There are two order-creating principles in classical physics, the thermo-statistical order-from-disorder principle, and the dynamic law based order-from-order principle, (ScE), (PlM). Bohm suggested a „wholeness, explicate & implicate order" principle for quantum theory, (BoD). The proposed Unified Field Theory (UFT) provides a deductive dynamic system structure enabling an order-from-order-creating principle starting from meta-physical (mathematical) dynamic quanta systems up to Hilbert space based classical partial differential equation (PDE) systems.

There are two building blocks of the UFT: building block 1 provides Krein space based dynamic quanta systems; building block 2 provides a Hilbert space based dynamic fluid system. 

The common denominator of the building blocks 1 & 2 is a „special“ Hilbert space, which we call „exponential decay“ Hilbert space. The „exponential decay“ Hilbert space as a common basis for all dynamic systems enables an order-from-order principle starting from the „ground state“ energy system layer up to the classical PDE model layer.

Note: The Laplacian operator is defined with domain H(2). Its self-adjoint Friedrichs extension is defined with domain H(1). Its discrete eigenpairs allow the definition of Hilbert scales H(a) for „a“ real. The „exponential decay“ Hilbert space includes all Hilbert scales H(a), a real, i.e., also all distributional Hilbert scales, a<0. It enables "Approximation theory in Hilbert scales", (NiJ), (NiJ1). We emphasis that each of such inclusions is a compactly embedded inclusion. In other words, the compactly embedded sub-Hilbert space provides „discrete energy knots“ to the considered overall Hilbert space. We also note, that the "exponential decay“ Hilbert space provides a solution to the still unsolved problem of appropriate domains for hyperbolic partial differential operators (e.g. d’Alembert operator). In this framework the wave equation shows the same appreciated shift theorems as for the potential equation operator and the heat equation operator.


The bottom-up structure of the „wholeness“ of dynamic systems

With Bohm’s notions of explicate and implicate order systems the deductive bottom-up structure of the UFT based on an a priori "ground state" (layer 0) up to "classical physics" (layer 4) may be described in a nutshell in the following form:


Layer 4: Classical dynamic laws

Key words: explicate order-from-explicate order principle, continuously differentiable functions, F=m*a, Laplace-, heat-, and wave equations, surface and volume forces, diffusion, symmetric Laplacian potential operator with Hilbert space domain H(2)


Layer 3: Variational dynamic laws and statistical L(2) Hilbert space

Key words: explicate order-from-explicate disorder principle, thermo-statistics, surface and volume forces, model case model cases diffusion, potential equation, self-adjoint Friedrichs extension of the Laplacian operator with Hilbert (energy) space domain H(1), self-adjoint Stokes operator with domain H(1)


Layer 2: Variational dynamic laws and extended H(-1/2) Hilbert space
Building block 2

Key words: explicate disorder-from-explicate & implicate order principle, dynamic fluid element, extended energy Hilbert space H(1/2), self-adjoint Stokes operator with domain H(1/2), bounded H(1/2) energy norm inequality of the non-linear, non-stationary 3D-NSE system, non-stationary Stokes operator governed by Fourier waves (superposition principle), non-linear compact (disturbance) operator governed by Calderon wavelets (self-organized coherent structure principle), coercive bilinear form, viscosity, friction, Mie pressure, and turbulence phenomena 


Layer 1: Dynamic quanta systems with explicate & implicate dynamics
Building block 1: physical reality

Key words: explicate & implicate order-from-explicate & implicate order principle, timeless, quanta number sequences >1, 1-component systems: free electroton, free magneton, atomic nucleus systems, 2-component system: "perfect electromagnetism" quanta system


Layer 0: Dynamic quanta systems with purely implicate dynamics
Building block 1: mathematical reality

Key words: explicate & implicate order-from-explicate & implicate order principle, timeless, quanta number sequences <1, two 2-component systems: „perfect plasma“ system and „ground state“ system.


Note re turbulence (layer 2; M. Farge et al.): A turbulent flow is a dissipative dynamical system, whose behavior is governed by a very large, even may be infinite, number of degrees of freedom. Each field, e.g., velocity, vorticity, magnetic field or current density, strongly fluctuates around a mean value and one observes that these fluctuations tend to self-organize into so-called coherent structures. The Fourier representation is well suited to study linear dynamical systems whose behavior either persists at the initial scale or spreads over larger ones. This is not the case for nonlinear dynamical systems for which the superposition principle no more holds. A wavelet representation allows analyzing the dynamics in both space and scale, retaining those degrees of freedom which are essential to compute the flow evolution.

Note re timeless (building block 1 layers; quotes from E. Schrödinger): 

"The great thing (of Kant) was the form the idea that this one thing - mind or world - may well be capable of other forms of apprearance that we cannot grasp and that do not imply the notions of space and time", (ScE) p. 145.

"Einstein has not - as you sometimes hear - given the lie to Kant's deep thoughts on the idealization of space and time; he has, on the contrary, made a large step towards its accomplishment", (ScE) p. 149.


References

(BoD) Bohm D., Wholeness and the Implicate Order, Routledge & Kegan Paul, London, 1980

(NiJ) Nitsche J. A., Lecture Notes 3, Approximation Theory in Hilbert Scales

(NiJ1) Nitsche J. A., Lecture Notes 4, Extensions and Generalizations

(PlM) Planck M., Dynamische und Statistische Gesetzmässigkeit, (transl., the Dynamical and the Statistical Type of Law). In: Roos, H., Hermann, A. (eds) Vorträge Reden Erinnerungen, Springer, Berlin, Heidelberg, (2001) 87-102

(ScE) Schrödinger E., What is Life? The Physical Aspects of the Living Cell with Mind and Matter, Cambridge University Press, Cambridge, 1967