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MHD
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LITERATURE

H(-1/2) variational Maxwell equations replacing the YME

The baseline model of Einstein’s special relativity theory are the Maxwell equations. For given distributions of electric charges and currents the Maxwell equations determine the corresponding electromagnetic field. The central underlying concept is the Lorentz transformation. The original inertia law (before Einstein's gravity theory) forced to attribute physical-objective properties to the space-time continuum. Analog to the Maxwell equations (in the framework of a short distance theory) Einstein considered the inertia law as a field property of the space-time continuum.

The unknown physical parameters of the Maxwell equations

The energy tensor for electromagnetic fields is unknown for elementary particles. The laws by which the currents and charges behave are unknown. Matter is built by electromagnetic particles, but the field laws by which they are constituted are unknown, as well. From (EiA) p. 52 we quote:  

However, the laws governing the currents and charges (in the Maxwell equations), are unknown to us. We know, that electricity exists within elementary particles (electrons, positive kernels), but we don’t understand it from a theoretical perspective. We do not know the energetical factors, which determine the electricity in particles with given size and charge; and all attempts failed to complete the theory in this directions. Therefore, if at all we can built on the Maxwell equations, we know the energy tensor of electromagnetic fields only outside of the particles“.

From (DiP) we quote:

"The Lorentz model of the electron as a small sphere charged with electricity, possessing mass on account of the energy of the electric field around it, has proved very valuable in accounting for the motion and radiation of electrons in a certain domain of problems, in which electromagnetic field does not vary too rapidly and the accelerations of the electrons are not too great.  .... The departure from electromagnetic theory of the nature of mass removes the main reason we have for believing in the finite size of the electron. It seems now an unnecessary complication not to have the field equations holding all the way up to the electron's centre, which would then appear as a point of singularity. In this way we are led to consider a point model for the electron."

(EiA) Einstein A., Grundzüge der Relativitätstheorie, Vieweg & Sohn, Braunschweig, Wiesbaden, 1992

(DiP) Dirac P., Classical theory of radiating electrons, Proc. Roy. Soc. London, 167, (1938) 148-169


The Yang-Mills Equations and the mass gap

The classical Yang-Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. As a classical field theory the Maxwell equations have solutions which travel at the speed of light so that its quantum version should describe the massless particles, the „gluons“. However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales.

The proposed Hilbert scale based quantum field model is about an generalized variational representations of the considered PDE (in this case the Maxwell equations) based on the H(1/2) inner (energy) product, which is a generalization of the H(1) Dirichlet integral inner product. The extended energy Hilbert space allows a decomposition into the compactly embedded (coarse grained) standard H(1) energy Hilbert space and a complementary closed sub-space of H(1/2). The corresponding dual Hilbert space H(-1/2) of H(1/2) contains the quantum elements carrying a sum of kinematical and potential energy, correspondingly governed by the related energy space decomposition in the considered physical situation/ PDE system.

The extended Maxwell equations in the proposed Hilbert scale framework provides the missing laws by which the currents and charges behave. This is very much in line with Mie's theory. The following statements are taken from

(WeH) Weyl H., "Philosophy of Mathematics and Natural Science", p. 171

(WeH1) Weyl H., "Space, Time, Matter" p. 206.

Mie's Theory

(WeH): " G. Mie in 1912 pointed out a way of modifying the Maxwell equations in such a manner that they might possibley solve the problem of matter, by explaining why the field possesses a granular structure and why the knots of energy remain intact in spite of the back-and-forth flux of energy and momentum. The Maxwell equations will not do because they imply that negative charges compressed in an electron explode; … The preservation of the energy knots must result from the fact that the modified field laws admit only of one state of field equilibrium  … The field laws should thus permit us to compute in advance charges and mass of the electron and the atomic weights of the various chemical elements in existence. And the same fact, rather than contrast of substance and field, would be the reason why we may decompose the energy or inert mass of a compound body (approximately) into the non-resolvable energy of its last elementary constituents and the resolvable energy of their mutual bond.   ….  At a certain stage of the development it did not seem preposterous to hope that all physical phnomena could be reduced to a simple universal field law (in the form of a Hamiltonian principle)."

From (WeH1) we note the essential differentiator between Lorentz’s equations of the theory of electrons and Mie’s equations. This is about the concept of an "electric pressure" in the ether:

- Lorentz equations: there is no law that determines how the potentials depend on the phase-quantities of the field and on the electricity; there is only a formula giving the density of the mechanical (ponderomotorische) force and the law of mechanics, which governs the motion of electrons under the influence of this force

- Mie equations: the requirement is that the mechanical law must follow from the field equations; the corresponding Mie equation is fully analogous to that of the fundamental law of mechanics. In the static case that is, the electric force is counterbalanced in the ether by an „electric pressure".


Mie's theory and the proposed model

THe physical notion "pressure" has the same unit of measure than a "potential difference". This is the common "ground" with the proposed NSE solution and the proposed YME solution. The common additional conceptual new element is the fact that a "potential difference" becomes now an intrinsic element of the corresponding PDE systems governed by the closed "potential energy" ("ground state", "internal energy") sub-space.


The harmonic quantum oscillator


Regarding the harmonic quantum oscillator the corresponding ground state energy model is provided in

                        
                     

Braun K. , A new ground state energy model


Futher related papers:



Braun K., An alternative Schroedinger (Calderon) momentum operator enabling a quantum gravity model


   

Braun K., Comparison table, math. modelling frameworks for SMEP and GUT


  

Braun K., Unusual Hilbert or Hoelder space frames for the elementary particles transport (Vlasov) equation


  

Braun K.,The Prandtl (hyper singular integral) operator with double layer potential


                   

Braun K.,Some commutator properties


             

Braun K., Quantum Gravity, homepage text, 2019
 

       

Braun K., An idea for a quantum gravitation theory, Nov. 2010


  

Braun K., A quantum gravity and ground state energy Hilbert space model


                                  

Riesz operators and rotations


                    

Music melodies signals between red and white noise


                       

Einstein A., Ether and the theory of relativity


                            

Einstein A., The World as I See it
 

                        

Einstein A., The meaning of relativity


         

Stein E. M., Conjugate harmonic functions in several variables
 

             

Stankewicz J., Quarternion algebras and modular forms


       

June 2013, Quantum gravity model related mathematical areas
 

                     

Braun K., Quantum gravity 2011-2015


           

Braun K., earlier 2020 data, about a fractional quantum field energy Hilbert space


     

Braun K., Quantum gravity, the form-fit-function vision, Dec. 2015