The H(-1/2) Hilbert space

The Millenium 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.

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 :

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

For space dimension 1 the newly proposed Schrödinger operator is equal to the Calderon operator C:=DH whereby H denotes the Hilbert transform operator and D:= -i(d/dx) ((MeY) 7.1): Let A denote the operator of pointwise multiplication by a function a(x), then Calderon showed that the commutator (A,C) is bounded on L(2)(R) if and only if the function a(x) is Lipschitz. This characterization opened the way to the study of the operators on Hardy spaces.

The related (Pseudodifferential) operators (i.e. the model (harmonic quantum oscillator problem) operators for the space dimension m=1) are given in (see also )


Braun K. , A new ground state energy model

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".

The new Hilbert scale concept enables also alternative modelling opportunities with respect to plasma (the "4th state of matter", beside solid, liquid, and gaseous states) dynamics (ChF),(DeR)):

" a plasma is a quasi-neutral gas of charged and neutral particles which exhibits collective behavior".

(DeR), Introduction: "One of the distinctive features of plasma physics is the fact, that plasma consists of an assembly of charged particles, interacting with each other through the Coulomb force. The movement of each plasma particle is governed by the local electric field; at the same time, the particle is also a source of electric field (!!!). In order to see what happens in various physical situations, we shall need to obtain solutions which simultaneously satisfy the equation of motion and Maxwell's equations. This is known as the requirement of self-consistency."

A cold plasma is an idealized assembly of an initially uniformly distributed set of electrons (with charge -e) and ions (atomic nuclei with charge +e), i.e. the plasma is initially electrically neutral everywhere, whereby the electrons and ions are initially motionless, i.e. there is no random thermal motion. There are two responses of plasma electrons to perturbation, whereby the ions in the plasma are affecting the dynamics of the electrons only by providing the overall electrical dynamics of the electrons only by providing the overall electrical neutrality of the plasma. These are oscillation with characteristic (electron plasma) frequency and charge screening with characteristic length scale. Additionally, there is also the case, where a given electron must have a close encounter with an individual ion. In this case the corresponding model is given by the Coulomb collisions between single electrons and single ions, which are treated as an instance of Rutherford scattering. As this is about the Coulomb force between two charges and its related total momentum the newly proposed concept of this page can be applied, as well.

The baseline for the 4th state of matter is the concept of temperature, the one-dimensional Maxwellian distribution, in order to model a gas in thermal equilibrium with particles (electrons and ions) of all velocities. A fundamental characteristic of the behavior of plasma is its ability to shield out electric potentials that are applied to it. The Debye shielding is a statistical concept. The picture of Debye shielding is only valid if there are enough particles in the charged cloud. There are different criteria for plasmas that an ionized gas must satisfy to be called plasma. One has to do with collisions and with the corresponding Boltzmann & Vlasov equations.


(ChF) Chen F. F., Plasma physics and controlled fusion, Plenum Press, New York, London, 1984

(DeR) Dendy R. O., Plasma Dynamics, Clarendon Press, Oxford, 1990

(JaS) Jaffard S., The spectrum of singularities of Riemann's function, Revista Mathematica Iberoamericana, vol 12, No 2, (1996) pp. 441-460

(MeY) Meyer Y., Coifman R., Wavelets, Calderon-Zygmund and multilinear operators, Cambridge studies in advanced mathematics 48, Cambridge University Press, 1996