The Millennium problem solution enabled by an alternative H(1/2) energy Hilbert space domain of a corresponding modified Schrödinger (Calderón) momentum operator
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.
We provide an alternative Schrödinger momentum operator enabling a quantum gravity theory :
The related (Pseudodifferential) operators (i.e. the model (harmonic quantum oscillator problem) operators for the space dimension m=1) are given in (see also http://www.quantum-gravitation.de/ )
The proposed 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 mathematical 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.
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 paradox) 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 collapse to a state of extremely high density.
E. Schrödinger: "Indeed there is no observation concerned with the geometrical shape of a particle or even with an atom".
The idea, that the "spin(1/2)-mass-particle" does not "look" the same after each kind of "rotation" sounds at least mysterious; on top of that this spin(1/2)-"matter" concept requires different (force/energy-type dependent) kinds of related massless "interacting-particles" with corresponding different spins. The framework is gauge theory, which per definition does not provide any geometrical structure. How in such a mathematical framework can mass be essentially the manifestation of THE vacuum energy?
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 (of even or odd space dimensions), 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".