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, amorphous gases/fluids, high viscosity, undefined melting point, 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 space-time problems (layer 4, quote from A. Einstein): „The normal adult never bothers his head about space-time problems. Everything there is to be thought about it, in his opinion, has already been done in early childhood. I , on the contrary, developed so slowly that I only began to wonder about space-time when I was already grown up. In consequence I delved deeper into the problem than an ordinary child would have done“, (UnA) p. 180.  

Note re the concepts of temperature & kinetic energy (layer 4): „Indeed, inertia, being since Newton the intrinsic characteristic of all masses, alllows us to quantify masses as inverse accelerations. Newton’s second law, F = m*a, is thereby incorporated as a defintion in the system of laws of nature.   …. In earlier times, the notions of temperature and kinetic energy were completely distinct issues. Later, it turned out that temperature was nothing else than the average kinetic energy of a particle, and the „law of nature,“ (m*v*v)/2 = k*T, was established“, (UnA) p. 181. Temperature on the macroscopic (gravitational potential) level is simply energy per mass, i.e. the combination of Einstein’s mass-energy conservation law (E=m*c*c) and the definition of temperature in the form 1/T = (k/W) * (dW/dE).  

Note re the Planck constant (layer 4, quote from H. Dehnen et al.): "The Planck constant is independent from any weak or strong gravitation field. It therefore somehow mirrors the fundamental difference of physical macro and micro world", (DeH).

Note re the problem in thermodynamics (layer 3, quotes from E. Schrödinger):

„There is, essentially, only one problem in statistical thermodynamics: the distribution of a given amount of energy E over N identical systems. Or perhaps better: to determine the distribution of an assembly of N identical systems over the possible states in which this assembly can find itself, given that the energy of the assembly is a constant . The distinguished role of the energy is, therefore, simply that it is a constant of the motion – the one that always exists, and, in general, the only one. The generalization to the case, that there are others besides (momenta, moments of momenta), is obvious; it has occasionally been contemplated, but in terrestrial, as opposed to astrophysical, thermodynamics it has hitherto not acquired any importance. “To determine the distribution” .. means in principle to make oneself familiar with any possible distribution-of-the-energy (or state-of-the-assembly), to classify them in a suitable way, i.e. in the way suiting the purpose in question and to count the numbers in the classes, as as to be able to judge of the probability of certain features or characteristics turning up in the assembly. The question that can arise in this respect are of the most varied nature, especially in relation to the fineness of classification. At one end of the scale we have the general question of finding out those features which are common to almost all possible states of the assembly so that we may safely contend that they „almost always“ obtain. In this case we have well-nigh only one class – actually two, but the second one has a negligibly small content. At the other end of the scale we have such a detailed question as: volume (=number of states of the assembly) of the „class“ in which one individual member is in a particular one of its states. Maxwell’s law of velocity distribution is the best-known example,“ (ScE) p. 1-2.

„To my view the ‚statistical theory of time‘ has an even stronger bearing on the philosophy of time than the theory of relativity. The latter, however revolutionary, leaves untouched the undirectional flow of time, which is presupposes, while the statistical theory constructs it from the order of the events. This means a liberation from the tyranny of old Chronos", (ScE1) p. 152

Note re amorphous gases/fluids (layer 3; quote from E. Schrödinger): „Every little piece of matter handled in everyday life contains an enormous number of atoms. Many examples have been devised to bring this fact home to audience, none of them more impressive than the one used by Lord Kelvin: Suppose that you could mark the molecules in a glass of water; then pour the contents of the glass into the ocean and stir the latter thorougly so as to distribute the marked molecules uniformly throughout the seven seas; if then you took a glas of water anywhere out of the ocean, you would find in it about a hundred of your marked molecules“, (ScE1) p. 6.

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 (layer 0 & layer 1; 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 appearance that we cannot grasp and that do not imply the notions of space and time", (ScE1) 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", (ScE1) p. 149.

Note re "matter & mind" (quote from Sir Charles Sherrington, "Man on his nature", P. 73): "Matter and energy systems seem granular in structure, and so does "life", but not so mind", (ScE1) p. 134.


References

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

(DeH) Dehnen H., Hönl H., Westphal K., Ein heuristischer Zugang zur allgemeinen Relativitätstheorie, Annalen der Physik, Vol. 461, No. 7-8, (1960) 370-406

(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., Statistical Thermodynamics, Dover Publications Inc., New York, 1989

(ScE1) Schrödinger E., What is Life? The Physical Aspects of the Living Cell with Mind and Matter, Cambridge University Press, Cambridge, 1967   (UnA) Unzicker A., Einstein’s Lost Key: How We Overlooked the Best Idea of the 20th Century, 2015