PREFACE
OVERVIEW
RIEMANN HYPOTHESIS
NAVIER-STOKES EQUATIONS
YANG-MILLS EQUATIONS
physical considerations
WHO I AM
LITERATURE

A quantum Yang-Mills theory allowing low energy scales

The classical Yang-Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges. For given distributions of electric charges and currents the Maxwell equations determine the corresponding electromagnetic field. The laws by which the currents and charges behave are unknown. The energy tensor for electromagnetic fields is unknown for elementary particles. Matter is built by electromagnetic particles, but the field laws by which they are constituted are unknown, as well.  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.

As a classical field theory the Maxwell equations have solutions which travel at the speed of light so that its quantum version should describe massless particles (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 problem is to establish rigorously the existence of the quantum Yang-Mills theory and a mass gap.

Based on a H(1/2) energy Hilbert space we propose (analog to the NSE solution) a corresponding (weak) variational Maxwell equation representation. Its corresponding generalization (as described above) leads to a modified QED model. In the same manner as the Serrin gap issue has been resolved (as a result of the reduced regularity requirements) the chromo-electromagnetic field /particles can now carry charges. The open "field law" question above and how "particles" are interacting with each other to exchange energy are modeled in same manner as the coherent/incoherent turbulent flows of its NSE counterpart. The corresponding "zero state energy" model is no longer built on the Hermite polynomials but on its related Hilbert transformed Hermite polynomials, which also span the L(2) Hilbert "test" space.

In

                         http://www.quantum-gravitation.de/ 

we provide an alternative Schrödinger momentum operator enabling a quantum gravity theory:

        

Braun K., An alternative Schrödinger momentum operator enabling a quantum gravity model

This framework is valid for all energy-momentum (energy density-pressure) related differential equations, i.e. including also the NSE, the Maxwell equations and the Einstein field equations. It enables universal field laws of atomic nuclei and electrons spreading out continuously and being subject to fine fluent changes, where e.g. the mass of an electron derives completely from the accompanying electromagnetic field. As a consequence there are no longer dynamical matter-fields (i.e. no laws of interaction between matter and field), neither generated by nor acting upon an agent spate from the field. This means that the mass gap "problem" of the YME does no longer exist; it is a mathemematical consequence of the non appropriate current mathematical model, not a physical "reality" issue.

The same situation is given for the 3D-non-linear, non-stationary Navier-Stokes equations and the (3D-) related "Serrin gap" issue. With an analogue change of the underlying operator domain (resulting in a modified momentum/pressure operator) the corresponding weak NSE representation gets well posed with bounded energy inequality, including the non-linear term. It is provided in

                    http://www.navier-stokes-equations.com/

The Maxwell equations are well posed. However, in order to achieve this, there is a mathematical element introduced into those equations leading to the interpretation of an "existing" sophisticated displacement current w/o any physical justification. The alternative Maxwell equations definition (domain with reduced regularity and corresponding weak variational representation analogue to the YME and the NSE) keeps to be well posed, but does not need a current displacement concept, as the corresponding mathematical term vanishes.

The current understanding of all known "particles" in the universe is, that there is a split into two groups of those "particles" to overcome the contact body problem (body-force interaction problem), which e.g. ended up into the (physical) Copenhagen interpretation of the particle-wave "dualism" (or paradoxon) of quantum mechanics and the inconsistency between the two mathematical model frameworks for the quantum field dynamics and the Einstein field equations :

1. spin(1/2)-"matter"-"particles", which are "objects" with a spin(1/2), i.e. those "objects" look the same only after the second rotation

2. spin(0,1,2)-"force"-"particles", which are "objects" with spins 0,1,2, interacting with spin(1/2)-"matter" "objects".

The first group goes back to Dirac, who introduced this purely mathematical concept to "explain" why spin(1/2)-"matter"-"particles", especially the electrons, can exist as "separate" "objects", while not merging to one big "soup" ("object"?). Dirac's theory enables consistency of the quantum mechanics and the special relativity theory.

In order to avoid the same "soup" (disaster) effect Pauli postulated his exclusion principle in order to ensure that spin(1/2)-"matter"-"particles" under the influence of spin(0,1,2)-"force"-"particles" do not collaps to a state of extremly high density.

The newly proposed mathematical concept above is based on an only single "particle/fluid" "object" concept, whereby its corresponding state is modelled as an element of the Hilbert space H(-1/2). We note that the regularity of the Dirac "function" depends from the space dimension (causing purely mathematical challenges for higher space dimensions, while those challenges are even independent from the two cases, that the Huygens principle is valid or not), while even for the space dimension m=1 the Dirac function is "only" an element of the Hilbert space H(-1/2-e), i.e. less regular than the newly proposed single "particle/fluid" "object".

The Gaussian function stands out since it minimizes the Heisenberg uncertainty principle (DaS). The corresponding windowed Fourier (integral) transform is e.g. applied in quantum physics, where it is used for defining and investigating coherent states. It is related to the Weyl-Heisenberg group, while the corresponding wavelet (integral) transform is related to the affine group. In (DaS) a new interpretation of the Mexican hat function (which is a wavelet, while the Gaussian function itself is not) is provided: the mexican hat function can be interpreted as a minimizing function of an uncertainty principle, in case its rotation invariant form A has a certain definition.

(DaS) Dahlke S., Maass P., The Affine Uncertainty Principle in One and Two Dimensions, Computers Math. Applic. Vol. 30, No. 3-6, pp. 293-305, 1995