Albert Einstein, " Wolfgang E. Pauli, " This homepage addresses the following three Millenium problems (resp. links to corresponding homepages):
The proposed framework also provides an answer to Derbyshine's question, ("Prime Obsession") The answer, in a nutshell:
Regarding the proposed alternative quantization approach we also refer to the Berry-Keating conjecture. This is about an unknown quantization . This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principle and the natural numbers as spectrum of the harmonic quantum oscillator. The Hamiltonian needs to be self-adjoint so that the quantization can be a realization of the Hilbert-Polya conjecture.H
The Navier-Stokes equations describe the motion of fluids. The In the context of the below we note that the unit of measure of the http://www.navier-stokes-equations.com In order to prepare the relationship to the below, which is concerned with quantum field theory and its relationship to a proposed quantum gravity model, we recall from the Maxwell equations the following: the Maxwell equations are about 4 PDE constituting a (one could also call it a mathematical artefact/trick). With the inclusion of the displacement current the Maxwell equations treat electric and magnetic fields on equal footing, i.e. electric fields can induce magnetic fields and vice versa. The displacement current is defined in terms of the rate of change electric displacement field. It has the same unit as electric current, and it is a source of the magnetic field just as actual current is. However it is not an electric current moving charges, but a time-varying electric field.displacement currentThe displacement current is a crucial additition that completed Maxwell's equations and is necessary to explain many phenomena, most particularly the With respect to the relationship to the Einstein field equations we note that the electromagnetic field is described by the Maxwell tensor (E,B). Its source is given by the totals of the charges and the currents (r,j). The gravitation field is described by the Weyl tensor. Its source is given by the Ricci (energy) tensor, which describes the tide effect impacting the form (not the volume) of the attracted mass.
Topic B. above is implicitly addressing the Serin gap problem; we propose an alternative mathematical framework for the Standard Model of Elementary Particles (SMEP), which replaces gauge theory and variational principles (with its underlying concept of exterior derivatives and tensor algebra) by (distributional) Hilbert scales (enabling an inner product) and variational principles. As a consequence, the vacuum energy becomes an intrinsic part of the variational principles, i.e. is identical for all considered Lagrange resp. Hamiltonian mechanisms, while the corresponding "force" becomes an observable of the considered minimization problem; the proposed fractional Hilbert space framework addresses the mass gap problem of the Yang-Mills Equations (YME). A quick summary of the central differentiators beween both mathematical frameworks are provided in (latest update,
The common denominator with the topics above is about a mathematical (variational) framework (where the 4 Nature forces are "pseudo-forces", only) for a gravity & quantum field model with (standard) test space L(2)=H(0), a quantum state space H(-1/2) and the related energy space H(1/2), enabling a single energy "field" with a truly infinitesimal geometric structure. The embeddings of the Hilbert spaces H(1/2) into H(0) into H(-1/2) is compact, whereby compactness is a kind of preferred "countable/discrete" space compared to the larger space. The proposed mathematical framework above is supposed to provide The YME are concerned with quantum field theory. The challenge is about an appropriate mathematical model to govern the "mass gap" (i.e. to end up with finite energy norms), which is the difference in energy between the vacuum and the next lowest energy field. In some problem statements of the YME there are basically two assumptions made (which are not clearly defined): 1. the energy of the vacuum energy is zero 2. all energy states can the thought of as particles in plane-waves. As a consequence the mass gap is the mass of the lightest particle. Our challenge of proposition 1 is about the measure of the vacuum energy, which gives the value "zero". While the energy norm in the standard H(1) Hilbert space might be zero, the value of the quantum state with respect to the energy norm of the sub-space H(1/2) still can be >0. Our challenge of proposition 2 is going the same way: a particle with mass can be measured (condensed energy), i.e. it is an element of the test space H(0), while there still can be "waves" in the closed complementary space H(-1/2)-H(0), where the test space is "just" compactly embedded. Those "waves" might be interpreted as all kinds of today's massless "particles" (neutrinos and photons) with related "dark energy". As a consequence there is no mass gap, but there is an additional vacuum energy governed by the Heisenberg uncertainty principle. http://www.quantum-gravitation.de/ A physical interpretation could be about "rotating differentials" ("quantum fluctuations"), which corresponds mathematically to Leibniz's concept of monads. Its mathematical counterpart are the ideal points (or hyper-real numbers). This leads to non-standard analysis, whereby the number field has same cardinality than the real numbers. It is "just" the Archimedian principle which is no longer valid. This looks like a cheap prize to be paid, especially as hyper-real numbers might provide at least a proper mathematical language for the "Big Bang" initial value "function" and its related Einstein-Hilbert action functional. Looking on hyper-real numbers from the "real" number perspective one must admit to classify the term "real" is a contraction in itself, if it is understood as We further emphasis that - the differentable manifold framework of Einstein's field theory does not allow singularities, as required to model black whole and dark energy phenomena - the hyper-real /ideal points /monads above map to the "proper and terminal indecomposable past-sets/ideal points in space-time (PIPs and TIPs)" in the context of the comological censorship and the existence of past and future time-space singularities (GeR). - the idea to apply Non-standard Analysis (and its related non-classical motion) to explore a Quantum-relative Universe is not new (PoP). - the model enables an alternative concept to current symmetry breaking and inflation model for the early universe by which the required energy to generate matter out of photons (w/o violating conservation laws) is released during the symmetry breaking process. (GeR) Geroch R., Kronheimer E. H., Penrose R., Ideal points in spacetime, Proc. Roy. Soc., London, A347, 545-567, 1972 (PoP) Poluyan P. V., Non-Standard Analysis of Non-classical Motion; do the hyperreal numbers exist in the Quantum-relative universe? http://www.oocities.org/quantum_math_poluyan/hy_nu/hy-nu.htm
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