and related mathematical and theoretical physics research areas. With respect to the below we mention that there are basically two kinds of required "energy" measures as part of the energy inequality which are defined by the H(1/2)- and the L(2)(H(1/2)) norms. The first one could be interpreted as the quantum energy measure while the second one could be interpreted as the (average) energy of this quantum energy on the "classical "phenomenon PDE level. Building on the "H(-1/2)-solution concept" a corresponding Ritz-Galerkin approximation theory for non-linear parabolic PDE with non-regular inital value data is provided. The Maxwell equations build a symmetric hyperbolic system. The Yang-Mills equations are a nonlinear generalization of the Maxwell equations which is semi-linear. The Euler and Einstein equations are quasi-linear, but not semi-linear. Even the simplest linear hyperbolic equation (the wave equation) shows different ('not optimal') properties of corresponding shift theorems. The latter ones indicate the appropriate definition of adequate weak energy inner products. Building on the "H(-1/2)-solution concept" this is addressed in http://www.navier-stokes-equations.com where we present an unusual 'optimal order' shift theorem for hyperbolic problem with a corresponding "unusual" energy norm. This indicates that classical energy measurements and corresponding norms are mathematical problem type (elliptic, parabolic, hyperbolic) dependent reflecting the corresponding physical model situation (and the corresponding observed phenomena) and not a physical model situation independent "truly" physical phenomenon which is valid across different model type situations. At the same time the common quantum energy across such mathematical model types (as proposed in this page, see C.) is measured by same quantum energy norm. We further emphasis the elegant role of the alternatively proposed H(1/2) energy Hilbert space in universal Teichmüller theory. The new mathematical model is enabled by Plemelj's extended Dirichlet (energy inner product) integral definition: the standard Dirichlet integral is related to the Laplacian operator by the Green formula. It defines the energy inner product of the L(2) Hilbert space with corresponding domain H(1). Plemelj's extended Green formula ((PlJ) I, §4,5) require reduced regularity assumptions to the potential function V, i.e. grad(V) does not need to exist on the boundary. As a consequence Plemelj's extended Green formula lead to a Plemelj Dirichlet integral definition with domain H(1/2). We emphasis that this definition can be also applied to define a weak representation of the NSE which motivated the provided solution of B. The concept finally leads to PDE classes (elliptic, parabolic, hyperbolic) specific energy norm definitions, whereby the energy norms for elliptic and parabolic are conceptually similar. The energy norm for hyperbolic PDE needs to be conceptually different to anticipate e.g. wave fronts phenomena and the Huygens (space dimension depending) principle. Additionally we propose an alternative quantum harmonic oscillator model based on an alternatively defined Schrödinger momentum operator. The proposed solution concept allows a generalization to space dimensions n>1 (in contraction to the today's Dirac function concept) enabled by the Riesz operators, which are the corresponding n>1 operators of the n=1 Hilbert transform operator.
The common denominator of the proposed solutions resp. solution concepts A, B, C is about the proposed distributional fractional Hilbert scale with related generalized waves/modes. Classical short distance solutions of PDE are approximations of corresponding weak (variation) short distance solutions and not the other way around. There is only one fractional energy Hilbert space. The different forms of "forces" are expressions of "energy/actions" per considered weak PDE. We don't discuss the impact on corresponding interpretations to quantum mechanics and the mind-body problem, but just want to refer in this context to the following books: (EaS) Easwaran S. E. The Upanishads, Nilgiri Press, Tomales, California, 1987 (NaT) Nagel Th., Mind and Cosmos: Why The Materialist Neo-Darwinian Conception of Nature is Almost Certainly False, Oxford University Press, 2012 (ScA) Schopenhauer A., Die Welt als Wille und Vorstellung, erste Auflage, Leipzig, 1818 (ScE) Schrödinger E., The Interpretation of Quantum Mechanics, Ox Bow Press, Woodbridge, Connecticut, 1995 (ScE1) Schrödinger E., My View of the World, Ox Bow Press, Woodbridge, Connecticut, 1983 (WeH) Weyl H., Gravitation und Elektrizität, Sitzungsberichte Akademie der Wissenschaften Berlin, 1918, 465-480 (WeH1) Weyl H., Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, New Jersey, 1949, 2009 From (ScE1) we quote: VIII, Consciousness and mneme: " IX, On becoming conscious: " With respect to the mentioned Schoperhauer quote above we refer to Schopenhauer's concept of "will" in "The world as will and representation". His definition of "will" is not about the capability of decision making, but it describes a principle of life which is about the energy of life as it is part of any anorganic object (matter) or organic object (organism). By this concept the human body (organism) is the visible form of apperance (manifestation/ image) of the not visible acting principle of life. Schopenhauer and Schrödinger were very much impressed from the Upanishads (EaS). From (EaS) we mention A. Huxley's quote (p. 24): " " In our distributional Hilbert scale framework the "differential" corresponds to an "energy" element: - with respect to the concept of causality between mind/body elements we quote from (ScA) §18, p.151: " - with respect to the laws of transformation of most physical quantities which are intimately connected with that of the differentials and differentiable manifolds we quote from (WeH1) p.86: " Schopenhauer gives four different forms of causality as part of the "world as representation" ("On the fourfold root of the principle of sufficient reason").With the "world as will" principle Schopenhauer (inspired by the universal force and energy of the Upanischaden which they called "Brahma") does not mean neither a "conscious will" nor a "reasonable will" or a "chosen will". It is the purely opposite/counterpart of the "world as representation" principle with no common denominator or overlap. Therefore it is also not governed by the law of causality and has neither mind nor reason or space & time. As a consequence from the later two ones the will can not be split into individual wills. In simple words: "causality" (and related "force" phenomena) is part of the "world as representation" principle (which is related to classical PDE with weak energy Hilbert space H(1)), while "energy" (which is related to singular integral (Pseudodifferential) equations with weak, fractional (energy) Hilbert space H(1/2)) is part of the "world as will" principle. We give a trial to summaries the four forms of truths which build the baseline of the principle of sufficient reason: - the logical truth (i.e. the formal logic; causality which is comprehends by mind with the three forms cause, appeal, motif) - the empirical truth - the transcendental truth - the meta-logical truth (formal condition of thinking: principle of the identity, principle of contradictionary, principle of excluded third identity, principles of sufficient reason).
With respect to Schoperhauer's two world principles our central idea can be phrased in the following way: Schopenhauer's "world as representation" principle (built on the four different forms of causality which are the root causes of the principle of reason) corresponds to the Lagrange world (" " ...
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