This homepage addresses the problem
& solution areas
A. The Riemann Hypothesis
(RH)
building on a common
Hilbert scale framework enabling a combined usage of spectral theory, variational methods for non-linear operators (VaM), Galerkin-Ritz approximation theory (with in this case, quantum state approximations in the compactly embedded sub-(Hilbert)-space H(0) of H(-1/2)), (VeW) and tools like Pseudo-Differential operators ((EsG), (LoA), (PeB)), degenerated hypergeometric functions (GrI), Hilbert (resp.
Riesz) transform(s) and wavelets (HoM). The link between PDO and the Galerkin-Ritz approximation theory is given by the Garding inequality going along with the concept of hypoellipticity ((GaL), (PeB)) enabling compactness arguments to support "quasi-optimal" approximation properties (AzA), between the quantum space H(-1/2) and its subspace H(0)=L(2), which is the physical relevant (statistics) "observation" space. In case of the Boltzmann-Landau equation, which is concerned with plasma ("particles") collisons phenomena, we refer to (LiP), (LiP1). The norms of the Hilbert scale H(a) can be enriched with an additional norm enjoying an “exponential decay” behavior. Each Hilbert space norm with a<0 is governed by the norm of the Hilbert space H(0) and this "exp-decay" norm (BrK5), (BrK6). Already in the context of the provided RH solutions the Hilbert
space H(-1/2) turned out to be the appropriate alternative to its
subspace H(0). Its norm
is governed by
the norm of the "observation" space H(0) and this "exp-decay" norm.
This property is proposed to be applied in the context of the decomposition of the Hilbert space H(-1/2) = H(0) + H(0,ortho)) ((BrK), (BrK1), (BrK3), (BrK7)). The story line of this homepage is as follows: an appropriate Hilbert scale framework is proposed to
enable several RH criteria, while its correspondence to classical
function spaces of the current Zeta function theory is ensured. One of
the RH critera is about the Hilbert-Polya criterion, which is about a convolution operator representation of the Zeta function requiring a properly defined domain. This
Hilbert space based operator representation with its underlying Hilbert scale frame (defining also the regularity of
the Zeta funtion) can be applied to answer
the Berry-Keating conjecture. The Berry-Keating conjecture puts the zeros of the Zeta function (on the critical line, if the RH is true) in relationship to the (energy level) eigenvalues associated with the classical Hermitian operator H(x,p)=x*p ~ x*(d/dx), where x denotes the position coordinate and p the conjugate momentum. The Friedrichs extension of the variational representation of the Zeta function (on the critical) with L(2)-test space indicated a H(-1/2) quantum state space with related H(1/2) energy space. The today's standard quantum state resp. energy spaces are H(0)=L(2) resp. H(1), i.e. the latter Hilbert spaces are compactly embedded subspaces of the proposed new ones. Applying the physical quantum (fluid) Hilbert (state) space H(-1/2) to the 3-D non-linear, non-stationary NSE enables a well posed variational representation of the NSE with appropriate valid energy inequality, closing the Serrin gap problem. The correspondingly variational representation of the Maxwell equations enables a quantum field model (modified YME), enabling a differentiation of "elementary particles" with and w/o mass (modelled by the orthogonal decomposition of the Hilbert spaces H(-1/2) = H(0) + H(0,ortho) resp. H(1/2) = H(1) + H(1, ortho). It enables the concept of orthogal projection, which can be interpreted as "mass generation process during observation". The "EP" are "acting", i.e. interaction between each other with 100% probability in the orthogonal space (which might be interpreted as "zero point energy pool", or the "wave package" resp. "eigen-differential" space). The macroscopic and microscopic states of quanta relate to frequencies of corresponding vibrations. The related action variable of the system ((HeW) II.1.c) determines the ("inverse") related kinematical (physical, H(0) based) and thermodynamical concept of " If a successfully applied least action
principle (being interpreted as a maxime of Kant's reflective judgment)
results into appropriate consistent mathematical-physical models, those
models can be declared as law of natures. The above is related to the
three "forces of nature" as modelled by the SMEP. The nature of those
elementary particles and the way
they move, is described by quantum mechanics, but quantum mechanics
cannot deal with the curvature of space-time. Space-time are
manifestations of a physical field, the gravitational
field. At the same time, physical fields have quantum character:
granular,
probabilistic, manifesting through interactions. The to be defined
common mathematical solution framework needs to provide a quantum
state of a gravitational field, i.e. a quantum state of space. The
crucial difference between the photons characterized by the Maxwell
equations (the
quanta of the electromagnetic field) and the to be defined quanta
of gravity is, that photons exists in space, whereas the quanta of
gravity
constitute space themselves ((RoC2) p. 148). The proposed mathematical
framework provides a common baseline to integrate quantum mechanics
& thermodynamics with gravity & thermodynamics. From a physical
model problem perspective this is about a common mathematical framework
for black body radiation ((BrK4) remark 2.6, Note O55, O71, O72) and
black hole radiation ((RoC3) p. 56, 60 ff)). The thermodynamics is the
common physical theory denominator with the Planck concept of zero point
energy of the harmonic quantum oscillator (BrK), (BrK1), and the
Boltzmann entropy concept. An integrated model needs to combine the
underlying Bose-Einstein and the Dirac-Fermi statistic. In this context
already Schrödinger suggested half-odd quantum numbers rather than
integers. " For all of the considered physical problem/solution
areas of this homepage the Hilbert
space H(-1/2) is suggested as
physical quantum (Hilbert) state space model accompanied by
correspondingly defined
variational (Differential, Pseudo Differential or singular integral
operator) equations. Beside the NSE and the YME problem areas the
following other related areas are considered: (1) plasma dynamics
(Landau damping phenomenon, i.e. " (CiI) 2.8: (CiI) 3.5: " With respect to the overall conceptual idea of this homepage a Hilbert space based geometrodynamics is proposed to be built on "space-time states", which are represented by elements of H(-1/2), while their corresponding "space-time energy" elements are represented by the corresponding "dual" (wavelets) elements in H(1/2). The Einstein field equations are proposed to be re-formulated as a weak (!) least action minimization problem by correspondingly defined variational equations representation. With respect to the Bianchi identities we emphasis that if ((u,v)) denotes the inner product of H(-1/2) the following relationships hold true: ((div(u),v)) ~ (u,v) ~ ((u,grad(v))). The methods of functional analysis are basically the same as those in the elasticity theory (MaJ1). With respect to the below we note that the Dirac theory with its
underlying concept of a Dirac "function" is proposed to be replaced by (fluid/quantum/... state) "elements" of the distributional Hilbert space H(-1/2). We note that the regularity of the
Dirac distribution "function" depends from the space dimension, i.e. it is an
element of H(-n/2-e) (e>0, n = space dimension). Therefore, the
alternative H(-1/2) quantum state concept avoids space dimension depending regularity assumptions for quantum mechanics "wave packages" / "eigen-functions" / "momentum functions" with corresponding continuous spectrum.
This OVERVIEW page is frequently updated; a former related (more mathematical formula based)
version (status August 2018) is provided in
In order to prove the
The corresponding formulas, when
replacing the Gaussian function by its Hilbert transform, are well known: the
With respect to the considered
distributional Hilbert spaces H(-1/2) and H(-1) we note that the Zeta function
is an integral function of order 1 and an element of the distributional Hilbert
space H(-1). This property is an outcome of the relationship between the
Hilbert spaces above, the
The The theory of
spectral expansions of non-bounded self-adjoint operator is connected with the
notions "Lebesgue-Stieltjes integral" and "
The Riemann entire Zeta function Z(s)
enjoys the functional equation in the form Z(s)=Z(1-s). The alternatively
proposed Dawson (baseline) function leads to an alternative entire Zeta
function definition Z(*;s) with a corresponding
The RH is equivalent to the
One proof of the Riemann functional
equation is based on the The density function J(x)
of the log( The
The density of prime numbers appears to
be the
The asymptotics of the special Kummer
functions K(a;x):=K(a,a+1;x)
The relationship of the considered
Kummer functions to the incomplete Gamma function is provided in (AbM) 6.5.12.
We further note, that the
Additionally, the above alternative Z(s)
resp.
For other related application areas of
Gamma(*,s/2) we refer to Ramanujan's chapter "Analogues of the Gamma
Function" ((BeB) chapter 8).
In (TiE) theorem 4.11, an approximation
to the
In ((BrK4) lemma 3.4, lemma A12/19) the
function P(x) is considered in the context of (appreciated) quasi-asymptotics
of (corresponding) distributions ((ViV) p. 56/57) and the Riemann mapping
theorem resp. the Schwarz lemma. The considered "function" g(x):=-d/dx(cot(x))
(whereby the cot-"function" is an element of H(-1)) is auto-model (or
regular varying) of order -1. This condition and its corresponding asymptotics
property ((BrK) lemma 3.4) provide the prereqisitions of the
In (GrI) 8.334, the relationship between
the the cot- and the Gamma function is provided. From (BeB) 8. Entry17(iii)) we
quote: "
may be formally established by differentiating the corresponding Fourier series
equation for (the L(2)=H(0)-function) "
((BrK4) remark 3.8). The proposed distributional Hilbert scales provide the
proper framework to justify Ramanujan's related parenthetical remark "-log(2sin(cx))for
the same limit" (in a H(-1)-sense).The element of H(-1), but not an element of
H(-1/2).Therefore, there is a characterization of the Zeta function on the
critcal line in the form ((Z,v)) for all v ex H(0). As the "test
space" H(0) is compactly embedded into H(-1/2) this shows that there is an
extended Zeta function Z(*)=Z+Z(#) (Friedrichs' extension) with the characterization ((Z(*),v))
for all v ex H(-1/2), where Z can be interpreted as orthogonal approximation of
Z(*) with discrete spectrum.
Riemann's "workaround"
function h(x):-x*d/dx(f(x) do have an obvious linkage to the "
The common Hilbert scale is about the Hilbert
spaces H(a) with a=1,1/2,0,-1/2,-1 with its corresponding inner products The The newly proposed "fluid/quantum state" Hilbert space H(-1/2)
with its closed orthogonal subspace of H(0) goes also along with a combined
usage of L(2) waves governing the H(0) Hilbert space and "orthogonal"
wavelets governing the H(-1/2)-H(0) space. The wavelet "reproducing"
("duality") formula provides an additional degree of freedom to apply
wavelet analysis with appropriately (problem specific) defined wavelets, where
the " We propose modified Maxwell equations with correspondingly extended
domains according to the above. This model is proposed as alternative to SMEP,
i.e. the modified Maxwell equation are proposed to be a "Non-standard
Model of Elementary Particles (NMEP)", i.e. an alternative to the
Yang-Mills (field) equations. The conceptual approach is also applicable for
the Einstein field equations. Mathematical speaking this is about potential
functions built on correponding "density" functions. The source
density is the most prominent one. Physical speaking the source is the root
cause of the corresponding source field. Another example is the invertebrate
density (=rotation) with its corresponding rotation field. The Poincare lemma
in a 3-D framework states that source fields are rotation-free and rotation
fields are source-free. The physical interpretation of the rotation field in
the modified Maxwell equations is about rotating "mass elements w/o
mass" (in the sense of Plemelj) with corresponding potential function. In
a certain sense this concept can be seen as a generalization of the Helmholtz
decomposition (which is about a representation of a vector field as a sum of an
irrotational (curl-free) and a solenoidal (divergence-free) vector field): it
is derived applying the delta "function" concept. In the context of
the proposed distributional Hilbert space framework, the Dirac function concept
(where the regularity of those "function" depends from the space
dimension) is replaced by the quantum state Hilbert space H(-1/2). The solution
u (ex H(1/2)) of the Helmholtz equation in terms of the double layer potential
is provided in ((LiI), 7.3.4). From the Sobolev embedding theorem it follows,
that for any space dimension n>0 the modified Helmholtz equation is valid
for not continuous vector fields.
The We note that the solution of the Navier-Stokes equation are related to the considered degenerated hypergeometric functions by its corresponding integral function representation (PeR1). With respect to the open Millenium A "3D challenge" like the NSE above is also valid, when
solving the monochromatic scattering problem on surfaces of arbitrary shape
applying electric field integral equations. From (IvV) we recall that the
(integral) operators A and A(t): H(-1/2) --> H(1/2) are bounded Fredholm
operators with index zero. The underlying framework is still the standard one,
as the domains are surfaces, only. An analog approach as above with
correspondingly defined surface domain regularity is proposed.
We propose an alternative
mathematical framework for the Standard Model of Elementary Particles (SMEP),
which replaces gauge theory and variational principles: The underlying
concepts of exterior derivatives and tensor algebra are replaced by
(distributional) Hilbert scales and (purely Hamiltonean) variational
principles. As a consequence, the vacuum energy becomes an intrinsic part of
the variational principles, i.e. it is identical for all considered Lagrange
resp. Hamiltonian mechanisms of all related differential equations, while the
corresponding "force" becomes an observable of the considered
(Hamiltonean) minimization problem.
In some problem statements of
the YME there are basically two assumptions made: 1. the energy of the
vacuum energy is zero As a consequence the mass gap
is the mass of the lightest particle. Our challenge of proposition 1
is about the measure of the vacuum energy, which gives the value
"zero". While the energy norm in the standard H(1) Hilbert space
might be zero, the value of the quantum state with respect to the energy norm
of the sub-space H(1/2) still can be >0. Our challenge of proposition 2
is going the same way: a particle with mass can be measured (condensed energy),
i.e. it is an element of the test space H(0), while there still can be
"wavelets" in the closed complementary space H(-1/2)-H(0),
where the test space is "just" compactly embedded. Those
"wavelets" might be interpreted as all kinds of today's massless
"particles" (neutrinos and photons) with related "dark
energy".
As a consequence there is no
YME mass gap anymore, but there is a new concept of vacuum energy (wave packages, eigen-differentials, rotation differential) governed by the Heisenberg
uncertainty principle. This is about
an alternative harmonic quantum energy model enabling a finite "quantum
fluctuation = total energy", while replacing Dirac's Delta function by H(-1/2) distributions enabling and an alternative
Schrödinger's momentum operator (BrK7). A physical interpretation
could be about "rotating differentials" ("quantum
fluctuations"), which corresponds mathematically to Leibniz's concept of
monads. The mathematical counterparts are the ideal points (or hyper-real
numbers). This leads to non-standard analysis, whereby the number field
has same cardinality than the real numbers. It is "just" the
Archimedean principle which is no longer valid. The proposed mathematical concepts and tools are especially
correlated to the names of The electromagnetic interaction has gauge invariance for the probability
density and for the Dirac equation. The wave equation for the gauge bosons,
i.e. the When one wants to treat the time-harmonic Maxwell equations with
variational methods, one has to face the problem that the natural bilinear form
is not coercive on the whole Sobolev space. On can, however, make it coercive
by adding a certain bilinear form on the boundary of the domain (vanishing on a
subspace of H(1)), which causes a change in the natural boundary conditions.
In
SMEP (
Reformulated Maxwell or gravitation field equations in a weak H(-1/2)-sense leads to the same effect, as dealing with an isometric
mapping g --> H(g) in a weak H(0)-sense (H denotes the Hilbert transform) alternatively to a second
order operator in the form x*P(g(x)) in a weak H(-1/2). This results into some opportunities as - the solutions of the Maxwell equations in a vaccum do not need any callibration transforms to ensure wave equation
character; therefore, the arbitrarily chosen Lorentz condition for the
electromagnetic potential (to ensure Lorentz invariance in wave equations) and
its corresponding scalar function ((FeR), 7th lecture) can be avoided - enabling alternative concepts in GRT to e.g. current (flexible") metrical affinity, affine connexions and local
isometric 3D unit spheres dealing with rigid infinitesimal pieces, being
replaced by
Replacing the affine connexion
and the underlying covariant derivative concept by a geometric structure with
corresponding inner product puts the spot on the
This conjecture asserts that any compact
3-manifold can be cut in a reasonably canonical way into a union of geometric
pieces. In fact, the decomposition does exist. The point of the conjecture is
that the pieces should all be geometric. There are precisely eight homogeneous
spaces (X, G) which are needed for geometric structures on 3-manifolds. The
symmetry group SU(2) of quaternions of absolute value one (the model for the
weak nuclear force interaction between an electron and a neutrino) is diffeomorph
to S3, the unit sphere in R(4). The latter one is one of the eight geometric
manifolds above (ScP). We mention the two other relevant geometries, the
Euclidean space E3 and the hyperbolic space H3. It might be that our universe
is not an either... or ..., but a combined one, where then the
"connection" dots would become some physical interpretation. Looking
from an Einstein field equation perspective the Ricci tensor is a second order
tensor, which is very much linked to the Poincare conjecture, its solution by
Perelman and to S3 (AnM). The
The single scalar equation for the Ricci
potential (CaJ) might be interpreted as the counterpart of the The link back to a Hilbert
space based theory might be provided by the theory of spaces with an indefinite
metric ((DrM), (AzT), (DrM), (VaM)). In case of the L(2) Hilbert space H, this
is about a decomposition of H into an orthonal sum of two spaces H1 and H2 with
corresponding projection operators P1 and P2 relates to the concepts which
appear in the problem of S. L. Sobolev concerning The potential criterion
p(x)=c>0 defines a manifold, which represents a
A selfadjoint operator B defined on all
of the Hilbert space H is bounded. Thus, the operator B induces a decomposition
of H into the direct sum of the subspaces, and therefore generates related
hyperboloids ((VaM) 11.2). Following the investigations of Pontrjagin and
Iohvidov on linear operators in a Hilbert space with an indefinite inner
product, M. G. Krein proved the
In an universe model with appropriately
connected geometric manifolds the corresponding symmetries breakdowns at those
"connection dots" would govern corresponding different conservation
laws in both of the two connected manifolds. The Noether theorem provides the
corresponding mathematical concept (symmetry --> conservation laws; energy
conservation in GT, symmetries in particle physics, global and gauge
symmetries, exact and broken). Those symmetries are associated with
"non-observables". Currently applied symmetries are described by
finite- (rotation group, Lorentz group, ...) and by infinite-dimensional
(gauged U(1), gauged SU(3), diffeomorphisms of GR, general coordinate
invariance...) Lie groups.
A manifold geometry is defined as a pair
(X,G), where X is a manifold and G acts transitively on X with compact point
stabilisers (ScP). Related to the key tool "Hilbert transform" resp.
"conjugate functions" of this page we recall from (ScP), that
Kulkarni (unpublished) has carried out a finer classification in which one
considers pairs (G,H) where G is a Lie group, H is a compact subgroup and G/H
is a simple connected 3-manifold and pairs (G1,H1) and (G2,H2) are equivalent
if there is an isomorphism G1 --> G2 sending H1 to a conjugate of H2. Thus
for example, the geometry S3 arises from three distinct such pairs, (S3,e),
(U(2),SO(2)), (SO(4),SO(3)). Another example is given by the Bianchi
classification consisting of all simply connected 3-dimensional Lie groups up
to an isomorphism.
(AbM) Abramowitz M., Stegun A., Handbook of mathematical functions, Dover
Publications Inc., New York, 1970 (BrK related papers)
www.navier-stokes-equations.com/author-s-papers (CoR) Courant R., Hilbert D., Methoden
der Mathematischen Physik II, Springer-Verlag, Berlin, Heidelberg, New York,
1968 (EdH) Edwards Riemann's Zeta Function, Dover
Publications, Inc., Mineola, New York, 1974 (ScE) Schrödinger E., Statistical Thermodynamics, Dover Publications, Inc., New York, 1989 (SeA) Sedletskii A. M., Asymptotics of
the Zeros of Degenerated Hypergeometric Functions, Mathematical Notes, Vol. 82,
No. 2, 229-237, 2007 | ||||||||||||||||||