A quanta field theory
Riemann Hypothesis
Kummer & Goldbach
Who I am


                                               R. Descartes

My present design, then, is not to teach the method which each ought to follow for the right conduct of his reason, but solely to describe the way in which I have endeavored to conduct my own. They who set themselves to give precepts must of course regard themselves as possessed of greater skill than those to whom they prescribe; and if they are in the slightest particular, they subject themselves to censure. But as this tract is put forth merely as a history, or, if you will, as a tale, in which, amid some examples worthy of imitation, there will be found, perhaps, as many more which it were advisable not to follow, I hope it will prove useful to some without being hurtful to any, and that my openess will find some favor with all“, (DeR2) iii


                                               E. Schrödinger

                                     The principle of objectivation

"Science aims at nothing but making true and adequate statements about its object. The scientist only imposes two things, namely truth and sincerity, imposes them upon himself and upon other scientists. In the present case the object is science itself, as it has developed and has become and at present is, not as it ought to be or ought to develop in future", (ScE1) p. 117


                                               E. Schrödinger

                        Form, not substance, the fundamental concept

„The new idea is that what is permanent in these ultimate particles or small aggregates is their shape and organization. The habit of everyday language deceives us and seems to require, whenever we hear the word „shape“ or „form“ pronounced, that it must be the shape or form of something, that a material substratum is required to take on a shape. Scientifically this habit goes back to Aristotle, his causa materialis and causa formalis. But when you come to the ultimate particles constituting matter, there seems to be no point in thinking of them again consisting of some material. They are, as it were, pure shape, nothing but shape; what turns up again and again in successive observations is this shape, not an individual speck of material“, (ScE3) p. 12


                                                 B. Russell

“Substance“, in a word, is a metaphysical mistake, due to transference to the world-structure of the structure of sentences composed of a subject and a predicate.“, (RuB1) p. 212

Hume had proved that the law of causality is not analytic, and had inferred that we could not be certain of its truth. Kant accepted the view that it is synthetic, but nevertheless maintained that it is known a priori. He maintained that arithmetic and geometry are synthetic, but are likewise a priori. He was thus led to formulate his problem in these terms:

How are synthetic judgements a priori possible?

The answer to this question, with its consequences, constitutes the main theme of The Critique of Pure Reason.

Space and time, Kant says, are not concepts; they are forms of „intuition“. (The German word is „Anschauung“, which means literally „looking at“ or „view“. The word „intuition“, though the accepted translation, is not altogether a satisfactory one)", (RuB1) p. 680

                               The proposed quanta field theory

The proposed quanta field theory provides an all-encompassing theory, where physical models of different physical areas are no longer decoupled and differently scaled according to their different levels of granularity.

Classical mechanics is concerned with kinematics and dynamics. The kinematics deals with the different forms of the movement of bodies in a space-time environment. Classical dynamics should explain the reasons of the connection of those different form of movements. The principle of transfer causality („Impetustheorie“, "impetus ~ momentum" (WoM)) states that a transfer of movements requires the mediation of a certain type of quantum, a "transfer force quantum".

The proposed deductive structure of quanta fields is based on coherently defined mechanical or dynamical quanta. The physical quanta are appropriately composed by two abstract (mathematical) quanta forming a "ground state energy" quanta field in line with the thoughts in (DaJ). It novates the three decoupled theories, which are called the Standard Model of Elementary Particles requiring multiple arbitrary (free) parameters w/o any physical meaning. The underlying concepts of fermion & boson quantum types (= quanta) are replaced by a deductive structured kinematical & dynamical quanta energy field system.

From a top down perspective (from classical physical models down to quanta dynamics) classical Partial Differential Equation (PDE) or Pseudo Differential Equation models become approximation models of their related (weak) variational representations governed by the standard mechanical energy Hilbert space H(1) equipped with the Dirichlet integral inner product D(u,v). Those variational, purely mechanical H(1)-energy Hilbert space based models become approximation models in an extended H(1/2)-energy Hilbert space, where H(1) is compactly embedded into H(1/2). The H(1/2) energy Hilbert space based models become approximation models in an (one mechanical component) atomic nucleus dynamics framework.

The next two layers down require the concept of Krein spaces. It provides the (two mechanical, i.e. electroton & magneton) electromagnetic quanta Maxwell-Mie dynamics, being followed by the (two dynamical, i.e. electron & positron) "plasma quanta dynamics". 

The "plasma quanta dynamics" model overcomes the current physical modelling issue of the observed Landau damping phenomenon, as there are two, a linear and non-linear, Landau damping models, which conceptually must arise from different physical effects: "Both kinds of Landau damping are independent of dissipative collisional mechanisms. If a particle is caught in the potential well of a wave, the damping phenomenon is called "trapping". Particles can gain or lose energy in trapping. Collisionless damping with trapping occurs, when a wave grows to a larger amplitute. One then finds that the wave does not decay monotonically; rather, the amplitude fluctuates during the decay as the trapped particles bounce back and forth in the potential wells. This is nonlinear Landau damping, i.e. trapping not lie within the purview of the linear theory. Since the linear Landau damping is derived from a linear theory, the nonlinear Landau damping must arise from a different physical effect. The question is: Can untrapped electrons moving close to the phase velocity of the wave exchange energy with the wave?" (ChF) p. 248-249.

The "plasma quanta dynamics" model builds the baseline modelling framework for all physical dynamics. This is in line with the significant about 99%-share of „plasma matter“ in the universe, (ChF) p.1.

The underlying purely mathematical "vacuum quanta dynamics" model of the physical "plasma quanta dynamics" model provides the axiomatically defined initial a priori baseline model accompanied by purely mathematical quanta. Their definition and related potential operator based dynamical energy space is based on the fundamental different mathematical Schnirelman densities of the odd and even integers governing the corresponding different "densities" of the related "electrino" resp. "positrino" forms (accompanied by a "neutrino" form) in this mathematical "vacuum" world. This is in line with the observed deviation from the iso-spin-symmetry in electrodynamics, which was taken bei Heisenberg as indication for an asymmetry of the ground state, (DüH).

In particular, the integrated physical layer modelling framework

(a) enables a solution of the 3D-Navier-Stokes Millennium Problem of the Clay Mathematics Institute in a H(1/2) energy Hilbert space framework

(b) overcomes the physical "YME mass gap" Millennium Problem of the Clay Mathematics Institute by making those equations obsolete resp. replaced by the electromagnetic Maxwell-Mie dynamics

(c) provides a single physical plasma dynamics theory "explaining" the Landau damping phenomenon overcoming the current issue of a linear and a nonlinear theory accompanied by two different physical root causes of the phenomenon

(d) provides a H(1/2) based overall energy Hilbert space in line, which is in line with the Teichmüller theory & the universal period mapping via quantum calculus, (NaS), and the appropriate domains of the double layer (Prandtl) potential operator as applied e.g. in aerodynamics, (LiI)

(e) supports the aspiration of A. Unzicker's "mathematical reality", to "form a consistent picture of reality by observing nature from the cosmos to elementary particles," (UnA2).

           Related forgotten thoughts, discoveries and theories

The proposed deductive structure of quanta fields suggests to revisit forgotten thoughts, discoveries and theories regarding

(1) forgotten or questionable physical-mathematical modelling approaches

(2) neglected physical discoveries

(3) physical-metaphysical views


(1) Weizäcker's understanding of the unity of physics; Heisenberg's mathematical formalism for an unified field theory with its cornerstones of an indefinite metric in a Hilbert space and the degeneracy (asymmetry) of the ground state, (DüH), (HeW); Schrödinger's concept of a heat bath in statistical thermodynamics, (ScE); Penrose's calculated absurdly tiny probability to produce the universe resembling the one in which we live, (PeR); Barbour's problems about time and the need for a quantum cosmology, (BaJ1); Klainerman's global nonlinear stability of the Minkowski space, (ChD); Bohm's wholeness and implicate order in physical laws, (BoD1); the current galactic kinematics assuming an universe, which is homogeneous and isotrop on large scales, modelled as an ODE depending from a cosmic time t, the Hubble function H(t), and a scale factor function a(t); the sophisticated CMBR providing the most important evidence supporting the big bang model, (BrK11)(LaM); Schauberger’s implosion theory for planetary movements, (BaA), (BrK11); Schauberger’s implosion theory as the initializer of the universe creation process, (DeK)

(2) Ehrenhaft’s discovery of the photophoresis phenomenon ("kleinste freischwebende Materieteilchen in einem konzentrierten Lichtstrahl bewegen sich auf Schraubenbahnen; zu der Bewegung um die Schraube kommt oft noch eine Bewegung um die eigene Achse hinzu"), (BrJ); Schauberger's "konzentrisch-spiralförmige Bahn von außen nach innen, deren Zentrum saugend ist, also eine zentripetale Massenbewegungsform ist", which he called implosion

(3) Platon's theoretical philosophy in the context of the relations between "idea & cosmos" resp. "time & astronomy", (BöG),(BöG1); Leibniz' vis viva and teleology concepts, (KnA); Goethe’s „data for a theory of color“ („Zur Farbenlehre“), e.g. (MüO), (NuI), (RiN); Schopenhauer’s „on vision and colours“, e.g. (HüA), (ScD) p. 33 ff.; Schopenhauer’s concept of "matter as mere visibility of the will", (ScD) p. 42; Bergson's and Shaw's rejection of "neo-darwinism & mechanism" in favour of "creative evolution (= functional adaption) & vitalism", (ShB), (BeH2); Nagel's teleology, (NaT); Bergson's view on "duration & simultaneity" regarding Einstein's relativity theory, (BeH), Schrödinger's key facts (e.g. the concept of "differentials") for the physical basis of consciousness, (ScE1).

Braun K., An integrated quanta field theory enabling a deductive structure of physics


                                             The Gordian knot

Braun, K., Current physical and mathematical realities regarding an unified field theory

                                       Supporting mathematics

Braun, K., A Krein space based quanta energy field model, supporting mathematics


                                   Some former related papers


Braun K., UFT related list of papers

     Relations to the Riemann Hypothesis and the Goldbach conjecture

The Krein space based hermitian (potential) operators governing the vacuum quanta field may provide an alternative (selfadjoint) operator to the Berry-Keating "quantized" classical Hamiltonian operator of a particle of mass m that is moving under the influence of a problem specific to be defined potential function V(x).

The primes (excluding the integer "2", the base number of the even numbers) are a subset of the odd integers. The conceptual design of the mathematical baseline quanta of the proposed quanta field theory, the electrinos and the positrinos forming the "ground state energy quanta field", is based on the different Schnirelmann densities of the odd resp. the even integers, a half resp. zero. Physically speaking, the Schnirelmann densities determines a kind of density distributions of the (odd integer related) electrinos and the (even integer related) positrinos of a mathematical "vacuum" quanta field. 

The  binary Goldbach conjecture states that every positive even number n>2 is the sum of two primes. The claim is, that the additional "prime number density function" with domain 0<x<1 provides an alternative two-semicircle method to the standard Hardy-Littlewood circle method to prove the binary Goldbach conjecture.

Braun K., A toolbox to solve the RH and to build a non-harmonic Fourier series based two-semicircle method


Polya Ueber eine neue Weise bestimmte Integrale in der Zahlentheorie zu gebrauchen

The physical Montgomery-Odlyzko law states that the distribution of the spacing between successive non-trivial zeros of the zeta function is statistically identical with the distribution of eigenvalue spacing in a "Gaussian Unitary Ensemble". The claim is that the above conceptual modelling components also enable an appropriate mathematical model, where the physical Montgomery-Odlyzko law becomes the (L(2)-space based) statistical relevant part of the zeros distribution of the zeta function on or close to the critical line. 


Braun K., The Montgomery-Odlyzko law, eigenvalue spacing in a collection of Gaussian unitary operators


(DeJ) Derbyshire J., Number theory meets quantum mechanics