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. Notere 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