Let grad(u) denotes the gradient operator applied to a function u and S(u) be the Calderon-Zygmund operator according to (BrK). In variational theory the Dirichlet integral D(u,v)=(grad(u),grad(v)) defines the energy inner product with (Sobolev) domain H(1) x H(1).
The key idea is to replace the gradient (energy) operator by the Calderon-Zygmund operator and the Dirichlet integral by the corresponding inner product (S(u),S(v)) with corresponding energy inner product with the domain H(1/2) x H(1/2). This then implies a physical state Hilbert space H(-1/2), enabling continuous spectrum in H(-1/2)-H(0), while still governed by the Heisenberg principle with respect to the sub-space H(0).
The (singular) Calderon-Zygmund integrodifferential operator enables less regularity assumption to its domain than (H(1/2)) than standard theory (H(1)). The Dirichlet integral goes along with modeling energy and momentum (H(1)), which requires the concept of space and extented bodies within this space (WeH, III, 22, d). The primary physical concepts and physical laws are the laws of conservation of energy and momentum.
Eskin G. I., "Boundary Value Problems for Elliptic Pseudodifferential Equations", AMS, Providence, Rhode Island, Trnaslations of Mathematical Monographs Vol. 52, 1981
Lifanov I.K., Poltavskii L.N., Vainokko G.M., "Hypersingular Integral Equations And Their Applications", Chapman & Hall/CRC, Boca Raton, London, New York, Washington D.C., 2004
Translating the equation
E = E(kin) + E(pot)
into quantum operator language, means, that the two domains of the operators on the right side of the equation above are orthogonal. In our proposed less regular Hilbert space Environment this will be no longer the case, i.e. there is an intersection between both domains, which is not equal to zero. This means there is no discrete "jump" from kinetic to potential energy anymore. But by orthogonal projection of the corresponding variational equation in H(-a) into the higher regular Hilbert space the sum above becomes the corresponding approximation solution.
The same argument is valid for Einstein's thermo-dynamic theorem, built as the sum of quadrates of energy variances, based on classical particle and wave theory ((HeW) V.7 (110)).
With respect to an alternative definition of a mass element "dm", we refer to the great book of
(PlJ) Plemelj J., "Potentialtheoretische Untersuchungen“, B. G. Teubner Verlag Leipzig, 1911
It provides a physical interpretation of a mass element "dm" which defines a new concept of a "mass element" creating a potential not only by a density of the mass, but by the element "dm" itself.
In case the mass element "dm" does have also a density it roughly holds:
((dm,dm))=(m,m), as it holds (Hm,Hm)=(m,m) .
This means, that the quantity of a quantum "dm" in the sense of quantum mechanics (as an element of the Hilbert space L(2)) corresponds to the norm of the mass element "dm" in our new ground state energy model, which is "just" and only the physical state of its energy (nothing more, but also nothing less!).
With respect to (complementary) variational methods (Friedrichs, Noble) we refer to
Arthurs A. M., "Complementary Variational Principles", Clarendon Press, Oxford, 1970
Velte W., "Direkte Methoden der Variationsrechnung", Teubner Studienbücher, 1976
We emphasis that the dual operator of T:=grad is given by T(*)=-div, while the dual operator of T:=curl is given by T(*)=curl.
The above enables a quantum gravity model, which supports
- the definition of a manifold, providing a purely inner geometry (1st fundamental forms only), building the Hamilton9an formalism on a H(-1)-based quantum mechanics model
- a purely field based theory enabling a purely infinitesimal geometry with proper linkages to differential forms. Those build the foundation of nearly all relevant physical models.The less regular Hilbert space framework H(-a) than the standard L(2)=H(0)-Hilbert space enables a differentiation between the Hamiltonean and the Lagrange formalism. The Legendre transformation proves the equivalence of both formalisms, in case the Legendre transformation is well defined. If this would be no longer the case, the Hamiltonean formalism (action minimization in H(-a), which is about purpose) keeps valid, but the Lagrange formalism (work minimization in H(0), which is about causality) is only be defined, if experiments gives results (to be modelled by probability theory), which needs to be reflected and validated by an appropriate physical model. The first one is beyond the trancendence border, while the causality model is part of the physics world.
In the context of "Emmy Noether's Wonderful Theorem" (D. E. Neuenschwander, The John Hopkins University Press, Baltimore, 2011) we quote:
Noether E. (Invariante Variationsprobleme): "The problems in variation here concerned are such as to admit a continuous group (in Lie's sense); .... what is to follow, therefore, represents a combination of the methods of the formal calculus of variations with those of Lie's group theory"Ramond P. (Field Theory: A Modern Primer (1981). "It is a most beautiful and awe-inspiring fact that all the fundamental laws of Classical Physics can be understood in terms of one mathematical construct called the Action. It yields the classical equations of motion, and analysis of its invariances leads to quantities conserved in the course of the classical motion. In addition, as Dirac and Feynman have shown, the Action acquires ist full importance in Quantum Physics."
The Calderon-Zygmund Pseudo Differential Operator
The singular (Calderon-Zygmund Pseudo Differential) operator with domain of (Cartan's) differential forms is proposed to be the non-standard alternative to the "standard", non-bounded (momentum) differential operator. It is basically an isomorphism from H(a+1) --> H(a) with a real. The requirements from physics determines the setting of the scale factor a: it is proposed to put a:=-1 in order to ensure that the range of S is isomorph to L(2)=H(0).