(Wikipedia)
physical states are described by vectors of a Hilbert space, i.e. physical states are mapped injective onto the radiancies of a Hilbert space."
The Higgs boson combines the existence of mass together with the action of the weak force. But why it provides especially to the quarks that much mass, is still a mystery."Indeed there is no observation concerned with the geometrical shape of a particle or even with an atom." (E. Schrödinger)DISCLAIMER
For an alternative introduction with respect to the Yang-Mills equations and the related mass gap problem we refer to http://www.yang-mills-equations.de
Schrödinger E., " 1. The spiritual bearing of science on life: ... " ... "There is the problem of matter. What is matter? How are we to picture matter in our mind? The first form of question is ludicrous. (how should we say what matter is - or, if it comes to that, what electicity is - both being phenomena given to us once only?) The second form already betrays the whole change of attitue: matter is an image in our mind - mind is thus prior to matter (notwithstanding the strange empirical dependence of my mental processes on the physical data of a certain portion of matter, viz. my brain)...Form, not substance, the fundamental concept: 4. ... "The situation is rather disconcerting. You will ask: What are these particles then, if they are not individuals? And you may point to another kind of gradual transition, namely that between an ultimate particle and a palpable body in our environment, to which we do attribute individual sameness. ... and if the latter lack individuality, how does, say, my wrist-watch come by individuality? What is the limit? How does individuality arise at all in objects composed of non-individuals?" ...7. The intricacy of the continuum: ... "It seems simple to us, because the idea of the continuum seems simple to us. We have somehow lost sight of thr difficulties it implies. That is due to a suitable conditioning in early childhood. Such an idea as 'all the numbers between 0 and 1' or 'all the numbers between 1 and 2' has become quite familar to us. We just think of them geometrically as the distance of any point like P and Q from 0. ... Among the points P and Q there is also the square(2). We are told that such a number as square(2) worries Pythagoras and his school almost to exhaustion. ...There worry was highly creditable. ... The idea of a continuous range, so familar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is really accessible to us." ...
The today´s well accepted ground state energy formula of the 1-dimensional harmonic quantum oscillator is just (!) a divergent series. Nobody seems to be concerned about this. Sophisticated renormalization techniques were developped to overcome this home made "issue", when building a quantum gravity theory. If the same nonchalance would be allowed to apply to prove the Riemann Hypothesis, this conjecture is simply proven by the only formally valid representation of Zeta(s)/(s(s-1)) as transform of the invariant self adjoint integral operator with the "density Theta function" G(x) (EdH 10.3, 10.5). The free energy of a system of interacting harmonic quantum oscillators to model the Planck blackbody radiation law contains same divergent series ((FeR) 10.85). The underlying still unsolved mathematical conceptual problem is similar to the non-vanishing constant Fourier coefficient of the Theta function for the Riemann Hypothesis (RH) duality problem. The given positive answer to the RH in in combination with remarkable properties of the Hilbert/Riesz transforms enable an alternative mathematical ground state energy model. This finally ends up in a Hilbert space based quantum gravity model, overcoming current handicaps of (metric space, differentiable (!)) manifolds and Weyl's affine connexion concept. The latter one leverages on Lie's concept of contact transforms (LiS). (DeJ) 18, VII: " We further mention also the paper from Aref'eva I. Ya., Volovich I. V., Quantization of the Riemann Zeta-Function and Cosmology, 2007
- not changing the way, "how to measure distances" ( Archimedean axiom), but changing the "what to be measured", i.e. the structure of the underlying field from an ordered to a non-ordered field- not increasing the "degree" of transcendence "complexity" (knowing that this is a question of yes/no, of course), if this is measured by Cantor's definition of cardinality (as the field of Non-Standard numbers *R does have the same cardinality than the field of real numbers R) - applying the Riesz and Caldéron-Zygmund Pseudo Differential Operators (PDO) with domains in Hilbert spaces H(-a), a>0 enabling convergent (!) quantum oscillator energy series in a Hilbert space H(-a), for appropriate a>0.The concept of affine connection is the weakest structure with which one can endow a manifold so that parallel displacement of vectors along curves is possible considerably less than a Riemannian structure is required. "Parallel displacement" is a "linear concept" along coordination (x, or y- or z-) axis; any space (or space-time) "point" does have infinitely directions of "displacements", which is nothing more than the whole "rotation" "ball" of a "mass element" dx, with x=(x,y,,z).- enabling an infinitesimal small geometry model with an differentiable (!) manifolds- enabling a truly infinitesimal geometry, alternatively to the affine connexions (affine, parallel infinitesimal displacements, only)A new Hilbert scale based truly infinitesimal small geometry is provided, which is proposed to replace the Semi-Riemannian (metrical space) 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 (of an appropriately defined Hilbert space). The latter one is consistent with existing quantum states modelling within a Hilbert space framework. The quantization technique from quantum mechanics transfers a physical "real" world´s Partial Differential Equations (PDE) model to a mathematical (transcendental) "quantum" world Hilbert space model. The new model operates the other way around: out of the a priori "quantum/ mathematical" distributional Hilbert space world H(-a) and corresponding Pseudo-Differential Equations (PDO) (variational form representation with respect to the H(-a) inner product) it defines and re-produces the today's "real / physical" PDE world (its variational representation within a Hilbert space H(0)) by orthogonal projection from H(-a) into H(0).
A. Eight mathematical and philosophical guides B. Four mathematical baseline facts C. Eight mathematical key characteristics
The four mathematicians and the four philosophers, who provide guidance and support on our journey, are B. Riemann, J. Plemelj, H. Weyl, E. Schrödinger and G. W. Leibniz, I. Kant, A. Schopenhauer, M. Heidegger. The baseline of Schopenhauer's philosophy concerning natural recognition is the philosophy of Kant. The common denominator of Schopenhauer's "
1. The topics "quantum gravity" and "ground state energy" are mathematical modelling challenges and not problems of physics. The purely experimental facts are given, its interpretations might be not adequate or even wrong due to not adequately applied and/or not appropriately formulated mathematical models. But the validation of physical observations and related measured data requires a common mathematical model, which is not existing today. 2. Physical (field) laws are formulated as Partial Differential Equations (PDE), which can be formulated in a weak form, which is about applying variational theory in a Hilbert space framework. Quantum mechanics is enabled and formulated in a specific Hilbert space framework (von Neumann, Heisenberg), which is (by purpose, by accident or by chance) the same framework, which enables probability theory. In order to define the "regularity" of the Dirac function, which is not Lebesgue integrable, it requires distributional Hilbert spaces. The Dirac function is an element of H(-n/2+e) with e>0 and n being the space-time dimension. 3. The Einstein field equations are not well-posed, non-linear PDE w/o appropriate or even missing initial or boundary value conditions. This is even valid for the Hilbert-Einstein action minimization representation of the field equations. This fact is "excused/justified/explained/anticipated" with (basically philosophical) open questions, e.g. concerning the (anyway transcendental) "appropriate" assumptions of the big-bang theory and related singularity (black whole) phenomenon. The "measurement" capability of today's (Riemannean) metric space model is built on (not only continuous (Riemannean)) differentiable manifolds, which is not appropriate to capture energy and/or action relationships between quantums to model the ground state energy phenomenon appropriately. 4. Newton's famous equation m * a = F has it's counterpart in Einstein's famous "tensor" field equations G(i,k) = T(i,k). The latter one is more exact than Newton's formula by a factor 10 * exp(-7). From a modelling perspective there is a huge conceptual difference between both equations: while Newton's equation is embedded in the Euclidean space and is purely describing the relationship between three isolated (a priori) physical terms (mass, velocity, force) as part of the Euclidean framework, the later one is describing relationships between the"geometrical" (framework) structure itself and its influencing /building matter / energies. The "geometrical" (framework) structure contains matter, which, at the same time, influences / builds / defines the framework:
This situation is not the result of an a priori defined physical requirement to be modelled or a new knowledge about physical "reality" derived from the model, but just the consequence of the chosen mathematical (tensor, manifolds, affine connexions, exterior derivatives) model.
1. Given the contradictorily scene / stage framework of the Einstein field equations the probability that the underyling mathematical model can be leveraged to a well-posed physical model in any future, is 0% 2. The weak (variational) representation of PDE is based on same Hilbert space framework than quantum mechanics; the probability that physical PDE models will meet quantum mechanics models in a common variational theory framework, is 100% 3. The probability (from our human (BEING) perspective) that the universe is discrete, i.e. that there is no continuum at all, is 0%, as long as mathematics is "existing" for human BEING 4. The probability that all (physical) Nature constants are transcendental, is 100% 5. Singular integral operators (PDOs of negative order) operate on Hilbert scale. Their domain is defined per usage and required regularity. The Riesz (PD) Operators have nice properties relative to rotations ((StE), III, 1.2), which enable the definition of an inner (operator) product with domain of the linear space of differentials, which corresponds to a Hilbert space with negative scale: that enables the building of a Hilbert space of n-forms. 6. The mathematical model of a single or double layer potential V(t) induced by "matter" is given by a Lebesgue integral with Newton kernel k(s,t) and a Lebesgue integral by a Stieltjes integral and the mass density u'(s)ds by du(s). ..." eine solche (mass density) Annahme erweist sich aber als eine derart folgenschwere Einschränkung, dass dadurch dem Potentiale V(t) der grösste Teil seiner Leisungsfähigkeit hinweg genommen wird". The challenge of the (contact-interacting problem) test particle does not appear in a distributional framework similar to the Dirac "function" definition (LiI), (LiI1).7. The Hilbert space L(2)=H(0) with e.g. the Hermite polynomials as orthogonal system cannot be the common denominator framework for a quantum gravity. It requires a less regular Hilbert space H(-a), with a>0 appropriately chosen (for the 1-dimensional harmonic quantum oscillator it seems to be sufficent to choose a := n/2). In the context of distributional Hilbert spaces we refer to (LiI), (LiI1). With respect to a corresponding alternative polynomial systems to the Hermite or Laguerre polynomials we refer to (SeA). As the H(0) Hilbert space is compact embedded in H(-a) for any a>,0 (and dense with respect to the H(-a) norm) "discrete" phenomenon in H(0). like quantum leaps and uncertainty inequality, could be modelled as orthogonal projection from H(-a) into H(0).of corresponding "continuous" phenomenon in H(-a). 8. In variational theory the Dirichlet integral D(u,v) (= the Lebesgue integral of the product of gradient(u) * gradient(v)) defines the inner product of the standard (energy) Hilbert space H(1). The Dirichlet / gradient operator, which defines the (energy) inner product D(u,v) is replaced by the Calderon-Zygmund operator to define an alternative (energy) inner product (Su,Sv)(a) with appropriately chosen scale factor "a". The above equalities put the norm of a “mass element” into relationship to an element of the H(0) Hilbert space, which is framework of today's quantum mechanics. What's remaining to do is an appropriate Interpretation and probably a revisiting of the current state axiom of quantum mechanics.
(AhJ) Ahner J. F., "
(DeJ) Derbyshine J., " (FeR) Feynman R. P., Hibbs A.R., " (LiS) Lie S., " (LiI) Lifanov I.K., Poltavskii L.N., Vainikko G.M., " (LiI1) Lifanov, I. K., Nenashev A. S.,
(PlJ) Plemelj J., " Potentialtheoretische Untersuchungen", B. G. Teubner Verlag, 1911 http://www.navier-stokes-equations.com/ (PoP) Poluyan P. V., " (ScE) Scholz E., " (ScE1) Schrödinger E., " (SeA) Sedletskii A. M., "
(SoH) Sohr H, " (TrE) Trefftz E., "
(WEH) Weyl H., " (WeH2) Weyl H, " | |||||||||||||||||