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.

we provide an alternative Schrödinger momentum operator enabling a quantum gravity theory. This framework is valid for all energy-momentum 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

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.