This preface has been edited at the end of a long journey, which started in 2010 until today (2017). During this period of time this homepage has been developped. The journey is still going on, but the main pillars and their underlying mathematical challenges and delivered solutions are established yet.
there are the three views on the considered problems, which are the physical, the mathematical and the philosophical views. Kant's critique of pure reason gives the rational for the interface and boundaries of those three areas, which is governed by the term "transcendence". We emphasis that the term "transcendental" in mathematics (number theory) is even beyong Kant's definition of the term "transcendental": the transcendental numbers are a subset of the set of the irrational numbers (from a mathematical (definition) point of view), but already the irrational numbers are trancendental in the sense of Kant. The mathematical terms "continuity" and "Riemann integral" are building on the concept of irrational numbers, i.e. they are also transcendental terms. The Lebesgue integral is defined as a generalization of the Riemann integral. In the framework of the Lebesgue integral concept the set of rational numbers is a so-called zero-set, only (!), i.e. the probability to pick a rational number out of the set of the real numbers is zero.
Our proposed mathematical model is building on the (Leibniz) mathematical transcendental term "differential". This means that there is no additional transcendence level added (which would be anyway a contradiction by itself), but the mathematical model becomes now applicable to all considered problem areas. The physical-mathematical modelling requirements (measurement/ observation/ test results validation) is still building on the test space L(2):
we "just" propose and show evidence of a consistent mathematical language (definitions, axioms) in an unusual distributional Hilbert space framework, which is less regular than the L(2)-test space, but still more regular than the domain of the Dirac function, while still applying standard functional analysis/spectral analysis/variational theory.
There are multiple handicaps regarding the usage of the Dirac "function" as a central concept in the quantum theory: let e denote an arbitrarily small positive real number and n denote the space dimension. The Dirac "function" is a distribution which is not an element of the quantum state Hilbert space L(2)=H(0). Its regularity depends from the space dimension n, i.e. the Dirac "function" is an element of the Hilbert space H(-n/2-e). Our approach builds on an alternative quantum state Hilbert space H(-1/2). Its definition is enabled by the Riesz and Calderon-Zygmund integrodifferential operators. We note that in case of space dimension n=1 the Riesz operators are identical to the Hilbert transform operator.
The considered (distributional) Hilbert space framework enables a truly infinitesimal geometry (WeH); as one first consequence the manifold concept of Einstein's field equations with its handicap of differentiable manifolds (which is a purely mathematical requirement without any physical meaning/justification) can be omitted. The approach also omits concepts like exterior tensor & exterior algebras and exterior differential forms, as well as corresponding gauge theories. It leads to a modified Einstein-Hilbert action functional newly based on a Stieltjes integral representation replacing the Lebesgue measure dx(4) by the corresponding Stieltjes integral measure dg(x(4)). Complementary variational principles can be derived from this, whereby the corresponding (classical) PDEs representation could be well defined without any boundary conditions (A. M. Arthurs, L. B. Rall).
A common Hilbert space framework for PDE field equations and quantum dynamics enables an integrated mathematical quantum and gravity field theory model, including a gravitational collapse and space-time singularity theory (R. Penrose).
The Berry-Keating (Hilbert-Polya refinement) conjecture is verified by a convolution representation of the Zeta function, enabled by the distributional Fourier series representation of the cot(x)-function (S. Ramanujan). This provides an answer to Derbyshine's question (in "Prime Obsession"):
The common distributional Hilbert space framework of classical field (PD) equations and quantum field equations and its corresponding classical and variational (weak) mathematical models require a change of a current paradigma: now the classical models become the mathematical approximations to the weak (Pseudo-) Differential Equations models and not the other way around.
Another consequence is, that the term "force" is only valid for classical PDE, when the Lagrange formalism is equivalent to the Hamiltonean formalism due to a defined Legendre transform. Another consequence is the fact that the energy inequality (with respect to the newly proposed H(1/2) energy space) of the non-linear, non-stationary NSE now also anticipates a contribution of the non-linear term, while, at the same time, enabling a global bounded energy inequality for the non-linear, non-stationary NSE in case of space dimension n=3.
Leibniz's monad concept is an extension of the real numbers to ideal/hyper-real numbers. Those are nothing more than another set of "transcendental numbers" in the sense of Kant (whereby the term "real" for the real numbers is already miss-leading); the properties of the set of the ideal numbers are identical to those of the real numbers (which are (in a physical sense) not "real" at all with 100% probability), except only one missing valid axiom, the Archimedean Axiom: this is related to physical measurement capabilites of a lenght by a given standard measurement length (!). The set of real numbers provides the baseline for standard analysis with the concepts of the Riemann and the Lebesgue integrals. The latter one is the fundamental concept to define the inner product of the test (Hilbert) space L(2) resp. the Dirichlet (energy) integral with its underlying domain, the (Sobolev) Hilbert space H(1), which is a subset of the test space L(2). The set of Leibniz's ideal numbers provides the baseline of the non-standard analysis. In this framework the Stieltjes integral can be interpreted as the counterpart of the Lebesgue integral going along with a reduced regularity requirements of the corresponding domain, which becomes the newly proposed energy Hilbert space H(1/2). The corresponding quantum state Hilbert space framework changes from the current test space L(2)=H(0) to the distributions Hilbert space H(-1/2) whereby the test space L(2) is compactly embedded. The complementary subspace H(-1/2)-H(0) is closed enabling the definition of an orthogonal projection operator from H(-1/2) onto the test space H(0). It therefore provides the framework to model also an additional continuous spectrum of the (energy) Hamiltonian quantum operator (Berry conjecture), as well as superconductivity, superfluids and condensates.
The fascination, motivation and energy to walk through this journey was and is primarily to contribute as much as possible to all those subject areas at that moment in time, when the one or the other idea popped up. The main drivers are “amazement” and “pursuit of new”, and not to follow academical career paths. In this sense
"prosit" (lat. "may it be useful") :
The relationship of current and newly proposed “ideal” (transcendental) mathematical objects to describe very large and very small physical phenomena (R. Penrose) is still affecting open, valid philosophical questions, as e.g. addressed in (RuB).
(KnA) Kneser A., Das Prinzip der kleinsten Wirkung von Leibniz bis zur Gegenwart, B. G. Teubner Verlag, Leipzig, Berlin, 1928
(RuB) Russel B., The Problems of Philosophy, Oxford university Press, Oxford, 1912
(ScE) Schrödinger E., Mind and Matter, Cambridge University Press, 1958
(ScE1) Schrödinger E., My View of the world, Cambridge University Press, 1964
(WeH) Weyl H. The Continuum, A Critical Examination of the Foundation of Analysis, Dover Publications, Inc. New York, 1994
(WeH1) Weyl H. Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, 1949, 2009