energy densities)
of all quantum-electromagnetic effects in the universe (including those in the
absense of real electrodynamic fields in a vacuum; Lamb shift) and the range of the total information volume size of all matter in the four dimensional universe (calculated from the matter density of the universe).
(*****) The vacuum is a homogeneous,
dielectric medium, where no charge distributions and no external currents
exist. It is governed by the dielectric
and the permeability constants, which together build the speed of light; the fine structure constant can be interpreted as the ratio of the circulation speed of the electron of a hydrogen atom in its ground state and the speed of light. This puts the spot on the Maxwell equations and the "still missing underlying laws governing the "currents" and "charges" of electromagnetic particles. ...The energetical factors are unknown, which determine the arrangement of electricity in bodies of a given size and charge", (EiA), p. 52:
The Yang-Mills-, Maxwell- and
Helmholtz-Equations, the PCT theorem, and simultaneously measurable observables
in a Hilbert space framework
The
Yang-Mills theory is the generalization of the Maxwell theory of
electromagnetism, where the chromo-electromagnetic field itself carries
charges.
As a classical field theory it has solutions which travel at the speed
of light
so that its quantum version should describe massless particles (gluons).
However, the postulated phenomenon of color confinement permits only
bound
states of gluons, forming massive particles. This is the mass gap. The
physical ("color") confinement challenge is that the
phenomenon that "color-charged" particles (such as quarks and gluons)
have not been isolated until today. Another challenge of „confinement“
is
asymptotic
freedom which makes it conceivable that quantum Yang-Mills theory exists
without restriction to low energy scales. Therefore, before going for
hunting massless bosons, we should either intensify to capture quarks,
or revisit the triple, "electron", "neutron", "proton", becoming the
only physically observable fermions in a SRT & CPT aligned quantum
theory of fields, based on the theory of Hilbert spaces with indefinite
inner product, ((DrM), (AzT), (DrM), (VaM)) accompanied with the
calculus of variations in a distributional Hilbert scale framework.
The integral form of the Maxwell equations
is more closely connected to the physical situation, which is basically the
combined Gauss‘ Electric Law and Ampere’s Law, where electromagnetic waves
propagate in the interior of a region in R(3) enclosed by a related region boundary.
The corresponding two variational representations of the standard „electric“ and
„magnetic“ boundary problem is enabled by the Green formulae accompanied with
corresponding „natural“ boundary conditions. When applying variational methods,
the natural bilinear form of time-harmonic Maxwell equations is not coercive on
the whole Sobolev space H(1). One can, however, make it coercive by adding a
certain bilinear form on the boundary of the domain (CoM). The extended „electric“
and „magnetic“ (well-posed) boundary value problem is accompanied by the extended Green formulae, replacing
the standard normal derivative concept by a „mass element“ and „flux“ concept,
which is purely defined on the surface only, omitting the unit vector tangent concept,
(PlJ). The proposed physical governing concept is the least action principle, (KnA).
We note that Maxwell equations hold only in
regions with smooth parameter functions, modelling conductivity, dielectricity
and permeability. Vacuum (i.e. the conductivity is zero) is a homogeneous,
dielectric medium, where no charge distributions and no external currents
exist. Therefore, the dielectric and the permeability tensors reduce to the dielectric
and the permeability constants, which together build the speed of light. In order to consider situations in which a surface separates two media from each other,
the constitutive parameters (modelling conductivity, dielectricity and permeability)
are no longer continuous (i.e. also not differentiable). The concepts „conductivity“,
„dielectricity“ and „permeability“ are proposed to become intrinsic elements of
the Hilbert space model, whereby the „transmission boundary conditions“, ((KiA),
p. 12) are governed by a least action principle. In scattering theory the
solutions live in the unbounded exterior of a bounded domain. In standard
theory the behavior of electromoagnetic fields at infinity has to be taken into
account (example: Hertz dipole).
The solution of
time-harmonic Maxwell equations in a vacuum leads to the Helmholtz equation.
The fundamental solution of the Helmholtz equation at the origin is given by
spherical wave fronts. The time-dependent magnetic field has the form of the Hertz
dipole centered at the origin, (KiA) p. 14. In this example the considered
solution functions tends to infinity when absolute(x) tends to infinity. In
order to overcome this challenge the wave number k is replaced by the wave
number -k. It leads to „outgoing“ and „ingoing“ wave fronts. For
the scattering of electromagnetic waves the scattering waves have to be "outgoing"
waves. Thus, it is required to exclude the second ones by additional conditions
which are called radiation conditions (KiA) p. 15. The concept of „ingoing“
wave fronts (accompanied by (the Sommerfeld or the Silver-Müller) radiation
conditions) are proposed to become intrinsic elements of the Hilbert space
model, whereby the „transmission boundary conditions“ are governed by a least
action principle.
The (mathematically proven) spin-statistics PCT theorem says that "any Lorentz invariant local quantum field theory with a
Hermitian Hamiltonian must have PCT (Charge-Parity-Time) symmetry". Mathematically speaking, the PCT theorem states, that the product of the
transformations „P“ („Parity transformation“ (resp. a "half
integer" and "integer" spin concept), „C“ („Charge
conjugation“ (resp. confinement of spin property), „T“ („Time reversal“
(resp. SRT (hyperbolic PDE) scope)), in one order or the other, is
always a symmetry in local fields, whether or not the individual factors are. It
is one of the only few mathematically proven quantum physical
"theorems". The two essential
assumptions in proving the spin/statistics relation are "relativity",
i.e. the physical laws do not change under Lorentz transformation, and" microcausality",
i.e. space-like separated fields either commute or anticommute. It can be made
only for relativistic theories with a time direction.
Both quantum
theories, the quantum mechanics and the quantum dynamics, are
physical systems consisting of measurable variables and observable data; the
mathematical model framework is about the same complex number based L(2)
Hilbert space for both theories; its elements have wave character, which
are built on Fourier series representation based on Hermite polynomials to
model observables like location, momentum, energy density and energy radiation.
To stablish the states of different observables in the same Hilbert space
framework only simultaneously measurable observables may be considered.
Further royal road
markings towards a single, geometric gravity and quantum field theory
The central mathematical labelling data for a royal road
marking towards a single (truly geometric) gravity and quantum field theory are
-
the kinematical energy Hilbert space of the fermions is the standard
weak Hilbert space H(1), equipped with the "Dirichlet integral" inner
product
- the Hilbert space H(1) allows the split into two
complementary spaces, providing a model for the physical concepts of
"half-odd-integer" and "even-odd-integer" spins with related repulsive
and attractive effects, mathematically governed by Riesz bases coming
along with the non-harmonic Fourier series theory
- the Friedrichs extension of the Laplacian operator is a
selfadjoint, bounded operator B with domain H(1). Thus, also the operator B
induces a decomposition of H(1) into the direct sum of two subspaces, enabling
the definition of a „potential“ and a corresponding „grad“ potential operator.
Then a potential criterion defines a manifold, which represents a hyperboloid
in the Hilbert space H(1) with corresponding hyperbolic and conical regions
((VaM) 11.2)
- the kinematical Hilbert space H(1) is compactly
(coarse-grained) embedded into an overall energy Hilbert space H(1/2),
whereby the closed complementary subspace of H(1) governs all
non-kinematical energies, i.e. the closed sub-space H(1,ortho) replaces
all current bosons-type elementary particles
- theoretical
physics models are defined as weak (distributional
Hilbert scale based) variational (PDE) representations of the considered
mechanical system, which is compactly (coarse-grained) embedded into an
overall dynamical system. A first proof of concept is given by the
Millennium problem of a well-posed initial-boundary value problem of the
3D non-linear, non-stationary Navier -Stokes Equations, where the
corresponding variational representation accompanied with an extended
H(1/2) energy Hilbert space norm is well-posed
- the physical concept of "potential"
is an intrinsic element of the geometric quantum field model, and is no
longer an appropriately interpreted physical phenomenon of the
considered physical situation; mathematically speaking, the concept of
"potential" is an intrinsic element of the geometric Hilbert scale
framework, and do not need to be chosen properly as a given functions of
the considered PDE
- the "electron potential" (:=
electric potential energy/unit charge) in the geometric quantum field
(H(1/2) energy related Hilbert space) model corresponds to the electric
potential as defined by the weak representation of the Maxwell
equations; the hyperboloid manifold
in the Hilbert space H(1) with corresponding hyperbolic and conical
regions enables a corresponding "proton potential" governed by
the related "spin type"; in other words, also the concept of repulsive
electrons and attractive positrons are intrinsic parts of the geometric
model; the differentiator from a physical perspective is about the fact,
that the Planck action constant marks the barrior between the
measurarable action of an electron and the action of a proton,which is
beyond the Planck action constant
- Fourier
transforms do not allow localization in the phase space, leading to the concept
of windowed Fourier transforms. From a group theoretical perspective windowed
Fourier transforms are identical to the wavelet transforms. The wavelet
admissibility condition puts the spot on the Hilbert space H(1/2), (LoA). From
a „mathematization“ perspective „wavelet analysis may be considered
as a mathematical microscope to look by arbitrary (optics) functions over the
real line R on different length scales at the details at position b that are
added if one goes from scale „a“ to scale „a-da“ with da>0 but infinitesimal
small“, (HoM) 1.2
- the one-dimensional counterpart of the
below considered S(3) unit sphere is the unit circle. For the continuous
wavelet transform of a function over the unit circle with respect to a
wavelet g we recall from (HoM): „Geometrically, the wavelet transform
of a function over the circle T is a function over the half-cylinder
R(+) x T. … The Poisson summation formula shows a vanishing constant
Fourier term and the positive and negative frequencies do not mix,
enabling a corresponding (+/-) split of the L(2) space. … the wavelet
transform over the circle conserves energy. .. The wavelet transform
with respect to a progressive, admissible wavelet is an isometry. …. In
case where g=h it is now an orthogonal projector on the image of the
wavelet transform.
The proposed Hilbert space based model ...
... overcomes the main gap of Dirac‘s quantum theory of radiation, i.e. the small
term representing the coupling energy of the atom and the radiation field becomes part of the H(1)-complementary (truly bosons) sub-space of the overall
energy Hilbert space H(1/2); the new concept replaces Dirac’s H(-n/2-e)-based
point charge model by a H(-1/2)-based quantum element model
... acknowledges the primacy of micro
quantum world against the macro (classical field) cosmology world, where
the Mach principle governs the gravity of masses and masses govern the
variable speed of light, (DeH)
... allows to revisit Einstein's
thoughts on ETHER AND THE THEORY OF RELATIVITY in the context of the space-time theory and the kinematics of the
special theory of relativity modelled on the Maxwell-Lorentz theory of the
electromagnetic field
... acknowledges the Mach principle as a
selecting principle to select the appropriate classical cosmology field model out of the
few current physical relevant ones, (DeH): the to be selected classical cosmology field equation model may be modelled as the Ritz approximation equation (= orthogonal projection onto the coarse-grained (energy) Hilbert sub-space H(1) of the overall variational representation in the overall H(1/2) (energy Hilbert) space) of an extended Newton model, accompanied with the Dirichlet integral based inner product, the and the three dimensional unit sphere S(3) based on the field of quaternions in sync with the Lorentz transformation
... aknowledges Bohm's property of a "particle" in case of quantum fluctuation, (BoD), chapter 4, section 9, (SmL).
The GRT describes
a mechanical system characterized by the (inert = heavy) mass of the considered
„particles“ and their related interacting „gravity force“; the mathematical
model framework is about a real number based (purely) metric space. This is
about the distance measurment of real number points, which are per definition
without any „mass density“, (as long as the field of real numbers is not
extended to the field of hyper-real number, i.e. as long as the Archimedian
axiom is still valid). For a mechanical system every real-valued function of
the location and momentum (real number) coordinates represents an observable of
the physical system. The underlying metric
space concept (equipped with an (only) "exterior" product of differential forms, because a metric space has no geometric structure at all) accompanied by a global nonlinear
stable, Minkowski space, (ChD), is
replaced by a H(1/2)-quantum energy Hilbert space concept equipped, where the H(1/2)-inner product is equivalent to a corresponding inner product of differential forms.
The Hilbert space framework supports also the solution to related challenges,
e.g. regarding the „first
mover“ question (inflation, as a prerequiste) of the „Big Bang“ theory, the symmetrical
time arrow of the (hyperbolic) wave (and radiation) equation, governing the
light speed and derived from the Maxwell equations by differentiation, no long
term stable and well-posed 3D-NSE, no allowed
standing (stationary) waves in the Maxwell equation, the mystery
of the initial generation of an uplift force in a ideal fluid environment
of aircraft wings, i.e. no fluids collisions with the wings surfaces, and a Landau
equation based proof of the Landau damping phenomenon.
Some more labelling data towards a single geometric gravity and quantum field model
- the Hilbert space L(2) is compactly embedded
into the larger generalized Hilbert space H(-1/2), (with its dual (energy)
Hilbert space H(1/2) with respect to the L(2)-norm = L(2) metric)
- the Hilbert space H(-1/2) governs specific
variables like (the not measureable) ground state energy
- weak variational (PDE) representations of real
valued functions in a Hilbert scale framework, where a „coarse-grained“ (Lebesgue
integral based) Hilbert space L(2) governs the modelling requirements of mechanical
systems
- the field equations may be represented by the Einstein-Hilbert
action functional, which is already a baseline concept in the variational calculus
- the simultaneous measurement of the variables in
the H(-1/2) framework is modelled as the orthogonal projections onto the
(thermodynamics) statistical („coarse-grained“) Hilbert space L(2), which is
compactly embedded into H(-1/2)
- complex numbers play an important role in quantum mechanics; the extension to quantum dynamics needs a somehow richer object than R(3) or C(3) leading to Hamilton's quaternions as a fundamental building block of the universe accompanied by the three dimension unit sphere (UnA1)
- the Hilbert scale structure conserves the mechanical
measurements in the standard L(2) Hilbert space framework, while the underlying
energy dynamics are governed by the (much larger) orthogonal, closed subpace of
L(2) in H(-1/2) with respect to the overall "quantum element" Hilbert space
H(-1/2)
- a Hilbert space is THE mathematical role model building
geometric spaces, i.e. the proposed „coarse-grained“ Hilbert scale structure is
the best possible mathematical option to model the universe, which is nearly
all about vaccuum, whereby most of the rest of the vacuum is nearly all about
the sophisicated notions of „dark matter“ and „dark energy“
- the metric space comes along with the need to choose between three so-called "geometries" (spherical, flat, hyperbolic) in combination with multiple possible, only locally relevant metrices (e.g. Schwarzschild, Robertson-Walker, Gödel), while indefinite inner product spaces provide an all-in-one package, which includes the concepts of a "potential", (independently defined from any coupling constant (!)) and a "potential operator", enabling the definition of a corresponding potential criterion (barrior) defining a manifold, which represents a hyperboloid with underlying hyperbolic and conical regions
- in a
4D-field based Hilbert scale framework the metric space based so-called "geometry" types are replaced by
(problem specifically considered) PDE types, which are elliptic (mathematical object behind the naming convention: ellipse), parabolic (mathematical object behind the naming convention: straight (not a parabola)) and
hyperbolic (mathematical object behind the naming convention: hyperbole) PDE; only the parabolic (time arrow) and the hyperbolic (time symmetry) PDEs require the concept of "time". The model problems for the three PDE types are (1) the Newton potential equation resp. the Helmholtz equation (elliptic, steady state), (2) the heat equation (parabolic) and (3) the wave (or radiation) equation (hyperbolic). In the context of the concept "wave" we note that "distortion-free progressing wave" (alternatively to "standing waves"), which are key for the theory of transmission of signals, are indeed a central subject in the theory of hyperbolic PDE. Relatively undistored spherical waves relate spherical waves to the problem of transmitting with perfect fidelity signals in all directions. For the related Courant-Hilbert conjecture, putting spherical waves for arbitrary time-like lines in case of two or four variables into a one-to-one relationship to the wave equation, we refer to (CoR) VI. §18. In order to
prove this conjecture one might consider the fact that the only possible domain
fields of the considered functions are the sets of the 1D real numbers, 2D
complex numbers, and the 4D quaternions
- without the
parabolic or hyperbolic space-time concept in the purely elliptic world, there
is no theoretical model from which the numerical values of the velocity of
light and the Planck constant can be derived, as the underlying units „meter“
and „second“ are out of scope, (UnA1) p. 138 ff: Anticipating the
Courant-Hilbert conjecture there is the need for an alterative 4D field model,
and further anticipating Hamilton’s quaternions. Quarternions are a four
dimensional number system enabling a tricky multiplication of complex numbers, leading
to a four dimension unit ball with its underlying three-dimensional unit
sphere, which is equivalent with rotations in three dimensional space. We note
that Plemelj’s concepts of „mass element“ and „flow“ in the context of the
Green formulae are intrinsic elements of a unit sphere without the need of a
„normal vector“ accompanied with the concept of a „mass density“. With respect
to the physical universe we recall the following two quotes, (UnA1) p. 155:
R. Feynman:
„Now you can look back and say that Pauli’s spin matrices and operators were
nothing but Hamiltion’s quaternions“
W. Hamilton:
„Somehow quaternions are a fundamental building block of physical universe“
- a coarse-grained mechanical Hilbert space L(2) embedded into the Hilbert space H(-1/2) allows a re-interpretation of the notion "entropy" in the context of the "discrete" Shannon entropy vs. the log(x)-function based "continuous entropy"; we note that the (still best known) relationship between the notions "volume V", "number of particles N", temperature T", "pressure P", "entropy S" and the related total energy E, as a function of the parameters S, V, N, is given by the total differential dE=TdS-PdV+c*dN
- the coarse-grained mechanical (statistical) Hilbert space L(2) embedded into the overall Hilbert space H(-1/2) puts the spot on the Hawking-Hartle (probability theory related) "Interpretation of "The Wave Function of the Universe"" in (DrW): "the ground state is the amplitude for the Universe to appear from nothing"; ... "Physical probabilities, as exemplified by radioactive decay,
start with something, a first situation (particle in space and time) becoming
another situation (other particles in space and time). The probability is the
chance that the transition from situation one to situation two happens during a
certain interval of time, or that a particle is found in a certain volume of
space, or something like that". In this context we note that the embeddedness of
L(2) into H(-1/2) relates to the embeddedness of the set of rational numbers
into the set of real numbers, whereby the countable set Q is a „zero quantity set“ with
respect to the L(2) inner product (i.e. the probability to pick rational numbers ot of the set real numbers is zero). We note that the set of the (infinite) solutions of the field equations has the same cardinality as the set of real numbers
- the model supports also Einstein's composition model of radiation (Physikalische Zeitschrift 10 (1909),
185–193), while at the same point in time indicates to revisit the related discourse note with W. Ritz (Physikalische Zeitschrift, 10, (1909), 323-324)
- the extended H(1/2) (energy) Hilbert space framework replacing the Dirichlet integral based inner product space H(1) (derived from the classical Newton potential equation accompanied with Green's identities) leads to a well-posed 3D-non-linear, non-stationary Navier-Stokes boundary-initial value problem accompanied with Plemelj's extensions of Green's identities based on intrinsic surface notions "mass element" and "flux"). At the same time, it omits the Yang-Mills system (the expansion of the Maxwell equations in the context of the SMEP) accompanied with the corresponding (physical) Yang-Mills mass gap problem
- the SMEP (with its underlying separation into the categories fermions (EPs with spin 1/2 (electron, neutrino, quarks) and bosons ("EPs" with spin 0 (Higgs), spin 1 (gluons, W/Z-boson, spin 2 (graviton)) is replaced by a purely fermion based EP model defined in the coarse-grained Hilbert scale pair ((H(0),H(1)), governed by the complementary closed sub-space with respect to the overall ((H(-1/2),H(1/2)) Hilbert scale framework. As a consequence of the ("color") confinement problem of quarks (the phenomenon that "color-charged" particles (such as quarks and gluons) have not been isolated until today, i.e. they have not been observed until today), the "quark" concept is omitted by the original proton
- fermions are
governed the Fermi statistics, whereby two identical fermions (EP with mass) cannot
occupy the same quantum state (in a corresponding mathematical world this means
that a mathematical „fermion“ object behaves like a rational number). The Fermi
oscillator is „a particularly simple system. It is a thing capable only for
two levels, zero and „epsilon““, (ScE) p.20. The Fermi oscillator is
basically the mathematical model for his famous „spin-concept“. „The so-called „spin
statistics theorem“ is one of the few theorems in theoretical physics, which
are proven based on very very little
assumptions. .. it states that EPs with integer spin are representated by symmetric wave functions, while EPs with half integer spin (k+1/2, k=0,1,2,...) are representated as anti-symmetric wave functions“ (HeW) p. 103. All bosons are governed by the Bose-Einstein
statistics, which is concerned with “photon gases”. A characteristic of the
Bose-Einstein „photon gases“ statistics is, that the concerned particles do not
restrict the number of them that occupy the same (continuous) quantum state . All
„photon gas“ particles can be brought into the energetically lowest quantum
state (below the critical temperature of “normal gas”), where they show the
same “collective” behavior. They occupy a single (continuous) quantum
state of zero momentum, while „normal gas“ particles all have finite momentum. The statistical thermodynamics model for "normal gases" is the Planck oscillator; in the proposed model the related Planck-Schrödinger statistics of the n-particle problem starts with k=1, while to case k=0, the ground state energy case, is interpreted as physical condition for an model adequate ground state energy value approximation, from which the reduced Planck action variable can be derived, (ScE)
- Pauli’s spin concept and its related spin
statistics theorem is about amathematical object with the 2-value attributes, "half integer" and
"integer" multipliers of the reduced Planck action quantum. The
"connection" model of those spins is mathematically given by „complex
number multiplication“. The proposed fermions model is about the three observable
EPs (electron, neutrino, positron) considered in a Hilbert space framework. The
underlying field is the three-dimensional unit sphere S(3) enjoying the
following properties
i) S(3) is isometric to "rotations in the three
dimensional space"
ii) S(3) relates to the multiplication of triplets, Hamilton’s
concept of quarternions, (UnA1) p. 148
iii) quaternionic multiplication of a spatiotemporal
derivative with „electromagnetic potential“ leads to two terms precisely
matching the known expressions for the electric and the magnetic fields of the
Maxwell equations, (UnA1) p.152
iv) the governing mathematical concept of the Maxwell
equations is the Lorentz transformation, which is a linear transformation
mapping space-time onto space-time, preserving the scalar product
- the S(3) model in combination with a complex Lorentz
group puts the spot on the famous Spin-Statistics-PCT theorem, (StR), pp. 9, 13: "The PCT
theorem is a fundamental symmetry of physical laws. It is the only
combination of C, P, and T transformations (see below) that is observed to be an exact symmetry
of nature at the fundamental level. In terms of Einstein's SRT, "the CPT
theorem says that CPT symmetry holds for all physical phenomena, or more
precisely, that any Lorentz invariant local quantum field theory with a
Hermitian Hamiltonian must have CPT symmetry""
- regarding general irreducible spinor fields and assumed "wrong" connections of spins with statistics of integer and half-odd integer spins, in which all fields either commute or anti-commute, we refer to the Spin-Statistics Theorem for General Spin, (StR) Theorem 4-10
- wave-mechanical vibrations correspond to the motion of
particles of a gas resp. the eigenvalues and eigen-functions of the harmonic (Planck)
quantum oscillator modelled in the Hilbert scales L(2)/H(1)). The alternatively
proposed energy space (H(1/2) = H(1)+H(1,ortho) indicates to revisit
Schrödinger's "purely quantum wave" vision, which is about
half-odd integer wave numbers, rather than integer quantum numbers. "The wave point of view in both cases (Bose-Einstein and Fermi-Dirac statistics) or at least in the Bose case (mathematically a simple oscillator of the Planck type) raises another interesting question: we may ask whether we ought not to adopt for the quantum numbers half-odd integers. .... On the other side, if we adopt straightaway, we get into erious trouble, especially on contemplating changes of the volume (e.g. adiabatic compression of a given volume of black body radiation), because in this process the (infinite) zero-point energy seems to change by infinite amounts!" (ScE) p. 50. We emphasis that the proposed model overcomes this challenge, as there is no n-particle problem anymore in the purely continous spectrum world of the considered complementary sub-space of the statistical Hilbert space L(2)
- „Hubble’s observations of a shift to the red in the
spectra of the spiral nebulae—the farthest parts of the universe—indicated that
they are receding from us with velocities proportional to their distances from
us“ (DiP). In the context of the spherical wave conjecture regarding
observable the space-time hyperbolic world) Dirac’s „new basis for cosmology“
((DiP) 1937), suggests „a model of the (physical) universe in which there is a
natural velocity (requiring the concept of time) for the matter at any point, varying continuously from one
point to a neighbouring point. Referred to a four-dimensional space-time
picture, this natural velocity provides us with a preferred time-axis at each
point, namely, the time-axis with respect to which the matter in the
neighbourhood of the point is at rest. By measuring along this preferred time-axis
we get an absolute measure of time, called the epoch.“
- An elliptic mathematical „reality“ model of the universe
governs two complementary subsets. There is the parabolic/hyperbolic "physical reality world“, (which is a very „small“ (in the sense of a zero quantity set) countable
subset of the overall mathematical „reality“ model), accompanied with physical-kinematical
notions like „time“, „velocity“, „action“ and e.g. the second law of
thermodynamics. Then there is the dominant closed, complementary subset of the
"physical reality“ model, accompanied with purely mathematical-not measurable
notions like „ground state energy“. In this mathematical reality, (UnA1), concepts like Dirac’s „epoch“
or Penrose’s „cycles of time“ (for an alternative steady state model to describe, what came before the big bang), may be
revisited to describe how elementary particles with mass are generated out of
the purely mathematical, non-kinematical „world“ and how a kind of "annihilation force" occurs measurable by the second law of thermodynamics. The guiding principle may be, that "time" (resp. only "time duration") as perceived by human consciousness aggregates and reflects the "actions" (accompanied with related "action variables", (HeW)) of observed physical systems, which are detectable and understandable (e.g. the scattering theory based methods or Einstein's composition of radiation model) only by statistical ((L(2)-Hilbert space based) measurement methods. Regarding the prospects success measuring the generation of an elementary particles with mass we note that the sum of countable zero quantity sets is again a zero quantity set
The spectrum of
a Hermitian, positive definite operator H in a complex valued Hilbert space H
with domain D(A), where its inverse operator is compact, is discrete
(measurable/stable). Without the latter property the concept of a continuous
spectrum is required (unstable meaurement results). The link back
to part A is given by the Berry-Keating conjecture with its mathematical
counterpart, the Hilbert-Polya conjecture. It is about the suggestion that the
E(n) of the non-trivial zeros ½+iE(n) of the Zeta function are all real,
because they are eigenvalues of some Hermitian operator H. Berry’s extensions
are that if H is regarded as the Hamiltonian of a quantum-mechanical system
then, (BeM)
i) the operator H has classical limits
ii) the classical orbits are all chaotic (unstable)
iii) the classical orbits do not possess
time-reversal symmetry.
While the full
H(-1/2) based quantum-mechanical system behaves chaotic from an observer
perspective (measured in the statistical Hilbert (sub) space L(2)), the underlying complementary sub-systems allow a differentiation
between those observable thermostatistical effects and the related, not statistically measurable mathematical
model parameters, which are physically governed by physical Nature constants, reflecting the border line to those complementary sub-systems (with purely continuous spectrum).
The proposed circle method above (part A) comes along
with a split of the integer domain N (of number theoretical functions) being bijectively mapped onto two
sets of integers („odd“ and „even“) on the boundary of the unit circle, both with
Snirelman density ½. The two sets may be used to revisit the elementary
„particle“ state numbers, n=2n/2 (n=1,2,3, ...) exdended by n=1/2, and Schrödinger’s half integer
state numbers n+1/2=(2n+1)/2 (n=0,1,2,...), (corresponding to the levels of the
Planck oscillator, see above resp. (ScE) pp. 20, 50)) to re-organize the chaotic model behavior of the current thermodynamics based purely kinematical quantum-mechanical systems.
Further referring to part A we note that the zeta function on the critical
line is an element of the sequences Hilbert space l(-1). The Shannon sampling operator is
the linear interpolation operator mapping the standard sequences Hilbert space
l(2) isomorphically onto the Paley-Wiener space with bandwidth „pi“ (i.e. a signals
f represented as the inverse Fourier transform of a L(2) function g can be
bijectively mapped to its related sequences f(k) and vice versa). We note that the generalized
distributional (polynomial and and exponential decay) Hilbert scales l(a) and l(t,a) allow the definition of corresponding generalized Paley-Wiener scales enabling a (wavelet basis based) convolution integral representation
of the zeta function, fulfilling appropriate admissibility conditions for wave functions to support general quantumâmechanical principles.We note that the Fourier transform does not allow localization in the phase space. In order to overcome this handicap D. Gabor introduced the concept of windowed Fourier transforms. From a group theoretical perspective the wavelet concept and the windowed Fourier transform are identical. With respect to the above Hilbert scale framework we note that the admissibility condition defining wavelets (e.g. (LoA)) puts the spot on the proposed H(1/2) (energy) Hilbert space.
From a physical perspective the wavelet transform may be looked
upon as a „time-frequency analysis“ with constant relative bandwidth. From a „mathematization“
perspective „wavelet analysis may be considered as a mathematical microscope
to look by arbitrary (optics) functions over the real line R on different length
scales at the details at
position b that are added if one goes from scale „a“ to scale „a-da“ with
da>0 but infinitesimal small“, (HoM) 1.2. Technically speaking, „the wavelet transform allows
us to unfold a function over the one-dimensional space R into a function over
the two-dimensional half-plane H of positions and details (where is
which details generated?). … Therefore, the parameter space H of the
wavelet analysis may also be called the position-scale half-plane since
if g is localized around zero with width „Delta“ then g(b,a) is localized
around the position b with width a*Delta“.
The related wavelet
duality relationship provides an additional degree of freedom to apply wavelet
analysis with appropriately (problem specific) defined wavelets in a Hilbert
scale framework where the "microscope observations" of two wavelet
(optics) functions g and h can be compared with each other by a "reproducing"
("duality") formula. The (only) prize to be paid is about additional
efforts, when re-building the reconstruction wavelet.
Braun K., Looking back, part B, (B1)-(B17) |
(C) Linkages between the proposed truly geometric gravity and quantum field model and philosophy
Some related „views of the world“ from physicists and philosophers:
Braun K., Looking back, part C, (C1)-(C8) |
(D) Appreciation
Officially accepted solutions of the considered research areas would be honored by several prizes. For hopefully understandable
reasons none of
the papers of this homepage are appropriately designed to go there. Therefore, after a 10 years long journey accompanied by four
main ingredients "fun, fun, fun and learning", it looks like a good point in time to share resp. enable more
fun to the readers‘ side, who showed their interest by more than
1 GB downloads per day (on average) during the last years. From (KoJ) p. 148 we quote:
„find a skillful motivation.
Then do the math and enjoy the creativity of the mind“
and, with the words of master Yoda:
"may the Force be with you", ...:) .
In order to support this some MS-Word
based source documents of key papers are provided below.
For people, who are familar with the german language and who
want to get some guidance to autonomous thinking in current grazy times we recommend
the latest book from A. Unzicker: „Wenn man weiß, wo der Verstand ist, hat
der Tag Struktur“.
 Braun K., Looking back, part A, (A1)-(A3) |
Braun K., Looking back, part B, (B1)-(B17), Dezember 2, 2020
Braun K., Looking back, part A, (A1)-(A3), October 26, 2020 |
Braun K., Looking back, part B, (B1)-(B17), November 29, 2020 |
Braun K., Looking back, part B, (B1)-(B17), July 6 2020 |
Braun K., Looking back, part C, (C1)-(C8), June 28, 2020 |
1_Braun K., RH, YME, NSE, GUT solutions, overview |
2_Braun K., RH solutions |
3_Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis and the Goldbach conjecture |
4_Braun K., 3D-NSE, YME, GUT solutions |
5_Braun K., Global existence and uniqueness of 3D Navier-Stokes equations |
 6_Braun K., A new ground state energy model |
7_Braun K., An alternative Schrödinger momentum operator enabling a quantum gravity model |
8_Braun K., Comparison table, math. modelling frameworks for SMEP and GUT |
9_Braun K., An integrated electro-magnetic plasma field model |
10_Braun K., Unusual Hilbert or Hoelder space frames for the elementary particles transport (Vlasov) equation |
11_Braun K., A distributional Hilbert space framework to prove the Landau damping phenomenon |
Nitsche J. A., Lecture notes, Hilbert scales and approximations theory |
Disclaimer: None of the papers of this homepage have been
reviewed by other people; therefore there must be typos, but also errors for sure.
Nevertheless the fun part should prevail and if someone will become famous at
the end, it would be nice if there could be a reference found to this homepage somewhere.
