This homepage addresses the three Millennium problems
A. The Riemann Hypothesis (RH)
building on a commonly applied mathematical framework. The proposed solutions are building on common mathematical solution concepts and tools, e.g. Hilbert scale,
Hilbert (resp. Riesz) transform(s), Hilbert-Polya conjecture. The proposed
mathematical framework is also suggested to build a unified quantum field and gravity field theory, while using the proposed mathematical framework to overcome open mathematical challenges of the
D. Loop Quantum Theory (LQT)
Solution(s) concept(s) walkthrough
In order to prove the Riemann Hypothesis (RH) the Polya criterion can not be applied in combination with the Müntz
formula ((TiE) 2.11).
This is due to the divergence of the Müntz formula in the critical
stripe due to the asymptotics behavior of the baseline
function, which is the Gaussian function. The conceptual challenge (not only in this specific case) is about the not vanishing constant Fourier term of the Gaussian function and its related impact with respect to the Poisson summation formula. The latter formula applied to the Gaussian function leads to the Riemann duality equation ((EdH) 1.7). A proposed alternative "baseline" function than the Gaussian function, which is its related Hilbert transform, the Dawson function, addresses this issue in an alternative way as Riemann did. As the Hilbert transform is a convolution integral in a correspondingly defined distributional Hilbert space
frame it enables the Hilbert-Polya conjecture (e.g. (CaD)). The corresponding distributional ("periodical") Hilbert space framework, where the Gaussian / Dawson functions are replaced by the fractional part / log(2sin)-functions enables the Bagchi reformulation of the Nyman-Beurling RH criterion.
The corresponding formulas, when replacing the Gaussian function by its Hilbert transform, are well known: the Hilbert transform of the Gaussian is given by the Dawson integral (GaW). Its properties are e.g. provided in ((AbM) chapter 7, (BrK4) lemma D1). The Dawson function is related to a special Kummer function in a similar form than the (error function) erf(x)-function resp. the li(x)-function ((AbM) (9.13.1), (9.13.3), (9.13.7). A characterization of the Dawson function as an sin-integral (over the positive x-axis) of the Gaussian function is given in ((GrI) 3.896 3.). Its Mellin transform is provided in ((GrI) 7.612, (BrK4) lemma S2). The asymptotics of the zeros of those degenerated hypergeometric functions are given in (SeA) resp. ((BrK4) lemma A4). The fractional part function related Zeta function theory is provided in ((TiE) II).
With respect to the considered distributional Hilbert
spaces H(-1/2) and H(-1) we note that the Zeta function is an integral
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 Dirichlet series theory (HaG) and the Hardy space
isometry as provided in e.g. ((LaE), §227, Satz 40). With respect to the
physical aspects below we refer to (NaS), where the H(1/2) dual space
of H(-1/2) on the circle (with its inner product defined by a Stieltjes
integral) is considered in the context of Teichmüller theory and the
universal period mapping via quantum calculus. For the corresponding Fourier series analysis we refer to ((ZyA) XIII, 11). The approximation by polynomials in a complex domain leads to several notions and theorems of convergence related to Newton-Gaussian and cardinal series. The latter one are closely connected with certain aspects of the theory of Fourier series and integrals. Under sufficiently strong conditions the cardinal function can be resolved by Fourier's integral. Those conditions can be considerably relaxed by introducing Stieltjes integrals resulting in (C,1) summable series ((WhJ1) theorems 16 & 17, (BrK4) remarks 3.6/3.7).
The RH is connected to the quantum theory via the Hilbert-Polya conjecture resp. the Berry-Keating conjecture. It is about the hypothesis, that the imaginary parts t of the zeros 1/2+it of the Zeta function Z(t) corresponds to eigenvalues of an unbounded self-adjopint operator, which is an appropriate Hermitian operator basically defined by QP+PQ, whereby Q denotes the location, and P denotes the (Schrödinger) momentum operator. The notion "unbounded"
is not clearly defined (the assumption is, that it is related to the L(infinity) resp. C(0) Banach space), as an operator is only well-defined by describing
the operator "mapping" in combination with its defined domain.The proposed fractional Hilbert scales above enables a bounded Hermitian operator definition with appropriately defined domain (but still unbounded in the above Banach spaces due to the Sobolev embedding theorem, see also below)). The corresponding Hilbert scale framwork plays also a key role on the inverse problem for the double layer potential. The corresponding model problem (w/o any compact disturbance operator) with the Newton kernel enjoys a double layer potential integral operator with the eigenvalue 1/2 (EbP).
The alternatively proposed Dawson (baseline) function leads to an
alternative entire Zeta function definition Z(*;s), enjoying same zeros as Riemann's entire Riemann
Zeta function, enjoying the same term (1-s) (which defines the li(x)-function, the critical term
of Riemann's density function J(x), ((EdH) 1.14) when
calculating the Fourier inverse of the original Zeta function ((EdH),
(1.12) ff). The density function J(x) can be reformulated into a
representation of the function pi(x), that is, for the number of primes
less than any given magnitude x ((EdH) 1.17). The challenge of the corresponding li(x) function approximation criterion
(i.e. li(x) - pi(x)=O(log(x)*squar(x)) = O(x*exp(1/2+e)),
e>0, (BrK4) p.10) is about the (exponential) asymptotics of the Gaussian function
((EdH) 1.16, (BrK4) note S25). The Dawson function enjoys an only polynomial asymptotics
in the form O(x*exp(-1)).
The Riemann entire Zeta function Z(s) enjoys the functional equation in the form Z(s)=Z(1-s). The RH is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of Z(s) evaluated at unity (LiX). This equivalence results from a necessary and sufficient condition that the logarithmic of the function Z(1/(1-z)) be analytic in the unit disk. The proof of the Li criterion is built on the two representations of the Zeta function, its (product) representation over all its nontrivial zeros ((HdE) 1.10) and Riemann's integral representation derived from the Riemann duality equation, based on the Jacobi theta function ((EdH) 1.8). Based on Riemann's integral representation involving
Jacobi's theta function and its derivatives in (BiP) some particular
probability laws governing sums of independent exponential variables are
considered. In (KeJ) corresponding Li/Keiper constants are considered. The proposed alternative entire Zeta function Z(*,s) is suggested to derive an analogue Li criteron. Its counterpart to the Riemann functional equation is given by the relationship Z(*,1-s) = Q(s) * Z(*,s), with Q(s):=P(s)/P(1-s), whereby P(x):=cx*cot(cx) and the constant c denotes the number "pi"/2.
One proof of the Riemann functional equation is based on the fractional part function r(x), whereby the zeta function zeta(s) in the critical stripe is given by the Mellin transform zeta(1-s) = M(-x*d/dx(r(x))(s-1) ((TiE) (2.1.5). The functional equation is given by zeta(s) = chi(s)*zeta(1-s), whereby chi(s) is defined according to ((TiE) (2.1.12). The Hilbert transform of the fractional part function is given by the log(sin(x))-function. After some calculations (see also (BrK4) lemma 1.4, lemma 3.1 (GrI) 1.441, 3.761 4./9., 8.334, 8.335) the corresponding alternative zeta(*,s) function is given by zeta(*,1-s) * s = zeta(1-s) * tan(c*s).
The above indicates an alternative Gamma function G(*,s/2):=G(s/2)*tan(cs)/s with identical asymptotics for x
--> 0. It is proposed to revisit Riemann's proof of the formula for J(x) and its related fomula for the prime number counting function, resulting into the famous Riemann approximation error function ((HdE) 1.17 (3)) based on the product formula representation of the Gamma function ((HdE) 1.3 (4), (GrI) 8.322). Corresponding formulas for the tan(x)- resp. the log(tan)-function are provided in ((GrI) 1.421,1.518). In (ElL) the Fourier expansion of the log(tan) function is provided, giving another note to the Hilbert space H(-1). In other words, the alternatively proposed Gamma G(*,s/2) function leads to an alternative Riemann approximation error function with improved convergence behavior (at least with respect to the proposed Hilbert space norms). For other related areas we refer to Ramanujan's chapter "Analogues of the
Gamma Function" ((BeB chapter 8).
In (TiE) theorem 4.11, an approximation to the zeta
function series in the critical stripe by a partial sum of its Dirichlet
series is given ((BrK4) remark 3.8). One proof of this theorem is built on a simple
application of the theorem of residues, where the zeta series is
expressed as a (Mellin transform type) contour integral of the
cot(cz)-function ((TiE) 4.14). As the cot and the zeta function are both elements of the distributional Hilbert space H(-1) the contour integral above with a properly chosen contour provides a contour integral representation for the zeta in a weak H(-1) sense. In (ChK) VI, §2, two expansions of cot(z) are compared
to prove that all coefficients of one of this expansion
(zeta(2n)/pi(exp(2n))) are rational. Corresponding formulas for odd
inters are unknown. In (EsR), example 78, a "finite part"-"principle
value" integral representation of the c*cot(cx) is given (which is zero
also for positive or negative integers). It is used as enabler to obtain
the asymptotic expansion of the p.v. integral, defined by the
"restricted" Hilbert transform integral of a function u(x) over the
positive x-axis, only. In case u(x) has a structure u(x)=v(x)*squar(x)
the representation enjoys a remarkable form, where the numbers n+1/2
play a key role.
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 RH Polya criterion ((PoG), (BrK5) theorem 6). The above quasi-asymptotics indicates a replacement of the differential d(logx)by d(log(sinx)). The cot(z) function expansions (ChK) VI, §2) in combination with Ramanujan's formula ((EdH) 10.10) resp. its generalization theorem ((EdH) p.220) is proposed to be applied to define an appropriate analytical (Mellin transform) function in the stripe 1/2<Re(s)<1.
In (GrI) 8.334, the relationship between the the cot-
and the Gamma function is provided. From (BeB) 8. Entry17(iii)) we
quote: "the indefinite Fourier series of the
cot(cx)-function may be formally established by differentiating the
corresponding Fourier series equation for (the L(2)=H(0)-function) -log(2sin(cx))"
((BrK4) remark 3.8). The proposed distributional Hilbert scales provide
the proper framework to justify Ramanujan's related parenthetical
remark "for the same limit" (in a H(-1)-sense).
With respect to the NSE and the YME the proposed mathematical concepts and tools are especially correlated to the names of Plemelj, Stieltjes and Calderón. The essential estimate for the critical non-linear
term of the non-linear, non-stationary 3-D NSE has been provided by
Sobolevskii. With respect to the YME the proposed mathematical
concepts and tools are especially correlated to the names of Schrödinger
and Weyl (e.g. in the context of "half-odd integers quantum numbers for
the Bose statistics" and resp. Weyl's contributions on the concepts of matter, the structure of the world and the principle of action (WeH), (WeH1), (WeH2)). It
enables an alternative (quantum) ground state energy model embedded
in the proposed distributional Hilbert scale frame of this
homepage covering all variational physical-mathematical PDE and Pseudo Differential Operator (PDO) equations (e.g. also the Maxwell equations).
The Dirac theory with its underlying concept of a "Dirac function" (where the regularity of the Dirac distribution "function" depends from the space dimension) is omitted and replaced by a distributional Hilbert space (domain) concept. This alternative concept avoids space dimension depending regularity assumptions for (quantum) physical variational model (wave package) states and solutions (defined e.g. by energy or operator norm minimization problems) and physical problem specific "force" types.
The until today not successfull attempts to define a quantum gravity model is about dynamics models coupling
gravity + matter system, simply defined by adding the terms defining the
matter dynamics to the corresponding field related (i.e. Dirac+Yang-Mills+Higgs+Einstein) hamiltonians ((RoC)
7.3)). The best case result (which is unlikely to be achieved anyway) is then about a "four different forces" model (not only four different force type phenomena) governed by the same "energy" based on corresponding energy least action principles (whereby only "the least action principle in his most modern general public is a maxime of Kant's reflective judgment"). The loop quantum theory (LQT) (C. Rovelli) is the choice of a different algebra of basic field functions: a
noncanonical algebra based on the holonomics of the gravitational
connections ((RoC) 1.2.1). The holonomy (or the "Wilson
loop") is the matrix of the parallel transport along a closed curve and spacetime itself is formed by loop-like states.
Therefore the position of a loop state is relevant only with respect to
other loops, and not with respect to the background. The state
space of the theory is a separable Hilbert space spanned by loop states,
admitting an orthogonal basis of spin network states, which are formed
by finite linear combinations of loop states, and are defined precisely
as the spin network states of a lattice Yang-Mills theory."
proposed distributional quantum state H(-1/2) with corresponding inner product admits and requires
infinite linear combinations of LQT "loop states" (which we "promoted" becoming "quantum fluid/quantum element/"truly orthogonal fermion & boson/rotating differential/ideal point/monad" states), i.e. it overcomes the current challenge of LQT
defining the scalar product of the spin network state Hilbert space
((RoC) 7.2.3). The physical LQT (kinematical) space (which is a quantum superposition of the QLT "spin networks") corresponds to an orthogonal projection of H(-1/2) onto H(0).
This othogonal projection can be interpreted as a general model for a "spontaneous symmetry break down".
In the following we briefly sketch the conceptual common solution elements motivating our terminology of a "common
Hilbert space framework" to solve the three considered Millenium
problems (and a few other related and considered ones).
1. The common Hilbert scale frame & its corresponding common solution idea
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 ((u,v)), (u,v), (u,v), ((u,v)), (((u,v))).
The RH is connected to the quantum theory via the Hilbert-Polya conjecture resp. the Berry-Keating conjecture. The latter one is about a physical reason, why the RH should be true. This would be the case if the imaginary parts t of the zeros 1/2+it of the Zeta function Z(t) corresponds to eigenvalues of an unbounded self-adjopint operator, which is an appropriate Hermitian operator basically defined by QP+PQ, whereby Q denotes the location, and P denotes the (Schrödinger) momentum operator. The notion "unbounded" is not well defined, as an operator is only well-defined by describing the operator "mapping" in combination with its defined domain. The Zeta function is an 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(#) 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.
The Gaussian function f(x) plays a key role in the
Zeta function theory, as well as in the quantum theory. Its Mellin
transform defines the factor function between the Zeta function and its
corresponding entire Zeta function, which builds the Riemann duality
equation. The Riemann duality equation involves an inner product
which is natural with respect to the additive structure of R(+), namely
d(log(x))=dx/x, rather than the multiplicative structure, namely the
L(2)=H(0) inner product. This structure jeopardizes all attempts so far
to represent the entire Zeta function as convolution integral, which
would prove the RH. There is a (formally only!!) self-adjoint operator
with transform being the entire Zeta function, but in fact this operator
transform at all, as the corresponding integral representation does not
converge for any complex s ((EdH), 10.3). The root cause is related to
the Poisson summation formula in combination with the fact that the
term of the Gaussian function is not vanishing. Riemann overcame this
challenge by replacing the Gaussian function f(x) by the product of the
variable "x" and its first derivative, i.e. x -->
h(x):=x*d/dx(f(x)). The corresponding Mellin transform "effect" is about a
multiplication with the factor -s, i.e it does not effect the factor
(s-1), which is the counterpart of the Li-function.
The central idea of our alternative approach is "just" to alternatively
replace the Gaussian function by its Hilbert transform (which is the Dawson function). Considering this
in a weak H(0) variational representation ensures that eigenvalues of
correspondingly defined convolution integral operator are identical to
the zeros of the entire Zeta function (as in a weak L(2) sense every
L(2) function g is identical to its Hilbert transform). In quantum theory
this goes along with an analysis of signals on R filtered by the
Hilbert operator (ZhB). The corresponding analysis with signals on T=R/Z then leads to a replacement of the Gaussian function by the fractional part function and its related Hilbert transform log(2sin)-function, which is linked to the cot-function, building the kernel function of the Hilbert transform for
periodic functions. We note that for signals on R the spectrum of the
Hilbert transform is (up to a constant) given by the distribution
v.p.(1/x), whereby the symbol "v.p." denotes the Cauchy principal value
of the integral over R. Its corresponding Fourier series is given by
-i*sgn(k) with its relationship to "positive" and "negative" Dirac
"functions" and the unit step function Y(x). In a H(-1/2) framework the Dirac "function" concept can be avoided, which enables a generalization to
dimensions n>1 without any corresponding additional regularity
requirements (the Dirac "function" is an element of H(-n/2-e), e>0).
"workaround" function h(x) do have an obvious linkage to the
"commutator" concept in quantum
theory. In this context the Gaussian function can be characterized as
"minimal function" for the Heissenberg uncertainty inequality. Applying
the same solution concept as above
then leads to an alternative Hilbert operator based representation in
H(-1/2), resp. to a H(-1) based definition of the commutator operator
with extended domain. The common denominator of the alternatively proposed Hilbert space framework H(-1/2) goes along with the definition of a correspondingly defined "momentum" operator (of order 1) P: H(1/2) --> H(-1/2) defined in a variational form. In the one-dimensional case the Hilbert transform H (in the n>1 case the Riesz operators R) is linked to such an operator given by ((Pu,v))=(Hu,v). With respect to quantum theory this indicates
an alternative Schrödinger momentum operator (where the
gradient operator "grad" is basically replaced by the Hilbert
transformed gradient, i.e. P:=-i*R(grad) and a corresponding alternative
commutator representation QP-PQ in a weak H(-1/2) form. We note that the Riesz operators R commute with
translations and homothesis and enjoy nice properties relative to
Conceptually, dealing with the isometric mapping Hilbert transform instead of a second order operator in the form x*P(g(x)) (or the commutator (P,Q)) goes along with a few other opportunities. For example, it enables a correspondingly defined variational representation of the Maxwell equations in a vaccum, whereby its solutions 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. At the same point in time it enables alternative concepts in GRT regarding concepts like current (flexible") metrical affinity, affine connexions and local isometric 3D unit spheres dealing with rigid infinitesimal pieces, being replaced by geometrical manifolds, enabling isometrical stitching of rigid infinitesimal pieces ((CiI), (ScP)).
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 "microscope observations" of two wavelet (optics) functions can be compared with each other (LoA). The prize to be paid is about additional efforts, when re-building the reconstruction wavelet.
In SMEP (Standard Model of Elementary Particles) symmetry plays a key role. Conceptually, the SMEP starts with a set of fermions (e.g. the electron in quantum electrodynamics). If a theory is invariant under transformations by a symmetry group one obtains a conservation law and quantum numbers. Gauge symmetries are local symmetries that act differently at each space-time point. They automatically determine the interaction between particles by introducing bosons that mediate the interaction. U(1) (where probability of the wave function (i.e. the complex unit circle numbers) is conserved) describes the elctromagnetic interaction with 1 boson (photon) and 1 quantum number (charge Q). The group SU(2) of complex, unitary (2x2) matrices with determinant I describes the weak force interaction with 3 bosons (W(+), W(-), Z), while the group SU(3) of complex, unitary (3x3) matrices describes the strong force interaction with 8 gluon bosons.
With respect to the open Millenium 3D non-stationary, non-linear NSE problem we note that the alternatively proposed "fluid state" Hilbert space H(-1/2) with corresponding alternative energy ("velocity") space H(1/2) enables a (currently missing) energy inequality based on existing contribution of the non-linear term. In the standard weak NSE representation this term is zero, which is a great thing from a mathematical perspective, avoiding sohisticated estimating techniques, but a doubtful thing from a physical modelling perspective, as this term is the critical one, which jepordized all attempts to extend the 3D problem based on existing results from the 2D case into the 3D case. The corresponding estimates are based on Sobolev embedding theorems; the Sobolevskii estimate provides the appropriate estimate given that the "fluid state" space is H(-1/2) in a corresponding weak variational representation.
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 generalization of the Maxwell
equations, can be derived by forming a gauge-invariant field tensor
using generalized derivative. There is a parallel to the definition of
the covariant derivative in general relativity. With respect to the
above there is an alternative approach indicated, where the fermions are
modelled as elements of the Hilbert space H(0), while the complementary
closed subspace H(-1/2)-H(0) is a model for the "interaction particles,
bosons". For gauge symmetries the fundamental equations are symmetric,
but e.g. the ground state wave function breaks the symmetry. When a gauge
symmetry is broken the gauge bosons are able to acquire an effective
mass, even though gauge symmetry does not allow a boson mass in the
fundamental equations. Following the above alternative concept the
"symmetry state space" is modelled by H(0), while the the ground state
wave function is an element of the closed subspace H(-1/2)-H(0) of
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.
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 geometrodynamics provides alternative (pseudo) tensor operators to the Weyl tensor related to H3 (CiI). In (CaJ) the concept of a Ricci potential is provided in the context of the Ricci curvature equation with rotational symmetry. The single scalar equation for the Ricci potential is equivalent to the original Ricci system in the rotationally symmetric case when the Ricci candidate is nonsingular.For an overview of the Ricci flow regarding e.g. entropy formula, finite extinction time for solutions on certain 3-manifolds in the context of Prelman's proof of the Poincare conjecture we refer to (KlB), (MoJ).
The single scalar equation for the Ricci potential
(CaJ) might be interpreted as the counterpart of the CLM vorticity equation as a
simple one-dimensional turbulent flow model in the context of the NSE.
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 Hermitean operators in spaces with indefinite metric ((VaM) IV). For x being an element of H this is about a defined "potential" p(x):=<<x>>*<<x>> ((VaM) (11.1)) and a corresponding "grad" potential operator W(x), given by
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.
2. An integrated substance & field least action functional framework for elementary particles and gravity "forces" phenomena
The central mathematical concepts of the GRT are differentiable manifolds, affine connexions with the underlying covariant derivative definition on corrresponding tangential (linear) vector spaces. Already the "differentiability" condition is w/o any physical justification. The only "affine" connexion concept and its corresponding locally defined metrical space framework jeopardizes a truly infinitesimal geometry, which is compatible with the Hilbert space framework of the quantum theory and the proposed distributional Hilbert space concept in (BrK). In sync with the above we propose a generalized Gateaux differential operator:
let H(1/2) = H(1) + H(*) denote the orthogonal decomposition of the alternatively proposed "energy/momentum/velocity" Hilbert space, whereby H(1) denotes the (compactly embedded) standard energy space with its inner product, the Dirichlet integral; "lim" denotes the limes for t --> 0 for real t. Then for x,y ex H(1/2) the operator VF(x,y) is defined by VF(x,y):=lim((F(x+t*y)-F(x))/t), whereby the limes is understood in a weak H(-1/2) sense. The operator is homogeneous in y; however, it is not always a linear operator in y ((VaM) 3.1).
The main tools used in geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing space-time. A Lorentz manifold L is likewise equipped with a metric tensor g, which is a nondegerated symmetric bilinear form on the tangential space at each point p of L. The Minkowski metric is the metric tensor of the (flat space-time) Minkowski space.
The least action principle can refer to the family of variational principles. The most popular among these is Hamilton's principle of least action. It states that the action is stationary under all path variations q(t) that vanishes at the end points of the path. It does not not strictly imply a minimization of the action.
The least action principle can be also seen as THE fundamental principle to develop laws of nature in strong alignment with Kant's philosophy: ((KnA), p. 55, p. 56): (translated) "the least action principle in his most modern general public is a maxime of Kant's reflective judgment. ... Offenbar haben wir beim Energieprinzip eine typische Entwicklung vor uns: wenn das Prinzip der reflektierenden Urteilskraft mit einer seiner Maximen vollen Erfolg gehabt hat, rückt sein Ergebnis aus dem Reich der Vernunft im Kantischen Sinne, zu welchem die reflektierende Urteilskraft gehört, in die Sphäre des Verstandes herab und wird zum allgemeinen Naturgesetz (law of nature)".
The Einstein-Hilbert action functional W(g) is about the scalar curvature S=scal (which is the Ricci scalar of the Ricci tensor "Ric") applied to the metric tensor g. It is the simplest curvature invariant of a Riemannian manifold. The scalar curvature is the Lagrangian density for the Einstein-Hilbert action. The stationary metrics are known as Einstein metrics. The scalar curvature is defined as the trace of the Ricci tensor. We note that the trace-free Ricci tensor for space-time dimension n=4 is given by Z(g):=Ric(g)-(1/4)*S(g)*g, and that Z vanishes identically if and only if Ric = l*g for some constant l. In physics, this equation states that the manifold (M,g) is a solution of Einstein's vacuum field equations with cosmological constant. We further note, that the Ricci tensor corresponds to the Laplacian operator multiplied by the factor (-1/2) plus lower order terms.
The Einstein-Hilbert functional is an invariant integral, which is a must to describe the field-action of graviation ((WeH), §28). From a physical perspective a field-action term should be based on a scalar density G, which is composed of the potentials g(i,k) and of the field-components of the gravitation field (which are the first derivatives of the g(i,k,), i.e., g(i,k);r): "it would seem to us that only under such circumstances do we obtain differential equations of order not higher than the second for our gravitation laws .... Unfortunately a scalar density G, of the type we wish, does not exist at all; for we can make all g(i,k);r vanish at any given point choosing the appropriate co-ordinate system. Yet, the scalar R, the curvature defined by Riemann, has made us familiar with an invariant which involves the second derivatives of the g(i,k)'s only lineary. ... In consequence of this linearity we may use the invariant integral (the Einstein-Hilbert functional) to get the derivatives of the second order by partial integration. ... We then get a sum of a truly field-action functional (with a scalar density G) plus a divergence integral, that is an integral whose integrand is of the form div(w). Hence for the corresponding variations of theh potential functions g(i,k) the variations of both funtionals are identical; therefore the replacement of the physically required scalar density G by the integrand of the W(g) is justified (as the essential feature of the Hamilton's principle is fulfilled with W(g))." This is where a alternative field-action functional of gravitation in a alternative framework (as proposed above) can be defined, based on a "scalar density" function in a "Plemelj" (Stieltjes integral) sense.
The electromagnetic field is built up from the co-efficients of an invariant linear differential form. The potential of the gravitational field is made up of the co-efficients of an invariant quadratic differential form. Replacing the Newtonian law of attraction by the Einstein theory is about discovering the invariant law of gravitation, according to which matter determines the components of the graviation fields. The topic of the chapter above is about the substance-action and the field-action of electricity and gravitation in the context of the least action principle. The substance-action is based on the mathematical concept "density", while the field action is based on the mathematical concept "potential (function)". The substance-action related "tensor density" of electricity can be easily extented to the substance-action related "tensor density" of gravitation ((WeH) §28). A corresponding field-action of gravitation based on an invariant integral and an approporate potential "scalar density" is not possible from a mathematical perspective, as by choosing the appropriate co-ordinates the field components of the gravitational field vanish. The alternatively proposed aproach of this page can be summarized as follows:
- replacing of the mathematical "density" concept by
Plemelj's "mass element" concept, which goes along with an alternative (more
general) "potential" function concept
- replacing the manifold concept by a (semi) Hilbert
space-based concept, where a non-linear invariant integral functional F(V(g)) is defined by a
distributional (semi-) inner product, which is equivalent to a corresponding
functional F(R(g)) of a related inner product (where R denotes the Riesz
operators (which commute with translations & homothesis having nice
properties relative to rotations)) plus a (non-linear) compact disturbance term; the concept enables variational methods of nonlinear operators based on Stieltjes and curvilinear integrals (VaM).
The Yang-Mills functional is of similar structure than the Maxwell functional regarding the underlying constant fundamental tensor. The field has the property of being self-interacting and equations of motions that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. The YME mass gap problem is about the energy gap for the vacuum state. Therefore, the above proposed model alignments for the "electricity & gravity forces" phenomena covers also the cases of the "weak & strong nuclear forces" phenomena.
To merge two inconstent theories requires changes on both sides. In the above case this is about a newly proposed common "mass/substance element" concept, alternatively to the today's "mass density" concept, while, at the same time, the linear algebra tensor tool (e.g. a "density" tensor) describing classical PDE systems is replaced by non-linear operator equations defined by weak (variational) functional systems. Those (weak) equations provides the mathematical model of physical phenomena, while its correspondin classical PDE systems (requiring purely mathematical additional regularity assumptions) are interpreted as approximation solutions, only.
As a shortcut reference to geometrodynamics is
given by (WhJ). For a review of discoveries in the nonlinear dynamics of curved spacetime, we refer to ((ScM). An introduction to the foundations and tests of
gravitation and geometrodynamics or the meaning and origin of inertia in
Einstein theory is provided in (CiI).
In ((CiI) 4.6) the Gödel model universe is discussed, which is a four-dimensional model universe, homogeneous both in space and time, which admits the whole four-dimensional simply transitive group of isometries, in other words, a space-time that admits all four "simple translations" as independent Killing vectors. As the Gödel model universe is homogeneous both in space and time it is stationary. In other words, in this model the cosmological fluid is characterized by zero expansion and zero shear. Thus the Gödel model runs into difficulty with the expansion of the universe.
"The specification of the relevant features of a
three-geometry and its time rate change on a closed (compact and without
boundary manifolds), initial value, space-like hypersurface, together
with the energy density and density of energy flow (conformal) on that
hypersurface and together with the expansion of the equation of state of
mass-energy, determines the entire space-time geometry, the local
inertial frames, and hence the inertial properties of every test
particle and every field everywhere and for all time."
The related clarifcations regarding the distortion tensor or gravitomagnetic field is provided in ((CiI) §5.2.6, § 5.2.7).
The Laplacian equation for the gravitomagnetic vector
potential W, in terms of the current J of mass-energy is discussed in
((CiI) 5.3). The Neumann problem and its related integral equations with double layer potential leads to the Prandtl operator, defining a well posed integral equation in case of domain H(1/2) with range H(-1/2) ((LiI) theorem 4.3.2).
whereby (((*,*))) defines the H(-1) inner product and
((*,*)) defines the H(-1/2) inner product of the corresponding Hilbert
scales building on the eigen-pair solutions of the Prandtl operator
equation with domain H(1/2).
The proposed alternative Hilbert space based framework provides
also a "variational wave equation/ function" based approach of the "evolution of
geometric structures on 3-manifolds" in the context of Thurston's
"geometrization conjecture" and its underlying Poincare conjecture (which have been established by Perelman),
where the Ricci flows play a central conceptual solution element to build "nice behavior" metrics in manifolds.
"The hypothesis that the universe is infinite and Euclidean at infinity, is, from a relativistic point of view, a complicated hypothesis. In the language of the general theory of relativity it demands that the Riemann tensor of the fourth rank shall vanish at infinity, which furnishes twenty independent conditions, while only ten curvarture components enter the laws of the gravitational field. It is certainly unsatisfactory to postulate such far-reaching limitation without any physical basis for it.
The possibility seems to be particularly satisfying
that the universe is spatially bounded and thus, in accordance with our
assumption of the constancy of the mass-energy density, is of constant
curvature, being either spherical or elliptical; for then the boundary
conditions at infinity which are so inconvenient from the standpoint of
the general theory of relativity, may be replaced by the much more
natural conditions for a closed surface" ((CiI) 5.2.1)
The wave equation can be derived from the Maxwell equations by applying the rot-operator. It results into the "light" phenomenon. A similar tranformation is not possible for Einstein equations, which results into the "gravitation" phnomenon. The "approximation" approach is about the split g(i,k)=m(i,k)+h(i,k), where m(i,k) denotes the flat Minkowski metric. The perturbance term h(i,k) admits a retarded (only) potential representation, representing a gravitational perturbance propagating at the speed of light ((CiI) 2.10). An alternative splitting with defined distortion tensor enabling an analogue approach with electrodynamics is provided in ((CiI) 5.2.7).
In ((CiI) (2.7.10)) an „energy-momentum pseudotensor for the gravity field“ is introduced representing the energy and momentum of the gravitation field. Then, using the corresponding "effective energy-momentum pseudotensor for matter, fields and gravity field", in analogy with special relativity and electromagnetism, the conserved quantities on an asymptotically flat spacelike hypersurface are defined by the sum of four-momentum, energy and angular momentum operators (2.7.19-21). Following an analogue approach, which lead to the modified Maxwell equation (as proposed in the above paper), leads to an alternative effective energy-momentum tensor for matter, fields and gravity field". As the Einstein (gravity) tensor is derived from the condition of a divergence-free energy-momentum tensor, this results to an alternative Einstein tensor. The additional term of this alternative Einstein tensor could be interpreted as "cosmologic term", not to ensure a static state of the universe (which is not the case due to Hubbles observations), but to model the "vacuum energy" properly. This then would also be in sync with the physical interpretation of the corresponding term in the modified Maxwell equations with its underlying split of divergence-free and rotation-free tensors. At the same point in time the approach avoids the affine connexion concept and the "differentiable" manifolds regularity requirement, which is w/o any physical justification.
There are eight 3-dimensional geometries in the context of "nice"
metrics. The nicest metrics are those with a constant curvature, but
there are other ones. Their classification in dimension three is due to
In (GrJ) philosophical aspects of the geometrodynamics are considered. We quote from the cover letter summary:
The above questions concerning
singularities and non-geometric manifolds can be revisited based on the above alternative conceptual framework; the corresponding physical interpretation of the geometrodynamics are in line with Schrödinger's vision (resp. critique about the common handicap of all western philosophy baseline assumptions, propagating instead a purely monoism) of a truly quantum field theory (see also www.quantum-gravitation.de).
In (CoR) there is a
conjecture formulated, that distortion-free families of progressing, spherical waves
of higher order exist if and only if the Huyghens’ principle is valid, and that
families of spherical, progressing waves only exist for space-time dimension
n=2 and n=4 ((CoR) VI, §10.2, 10.3). In combination with Hadamard conjecture
(that the wave equations for even space-time dimension are the only partial
differential equations, where the Huyghens’ principle is valid) this would lead
to an essential characterization of the four-dimension space-time space with
its underlying Maxwell field theory.
We mention that the existing
electromagnetic phenomena on earth are the result of plasma physics phenomena
underneath the earth crust. Those “activities” are all triggered by gravitation "forces".
The above (distributional) Hilbert space based
alternative geometrodynamic modelling framework provides an alternative
approach to Penrose's "cycles of time" concept of a "conformal cyclic cosmology",
addressing e.g. the "collapsing of matter" of an over-massive star to a
black hole problem (PeR) and "the problem of time" (AnE).
"What characterizes the loop quantum theory (LQT) is the choice of a different algebra of basic field functions: a noncanonical algebra based on the holonomics of the gravitational connections ((RoC) 1.2.1). The holonomy (or the "Wilson loop") is the matrix of the parallel transport along a closed curve. ... In LQT, the holonomy becomes a quantum operator that creates "loop states" (to overcome the issue of current dynamics model of coupled gravity + matter system, simply defined by adding the terms defining the matter dynamics to the gravitational relativistic hamiltonian ((RoC) 7.3)). ... Spacetime itself is formed by loop-like states. Therefore the position of a loop state is relevant only with respect to other loops, and not with respect to the background. ... The state space of the theory is a separable Hilbert space spanned by loop states, admitting an orthogonal basis of spin network states, which are formed by finite linear combinations of loop states, and are defined precisely as the spin network states of a lattice Yang-Mills theory." The proposed distributional quantum state H(-1/2) above admits and requires infinite linear combinations of those "loop states" (which we call "quantum fluid" state), i.e. overcomes the current challenge of LQT defining the scalar product of the spin network state Hilbert space ((RoC) 7.2.3). The physical space is a quantum superposition of "spin networks" in LQT corresponds to an orthogonal projection of H(-1/2) onto H(0). This othogonal projection can be interpreted as a general model for a "spontaneous symmetry break down".
4. Plasma physics, Maxwell equations & non-linear Landau damping
Plasma is the fourth state of matter, where from general relativity
and quantum theory it is known that all of them are fakes resp. interim
specific mathematical model items. An adequate model needs to take into account
the axiom of (quantum) state
(physical states are described by vectors of a separable Hilbert space H)
and the axiom of observables (each
physical observable A is represented as a linear Hermitian operator
the state Hilbert space). The corresponding mathematical model and its
solutions are governed by the Heisenberg uncertainty inequality. As the
observable space needs to support statistical analysis the
Hilbert space, this Hilbert space needs to be at
least a subspace of H.
At the same point in time, if plasma is considered as
sufficiently collisional, then it can be well-described by fluid-mechanical
equations. There is a hierarchy of such hydrodynamic models, where the magnetic
field lines (or magneto-vortex lines) at the limit of infinite conductivity is
“frozen-in” to the plasma. The “mother of all hydrodynamic models is the continuity equation treating observations
with macroscopic character, where fluids and gases are considered as continua.
The corresponding infinitesimal volume “element” is a volume, which is small
compared to the considered overall (volume) space, and large compared to the
distances of the molecules. The displacement of such a volume (a fluid
particle) then is a not a displacement of a molecule, but the whole volume
element containing multiple molecules, whereby in hydrodynamics this fluid is
interpreted as a mathematical point.
The common distributional Hilbert space framework is also proposed for a proof of the Landau damping alternatively to the approach from C. Villani. Our approach basically replaces an analysis of the classical (strong) partial differential (Vlasov) equation (PDE) in a corresponding Banach space framework by a quantum field theory adequate (weak) variational representation of the concerned PDE system. This goes along with a corresponding replacement of the “hybrid” and “gliding” analytical norms (taking into account the transfer of regularity to small velocity scales) by problem adequate Hilbert space norms H(-1/2) resp. H(1/2). The latter ones enable a "fermions quantum state" Hilbert space H(0), which is dense in H(-1/2) with respect to the H(-1/2) norm, and its related (orthogonal) "bosons quantum state" Hilbert space H(-1/2)-H(0), which is a closed subspace of H(-1/2).
The Boltzmann equation
is a (non-linear) integro-differential equation which forms the basis for the
kinetic theory of gases. This not only covers classical gases, but also
electron /neutron /photon transport in solids & plasmas / in nuclear
reactors / in super-fluids and radiative transfer in planetary and stellar
atmospheres. The Boltzmann equation is derived from the Liouville equation for
a gas of rigid spheres, without the assumption of “molecular chaos”; the basic
properties of the Boltzmann equation are then expounded and the idea of model
equations introduced. Related equations are e.g. the Boltzmann equations for
polyatomic gases, mixtures, neutrons, radiative transfer as well as the
Fokker-Planck (or Landau) and Vlasov equations. The treatment of corresponding
boundary conditions leads to the discussion of the phenomena of gas-surface
interactions and the related role played by proof of the Boltzmann H-theorem.
The Landau damping (physical,
observed) phenomenon is about “wave
damping w/o energy dissipation by collisions in plasma”, because electrons
are faster or slower than the wave and a Maxwellian distribution has a higher
number of slower than faster electrons as the wave. As a consequence, there are
more particles taking energy from the wave than vice versa, while the wave is
damped. The (kinetic) Vlasov equation is collisions-less.
In fluid description of
plasmas (MHD) one does not consider velocity distributions. It is
about number density, flow velocity and pressure. This is about moment or fluid
equations (as NSE and Boltzmann/Landau equations). The corresponding situation
of the fluid flux of an incompressible viscous fluid leads to the Navier-Stokes
equations. They are derived from continuum theory of non-polar fluids with
three kinds of balance laws: (1) conservation of mass, (2) balance of linear
momentum, (3) balance of angular momentum.
The NSE are derived from
the (Cauchy) stress tensor (resp. the shear viscosity tensor) leading to liquid pressure force. In electrodynamics &
kinetic plasma physics the linear resp. the angular momentum laws are linked to
the electrostatic (mass “particles”, collision, static, quantum mechanics,
displacement related; “fermions”) Coulomb potential resp. to the magnetic (mass-less
“particles”, collision-less, dynamic, quantum dynamics, rotation related;
“bosons”) Lorentz potential.
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
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
The mathematical tool to
distinguish between unperturbed cold and hot plasma is about the Debye length
and Debye sphere. The corresponding interaction (Coulomb) potential of
the non-linear Landau damping model is based on the (Poisson) potential
equation with corresponding boundary conditions. A combined electro-magnetic
plasma field model needs to enable “interaction” of cold and hot plasma
“particles”, which indicates Neumann problem boundary conditions.
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