This website presents the results of a mathematician’s long-standing hobby, pursued during his even longer career in industry. The three topics of this hobby project concern the Riemann Hypothesis, the irrationality of the Euler-Mascheroni number, and the broad range of problems from a physics perspective that lie behind the concepts of experimental and theoretical physics, “wave-particle dualism, matter, field, vacuum,” etc.
Quantum Dynamics
The two new concepts of the proposed theory of quantum dynamics are:
(1) an extension of the statistical Hilbert space L(2) of quantum mechanics to H(-1/2)
(2) two Krein-space representations of the Hilbert space H(-1/2), which define the concept of a physical ether, consisting of two interconnected ground-state quantum fluctuation and plasma quantum fluctuation models with different superposition and non-superposition states.
The metaphysical prerequisite for both models is a mathematically justifiable “density” assumption regarding this concept of a „physical ether“. It includes a concept of dynamic energy that is independent of mechanical energy and independent of the concepts of the theory of motion - that is, in particular, independent of the concepts of space, time, and continuity. The laws of classical mechanics may be interpreted as aproximation solutions in Hilbert scales of underlying variational representions of quantum-mechanical laws governed by the so-called energy method.
What at first glance appears to be a cheap magic trick (the extension of the standard Hilbert space L(2) of the quantum mechanics to the quantum-mechanical Hilbert space H(-1/2)) proves to be a valid step toward resolving the problem of a missing physical interpretation of a mathematical “world-point state. The compact embedding of H(-1/2) into H(-n/2-e), e>0, which is independent of all spatial dimensions, ensures - by virtue of Sobolev’s embedding theorem - that continuous fields in classical physics can be generated by quantum-mechanical causes, regardless of the dimension of the underlying Euclidean space. The systems of this extended Hilbert space can thus accommodate the mathematical concept of the Newtonian or Coulomb potential, both of which, for mathematical reasons, only make physical sense in 3D Euclidean space. The concept of phenomenon-specific “point charges” that generate continuous fields throughout 3D Euclidean space can thus be completely replaced.
Furthermore, it has been shown that the variation equations of the 3D NSE initial boundary value problem, based on a “test space” H(-1/2), avoid existing “blow-up” solutions by means of L(2) test functions. In classical physics, the systems of the extended “test space” H(-1/2) could therefore be physically interpreted as “dynamic fluid/gas elements.”
Riemann Hypothesis
Attempts to find an analytical solution to the Riemann Hypothesis (RH) are based on the identity between the product representation of the holomorphic zeta function and a sum consisting of the Riemann distribution function of prime numbers in relation to the proper zeta function and “residual terms” arising from the product representation (EdH). The conceptual problem with this approach is that the product representation of the holomorphic zeta function reflects the axis-symmetric (critical line) distribution of its nontrivial solutions, whereas the sum representation introduces additional system parameters (e.g., the nontrivial zeros of the holomorphic zeta function, the presence of a symmetry in the distribution of zeros of the complex sine function, and its absence for the inverse of the gamma function, (LeB)). The proposed solution concept is based on a sum representation of the meromorphic zeta function, by means of which the zeros of the proper zeta function can be characterized in the form of the identity of two alternating power series, which contradicts a non-valid RH.
Euler–Mascheroni Constant
There is a general method for proving transcendence of the solutions to linear differential equations for “E-type functions” (SiC). The solutions to the Bessel differential equation provide such functions. Certain Bessel functions have a special integral relationship with the Euler constant, comparable to the relationship between the exponential function and the Euler constant. The basic idea behind the proof is to use these relationships to construct a line of reasoning similar to that underlying Niven’s proof of the irrationality of the circular constant “pi.”