The foundation of deductive theoretical physicsDiscussion & Collaboration Forum
Without the mathematical concept of “continuously differentiable functions,” there would be no theories of dynamics in the natural science of theoretical physics. Without the concept of “separable Hilbert spaces,” there would be no theory of quantum mechanics. Thus, the physical concepts underlying these two theoretical groups are grounded in terms that are ultimately based on mathematical (a priori) axioms. Accordingly, interpretations derived from them, such as “particle-wave dualism” or “geometrodynamics” (“the law without law at the basis of physics,” (CiI) p. 49), and concepts derived from them such as “massless particles,” “Big Bang,” “dark matter,” “dark energy,” etc., are metaphysical concepts that are ultimately grounded in mathematical (a priori) axioms. In the case of GRT and QM, key conclusions derived from them are, as is well known, incompatible - i.e., contradictory - and thus unsuitable for a physical understanding of physical phenomena from a theoretical standpoint. At least as long as one acknowledges that concepts such as singularities and infinitely small points of contact do not occur in nature.
The proposed mathematical concept allows for the joint use of the two concepts “continuously differentiable functions” and “separable Hilbert spaces.” A common, genuine metaphysical basic assumption that seeks to theoretically account for phenomena must therefore, in some way, make “facts” such as “action at a distance in a vacuum via massless photons” theoretically (i.e., mathematically) explainable. The new, mathematically grounded a priori physical concepts proposed here in a Hilbert-Krein space setting are (1) “dynamic energy” in addition to the existing “mechanical energy”, and (2) “vacuum” with the two features “ground state” and “plasma”.
The two new mathematical concepts of the proposed theory of quantum dynamics
The two new mathematical 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.”
The original work from 2023, revised in July 2026
According to Kant's „Theory of Natural Science“ in order to establish a meta-physical foundations of physics it requires an a priori conception of matter, i.e., its „form“, its necessary and universal determinations and laws that govern (predictively describe) its behavior, (PlP) p. 29. "The merely empirical concept of motion "reality" is conceived as objective thinghood (Sachheit). Here one speaks of „motion as an accient (Akzidens) of matter“, thus not a mere act (Actus) of the productive imagination, i.e., motion as description, the describing of a space (moreover, where time, specifically as quantity, is not taken into consideration at all", (PlP) p. 297.
Discuss this approach
Critical examination, alternative derivations and further development of this approach are welcome in the Discussion & Collaboration Forum.