Riemann Hypothesis, Unified field theory, Euler-Mascheroni constant

www.unified-field-theory.de

A.Einstein, "For us believing in physicists, the distinction between past, present and future is only a stubbornly persistent illusion".

H. Weyl, "We are not surprised that a concrete chunk of nature, taken in its isolated phenomenal existence, challenges our analysis by its inexhaustibility and incompleteness; it is for the sake of completeness, as we have seen, that physics projects what is given onto the background of the possible", (WeH) p. 220.

The phenomenological physical world

About 95% of the universe is about the phenomenon „vacuum“. The same proportion applies to the emptyness between a proton and an electron. The remaining 5% of universe’s vacuum consists roughly of 5% matter, of 25% dark matter, and of 70% dark energy. Nearly all (about 99%) of the 5% matter in the universe is in "plasma state". The presumed existence of „dark matter“ provides the baseline for a physical model of the galaxies‘ spiral shapes phenomenon. The presumed existence of „dark energy“ provides the baseline for a physical model of the cosmic microwave background phenomenon (CMP).

The dominance of the vacuum (in the macro case, as well as in the micro case) in opposite to the small amount of matter is addressed by an appropriate Hilbert space based definition of an „ether/ground state energy“ space, which is „orthogonal“ to a „kinematical energy“ space.

A "kinematical energy" Hilbert space based two-type plasma particle model enables a corresponding MHD based two-type-plasma particles field interaction model governed by three kinds of balance laws:

1.    conservation of mass
2.    balance of angular momentum
3.    balance of linear momentum.

The phenomenon of spiral shapes (in the macro case, in our "human being" case, (BaA), (HaG), and, as well as in the micro case) is governed by a balance law of angular momentum.

The phenomenon of the cosmic microwave background radiation is governed by a balance law of linear momentum.

In (ArI) a quaternionic unification of electromagnetism and hydrodynamics is provided. „Employing quaternionic Newton’s law, it is shown that the energy conservation equation is the analog of Lorentz gauge in electromagnetism. This Newton`s law yields directly the Euler equation and other equations governing the fluid motion. With this formalism, the pressure contributes positively to the dynamics of the system in the same way mass does. Hydrodynamics equations are derived from Maxwell`s equations by adopting an electromagnetohydrodynamics analogy.“

In (LeS) the isomorphism between unitaty quaternions and space time rotations is extended to Lorentz boosts. From the transformation properties of two-component spinors a quaternionic representation for the space-time algebra is derived. Additionally, a quaternionic bi-dimensional version of the Dirac equation is derived.

A reversed two-stage de-quantification process

In classical theory but also in quantum theory, symmetry groups are applied to derive conservation laws for energy, translation and angular momenta. However, in the current quantum theory translation and rotation operators are not interchangeable, which is a consequence of the quantification process of classical partial differential equations. Therefore, in the current quantum theory framework in order to characterize the angular momentum of a system about an axis by a quantum number it is neccessary that the perpendicular translation momentum vanishes or is unknown, (DüH).

In the below proposed Hilbert scale and MHD based unified plasma, quantum and gravity field theory the „quantification process“ is reversed to the current one.

The most finest "world model" is built by the mathematical „quantum world reality", (UnA2), accompanied by an energy Hilbert space H(1/2); the 1st stage of the de-quantification process is based on the more granular physical „quantum world reality“ accompanied by the standard variational kinematical (physically speaking, a coarse grained embedded) energy Hilbert space H(1), which is (mathematically speaking, compactly) embedded into the overall (ground state energy containing) energy Hilbert space H(1/2).

The conservation laws in the Hilbert spaces H(1) and H(1/2) are described by variational (pseudo) differential equations enabled by the related inner products. The variational approximating PDE in the coarse grained kinematical Hilbert space are interpreted as the variational form of the classical (2nd stage) PDE.

The kinematical Hilbert space H(1) provides the framework to model translation and angular momenta.

The decomposition of the energy space H(1/2) into H(1) and its complementary closed sub-space of H(1/2) enables the definition of an inner product supporting the approach in (UnA2):

„... to form a consistent picture of reality by observing nature from cosmos to elementary particles, based on investigating constants of nature and questioning their origin, while searching for mathematical objects whose properties describe the various physical phenomena in purely mathematical terms“, (UnA2).

The 2nd stage of the de-quantification process leads to PDE like the classical Maxwell equations or formulas like F=m*a, U=R*I, dE=t*dS-p*dV. Those formulas are mathematically valid in case the variational H(1)-based representation of the PDE are equivalent to corresponding classical PDE; this is the case if appropriate regularity assumptions are made/fulfilled.

From a philosophical perspective there is a double origin question concerning the term "time", (OrE) S. 9:

- „our self-understanding and understanding of the world is based/originated on an understanding of time

- our perception of time raises the question of our understanding of time (i.e. the original sense of time) and its origin.

Either the „origin“ of time is a purely functional (structure) formula (like the differentiation between past, present and future), which is not originally or the supposedly original is only any paradigmatic form of time, which then is as such not original.“

We note that the two-stage de-quantification process is in line with Husserl’s concept of an „internal time consciousness“:

Husserl differentiates between, (1) an "objective time of appearing objects", (2) "a subjective or pre-empirical time of acts", and (3) "experiences and a pre-phenominal absolute flow of the internal time consciousness", (ZaD) chapter 3.

Husserl’s conception of an „internal time consciousness“ provides an explanation of „time“ as a paradigmatic form of perceived time. Heidegger’s later conception of „time“ is based on his conceptions of „being, understanding of being, historicity“, i.e. the origin of time is a functional (not orginal) formula. Therefore, he denied Husserl’s conception.

In (BoA) "the problem of time and time of consciousness at Edmund Husserl" is analysed, to understand the present as a unity between past, present and future. It is argued that both, objective time of nature and historical time are constituted in the immanent time of consciousness.

The consistency of Husserl’s „internal time consciousness“ theory and the proposed reversed two-stage de-quantification process provides another push back argument for Heidegger’s time conception.

The mathematical modelling framework

Braun, K., Quaternionic Hilbert spaces and the quarternion rotation operator

August 14, 2022

The main changes dressing the emperor

The two main changes to the today's inconsistent quantum field theory and the General Relativity Theory are:

Change 1

the role model of all current types of „elementary particles“ is Dirac's electron. Its conceptual modelling issue is about the fact, that an electron is an „a priori“ existing object independently from Dirac's related radiation theory,(*). This issue is passed on to today's zoo of "elementary particles", which is "required" for just 0,05% of the overall mathematical-physical modelling world. Basically, the proposed change is about a replacement of the distributional Hilbert space H(-n/2-e), e>0, hosting the Dirac (Delta-) "function", by the (quantum element) Hilbert space H(-1/2). Its related dual (quantum energy) Hilbert space H(1/2) is composed by the standard kinematical Hilbert space H(1) (defined by the Dirichlet integral inner product) and a corresponding orthogonal (ground state energy) sub-space of H(1/2), (**).

Change 2

the space-time model of the GRT is a 3+1 dimensional Riemann manifold equipped with the Einstein metric, (***). Its conceptual modelling issue is about the fact, that the (M,g) structure has no geometric structure at all. It is a purely metric space. The proposed change is basically to replace this purely metric space by appropriate Hilbert scales equipped with corresponding inner products. For the kinematical Hilbert space H(1) this results into a replacement of the metric space (M,g) by a modified Minkowski space, which is newly governed by the complex Lorentz transform, (****), and where the field of complex numbers is replaced by the field of quarternions, (*****).

(*)
(FeE): „Dirac‘s theory of radiation is based on a very simple idea; instead of considering an atom and the radiation field with which it interacts as two distinct systems, he treats them as a single system whose energy is the sum of three terms: one representing the energy of the atom, a second representating the electromagnetic energy of the radiation field, and a small term representing the coupling energy of the atom and the radiation field.“

We note that there is a purely electricity field theory based on manifolds, which is called the Mie-Theory, (HeW1).

(**)
(BoD): Bohm’s conception of explicate and implicate orders is based on an „undivided wholeness of modes of observation, instrumentation and theoretical understanding“ considering the difference between a lens and a hologram (a kind of mathematical lens), (BoD).

(***)
The "alignment" tool between the 3+1 dimensional manifold M based field equations and the related Minkowski space based field equations is provided by the comparison theorem building on the following conceptual elements: "electric-magnetic decomposition", "null-decomposition of a Weyl field", "null-structure equations of space-time", "exterior and interior optical functions", "rotation vector fields", (ChD).

(****)
(ArF): „The complex Lorentz transform is the main tool to prove the CPT (Charge-Parity-Time) theorem. The PT transformations are (physical) geometric transformations, while the C transformation is not. The CPT theorem becomes a purely geometric PT theorem if one restricts to classical tensor theory, quantum tensor theory and quantum spinor theory. The fact that there is a purely geometric PT theorem for quantum field theories suggests that space-time has neither a temporal orientation nor a spatial handedness“. Put simple, there is no need for a "big bang" "event" in a time-spatial environment, (+).

(*****)
The replacement of the field of complex numbers by the field of quarternions is accompanied by corresponding alignments of affected physical PDE models, e.g. (ArA), (UnA). The link to the Hilbert space based proposed UFT is given by quaternionic inner product spaces V, accompanied by a non-degenerated indefinite inner producton V/V(0), where V(0) denotes the isotropic part of V, (AlD). In the context of the concepts “spin”, “dual turns”, line geometry & kinematics, and “quaternions” we refer to Study’s “movement indicator”, (BlW)§58.

The perhaps primary application of quaternions is the quaternion rotation operator. This is a special quaternion triple-product (unit quaternions and rotating imaginary vector) competing with the conventional (Euler) matrix rotation operator. The quaternion rotation operator can be interpreted as a frame or a point-set rotation, (KuJ). Its outstanding advantages compared to the Euler geometry are

- the axes of rotation and angles of rotation are independent from the underlying coordinate system and directly readable

- there is no need to to take care about the sequencing of the rotary axes.

The Hilbert scale framework in a quaternionic setting also puts the spot on the proposed Kummer function based zeta-function theory, on a related (distribution valued complex functions based) verification of the Berry-Keating conjecture, (BrK2), (PeB), §15, and, on the Kummer conjecture, (HaH), (KuE5), in a Hurwitz quaternionic setting, (BrK2), (++).

In the context of ergodic theory (physically speaking, the theory of the course of movements of mechanical systems; mathematically speaking, the solutions of corresponding hyperbolic PDE, (+++)) we note that the point spectrum of a flow builds an additive modul, (HoE) S. 27. For every complete area withconstant negativ curvature and finite surface the geodesic flow is metrically transitiv, (HoE) S. 69. This main theorem can be reformulated as a theorem over Fuchsian groups, (HoE) S. 72. In this context we note that a Fuchsian group Fcan be mapped to a quaternion algebra, which contains F in its 1-norm-subgroup,(MaC).

(+)
The root of evil is sometimes called the "problem of time", (Callender C., TheOxford Handbook of Philosophy of Time, Oxford University Press, Oxford, 2013). In the proposed variational MHD-based unified plasma-quantum-gravity field theory classical PDEare interpreted as approximation to underlying two layers of variational PDO equations based on two "layers" of (energy) Hilbert spaces accompanied by two "energetical" inner products. Those layers are accompanied by two models of "time" characterizing two borderlines A & B between (1) the mathematical (action-free) ground state energy "world", (2) themathematical-physical (cosmic time) "world", and (3) Husserl's mathematical-phenomenological (internal time-consciousness) view of the world.

In the context of Husserl's "ideas of a pure phenomenology" and philosophy as an "essence (eidetic) science", (like pure logic, pure mathematics, space and time theory, kinematics, etc. ((HuE1) p. 20)), the crossing between the two layers A and B might be interpreted as "transcendental reduction" resp. "eidetic reduction", (HuE1).

With respect to the "problem of time" dispute between A. Einstein and H. Bergson resulting into the famous "twin paradox" formulation and its resolution by H. Bergson we refer to (BeH).

(++)
the theory of algebraic integrals containing the square root of functions of 1st or 2nd order are built on trigonometric functions and the irrelated circle functions. The theory of algebraic integrals containing the square root of functions of 3rd or 4th order (e.g. the Fagnano-integral to calculate the value of Gamma(1/4)) are built on elliptic functions and their related elliptic integrals. The conceptual new property, which is unique in the whole area of elementary functions, is the double-periodicity of the elliptic functions. This property, in combination with the change from the set of zeros of the trigonometric functions to the set of the +/- zeros of the Kummer function (accompanied by a change from harmonic Fourier series to non-harmonic Fourier series, cohomology groups interpreted as homology groups of negative order, (NeJ), and the H(1/2) space as first cohomology, (NaS)), puts the spot on the concept of the n-Kummer-Galois extension of the field containing a primitive nth root of unity and the Kummer’s prime number irregularity theorem in the context of cyclotomic fields, zeta-function values, the Kummer pairing and on the Kummer conjecture, (HaH), (KuE5). The new elements from the proposed Kummer function based zeta function theory are given by the decomposition of the li(x) function resp. the unit circle into a sum of two number distribution function resp. into two-semicircles.

(+++)
the role model for elliptic PDE is the Laplace equation; the role model for hyperbolic PDE is the (time-symmetrical) wave equation; the role model for parabolic PDE is the heat equation with its underlying positive time arrow. We note that the three characterizing "discriminants" for those three PDE types describe elliptic, hyperbolic and linear curves. Applying those types to model physical phenomena means different impact on the the corresponding degree of complexity: parabolic PDEs show reduced mathematical complexity compared to corresponding elliptic or hyperbolic PDEs. Hyperbolic PDEs (governed by the real proper Lorentz transformation) show a mathematical complexity increase compared to corresponding elliptic PDE with respect to the underlying domains (e.g. Minkowski space-time in R(4) vs. simple connected domains in R(4)).

A famous example for different interpretations as a consequence of the imbalances of physical phenomena complexity vs. mathematical modelling complexity (restricted to only hyperbolic PDE) are the different opinions of W. Ritz and A. Einstein (Physikalische Zeitschrift 10 (1909) 323-324) regarding the radiation problem with the possible different mathematical forms modelling "an electromagnetic process remaining restricted to a finite space", and correspondingly different required physical modelling assumptions (one of the roots of the second law (the mathematician) vs. due to reasons of probability (the physicist)).

The real Lorentz group has to do with the symmetry of vectors (tensors). The "covering group" of the real Lorentz group is SL(2,C), having to do with that of spinors.

The complex, homogeneous Lorentz group, L(C), (the set of (4x4) complex matrices) has two connected components L(+,C) and L(-,C), where L(+,C) is the only sub-group of L(C). It leaves the complex Minkowski scalar product z*z invariant, without complex conjugation of one of the complex four-vectors z. We note that there is a two-to-one homomorphism from SL(2,C) x SL(2,C) onto L(+,C), which is a proper Lorentz transform, (WiD).

For quaternionic Lorentz group representations in relation to the Dirac equation we refer to (LeS).

For quaternionic analysis and elliptic boundary value problems we refer to (GuK).

The proposed Hilbert scale & MHD based unified quantum, plasma and gravity field model is built on variational elliptic PDE with domains governed by the Hurwitz quaternions and an "extended" H(1/2) energy inner product, which are approximated by variational hyperbolic PDE governed by a "coarse grained" (i.e. compactly embedded) kinematical H(1) approximation energy sub-Hilbert space of H(1/2) and the complex Lorentz sub-group L(+,C). Thereby, the two-to-one homomorphism from SL(2,C) x SL(2,C) onto L(+,C) is in line with the proposed "two-plasma-component-model" H(1) decomposition.

Braun K., A Hilbert scale and MHD based unified plasma, quantum and gravity field theory

                                                 Feb 4, 2022

Braun K., A geometric Hilbert scale based integrated gravity and quantum field model, Dec 2021 overview

                                                Dec 19, 2021

related 2018-2021 papers:

Braun K., A geometric Hilbert scale based Mie electricity field theory accompanied by a complementary 0-point energy space, Sept 2021

Braun K., A validation approach of the proposed gravity and quantum field model, June 2021

Braun K., A geometric Hilbert scale based gravity and quantum field model, the modelling landscape, May 2021

Braun K., A geometric gravity and quantum field model, some royal road markers, April 2021