This section applies the proposed solution concept of the YME section to the harmonic quantum oscillator. The today's 1-dimensional harmonic quantum energy model, which is the Helmholtz free energy, provides the simplest model of a ground state energy definition. The underlying defining function -log2sinh(x) plays also a key role in the vaccum energy of electromagnetic fields and the Planck body radiation law. The total energy of the harmonic quantum oscillator model is represented by a divergent series. The corresponding "re-normalization technique" is by the argument that "
It is built on the representation of the Hilbert space H(-1/2) in the form = The transition from the classical harmonic oscillator model to a corresponding quantum mechanics description is enabled by the Schrödinger momentum operator P:=-i*h*d/dx. The challenge is that by this transition the commutator (x,P) no longer vanishes, as it is the case for the classical model (x,p). The operator A (and the underlying definition of the inner product of H(-1/2)) relates to the Hilbert transform basically by the equation A(d/dx)(u) = H(u) . This enables an alternative Schrödinger momentum operator definition in the form
As a consequence of the above the commutator for quantum states related to the space We note that the Riesz operators are the n-dimensional analogues of the 1-dimensional Hilbert transform operator; those operators enable the analog approach for space dimensions n>1. We further note the relationship of the wave equation with the Lalesco-Picard integral equation (TrF): let l and m denote the greek letters "lamda" and "mue" and let A, B denote arbitrary constants. Then the solutions of the wave equation a*exp(m*x)+B*exp(-m*x) solve the (homogenous) Lalesco-Picard integral equation with Parameter l. provided that the absolute value of Re(m):=Re(squar(1-2*l)) is smaller than 1 which, for real l, implies that l>0. For the fields of real numbers the spectrum of the L-P integral equation covers the infinite segment l>0. Each point of this segment is an eigenvalue of index 2 of the L-P equation.
http://www.quantum-gravitation.de/44250/43532.html http://www.navier-stokes-equations.com/author-s-papers In the following we give some additional context to the related topics of the other sections of this homepage:
(DeJ) 18, VII: " The today´s well accepted zero energy formula of the quantum oscillator is (just (!)) a divergent series. Nobody seems to be concerned about this. Sophisticated renormalization techniques were developed to overcome this home made "issue", when building a quantum field theory. The free energy of a system of interacting oscillators to model the Planck blackbody radiation law contains same divergent series (Feynman R., P., Hibbs A. R., "Quantum Mechanics and Path Integrals", (10.85)). At this point we just mention that from a purely mathematical logic point of view it holds that every assertion/conclusion from a wrong assumption is always true. Therefore, from the obvious wrong assumption above one could e.g. conclude that the universe is a stack of turtles, end of story. This recalls the story of " The underlying still unsolved mathematical conceptual problem is similar to the non-vanishing constant Fourier coefficient of the Theta function for the RH duality problem. The above solution of the RH in combination with remarkable properties of the Hilbert/Riesz transforms enables an alternative mathematical ground state energy model.
We notice that there is a well-established theory of vortex dynamics for incompressible fluid flows (e.g. (MaA). The link to the rotation invariant Riesz operators is given by a vorticity-stream formulation of the Euler and NSE. The celebrated Kelvin's conservation of circulation ( circulation around a curve moving with fluid is constant in time) resp. Helmholtz's conservation of vorticity flux (vorticity flux through a surface moving with the fluid is constant in time) to the Euler equation provides the relationship to the alternatively proposed Plemelj normal derivative.We claim that the provided NSE solution which is addressing the too strong/restrictive mathematical model assumption of the NSE can be also applied to the Maxwell equations, as well as to the zero state energy "problem" of the harmonic quantum oscillator. The d'Alembert paradox (fluid vs. velocity/momentum) is related to this too restrictive assumptions for the NSE, while the Schrödinger paradox (particle vs. wave/momentum) is related to the latter one. The proposed approach for a quantum gravity model is about rotation-invariant differentials as elements of a Hilbert space with negative (distributional) Hilbert scale defined by the eigenpairs of the Hamiltonian operator.The link between the RH-proof-P2 and an appropriate vacuum energy model is given by the Hilbert space H(-1/2) with its corresponding energy Hilbert space H(1/2). The (to be extended tool set) of "quantization" operators are Pseudo-Differential Operators with domains H(-a), a>0 ((AhM) II § 13). The link to the Yang-Mills theory (Coloumb potential, Yukawa potential, Dawson function, confluent hypergeometric functions) is given in Braun K., " An alternative trigonometric integral representation of the Zeta function on the critical line", Note 34Plemelj's alternative concepts of a "mass element" and its related potential enable 1-forms as domain of Pseudo-Differential Operators (PDO), defining graph and energy norms of corresponding Hilbert spaces. Same (resp. isometric) Hilbert spaces (with not positive Hilbert scale factor) build the framework of variational theory to solve Partial Differential or Pseudo Differential equations. Our baseline proposition is to declare the weak representations of PDE or PDO equations as the "truly" physical (world) models (not the strong PDE model) and declare the strong PDE as related (macro world) approximations to the weak representations. The prize to be paid for the macro world approximation is a more restrictive regularity assumption than physically necessary. Then there is a consistent (Hilbert space) model (framework) between quantum & macro world equipped with the geometrical properties of a Hilbert space, providing e.g. an inner product, alternatively to the external product of differentiable manifolds, and enabling e.g. spectral theory. Therefore, the proposed solution approach is based on 1. a problem adequately defined, fractional scaled (energy) Hilbert space, singular integral operators and Plemelj's alternative "potential" definition. 2. a relationship between the asymptotic behavior of the imaginary parts of the Riemann zeta function on the critical line and the large complex zeros of the Jost function in the complex wave number-plane for s-wave scattering by truncated potentials. ad 1: Plemelj's concept is about an alternative definition of a "mass element", which requires less regularity assumptions as the definition of a "mass density". At the same time Plemelj's alternative normal derivative definition, which he called a "current" (" continuity" instead of "differentiability" assumptions) ensures a valid Green formula. Plemelj's "mass element" is one-to-one linked to a differential, i.e. a (truly infinitesimal) Hilbert space framework can be designed for 1-forms. This solution concept is built on the solution concepts of P1 and P2.ad 2: in (JoS) a variant of the Hilbert-Polya conjecture is proposed in this context and considerations about the Dirac sea as “virtual resonances” are discussed. We note that the regularity of the "Dirac sea" is about the distributional Hilbert space H(-n/2+a), a>0, while "our" distributional Hilbert space is about H(-1/2). One key differentiator to standard theory framework is about the fact, that the Legendre transformation is no longer applicable. Therefore, the Lagrange and the Hamiltonian formalism are no longer equivalent; in fact, the Lagrange formalism is no longer defined in the infinitely small "area". A new Hilbert scale based truly infinitesimal small geometry is provided, which is proposed to replace the Semi-Riemannian (metrical space) differentiable (!) manifold concept (and its underlying gauge theory to enable the Standard Model of Elementary Particles) by a Hilbert space based quantum (differential forms) ground state energy model, building on an intrinsic ground state energy scalar product. We consider the 1-dimensional periodic L(2)-functions on the unit circle. Let T denote the normal derivative operator of the double-layer potential, which is a bounded, selfadjoint integral operator ((KrR), theorem 8.21). We apply Plemelj's alternative definitions of a "mass element" and his alternative definition of a potential (PlJ) to define an inner product for 1-form du, dv in the form ((du,dv)):=(Tu,Tv) . The inner product is well defined for u, v being elements of H(0)=L(2), only. Plemelj's alternative normal derivative definition, which he called "current", ensures compatibility with the Green formula, even for less regular functions (L(2)-functions) u and v. Based on the above the Yang-Mills functional can be formulated in a Hilbert space framework as (standard) minimization problem with respect to the energy or operator norm. For u being an element of H(1/2) the energy norm (Tu,u) and operator norm (Tu,Tu) are equivalent to the norms of H(1/2) and H(0)=L(2). In case of the reduced regularity assumption that u is only an element of H(0)=L(2), this "generates" a energy norm and a operator norm, which are equivalent to H(0) and H(-1/2), only. The linkage to quantum mechanics is given by the fact, that the "quantum elements" are represented as elements u and v of the Hilbert space H(0)=L(2). In case of the standard regularity assumptions, i.e. if u and v are elements of H(1), the inner product is identical to the Dirichlet integral. This is due to corresponding properties of the Hilbert transform, i. e. it holds ((du,dv))=(Tu,Tv)=(H(dx)(u),H(d/dx)(v)=((dx)(u),(d/dx)(v) i.e. T = H(d/dx).
(BrK) Braun K., Interior Error Estimates of the Ritz Method for Pseudo-Differential Equations, Jap. Journal of Applied Mathematics, 3, 1, 59-72, (1986) (DeJ) Derbyshine J., Prime Obsession, Joseph Henry Press, Washington D.C., 2003 (ElE) Elizalde E., Zeta functions: formulas and applications, J. Comp. and Appl. Math. 118 (2000) p. 125–142 (EsG) Eskin G. I., (MaA) Majda A. J., Bertozzi A. L., Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002 (ShM) Shubin M.A., Pseudodifferential Operators and Spectral Theory, Springer Verlag, 1987 (TrF) Tricomi, F. G., Integral Equations, Dover Publications, Inc., New York, 1985 | |||||||||||||