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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 |
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