Introduction
ground state energy
quantum gravity
mass & vacuum
Weyl affine connexions
Einstein universe
Einstein action minimization
alternative concepts
Hyper-real universe
affected areas
who I am
literature


which is about a Hilbert space of differential forms with appropriately defined inner product replacing an exterior algebra (with exterior derivatives) over differential forms.

Prolog

(RoC) xi, "The problem of what happens to classical general relativity at the extreme short-distance Planck scale of 10*exp(-33) cm is clearly one of the most pressing in all of physics. It seems abundantly clear that profound modifications of existing theoretical structures will be mandatory by the time one reaches that distance scale. There exists several serious responses to this challenge. These include effective field theory, string theory, loop quantum gravity, thermo-gravity, holography, and emergent gravity. .....

.... it is probably that all these ideologies, including my own (which is distinct from the above listing), are dead wrong. The evidence is history: from the Greeks to Kepler to Einstein there has been no shortage of grand ideas regarding the basic questions."

(RoC) Rovelli C., "Quantum Gravity", Cambridge University Press, 2004

(VeW) Velte W., "Direkte Methoden der Variationsrechnung", Teubner Studienbücher, 1976


Rational

The characteristic of an affine geometry is the fact, that only parallel distances can be measured against each other, i.e. other kinds of infinitesimal small "actions" between not parallel "objects" are not considered in this kind of "continuum". Vectors are the mathematical model of such translations (resp. parallel displacements) and the underlying (affine) geometry is mathematically described by the group properties of vectors (WeH). An affine geometry with space dimension n is the “same” as its related (n-1)-dimensional projective group. The “enrichment” of the today's n-dimensional space-time affine geometry (manifolds and affine connexions, (ScE), (WeH1) and quotes § 18 below) goes along with the concept of exterior derivatives to allow “measurements” and the definition of appropriate metrics resp. to link to the Riemannian metric and the concept of curvature. 

The Sobolev H(1/2) space on the circle plays a key role of universal period mapping universal Teichmüller parameter space for all Riemann surfaces via quantum calculus:

Biswas I., Nag S., Jacobians of Riemann Surfaces and the Sobolev Space H(1 2) on the Circle

Nag S., Sullivan D., Teichmüller theory and the universal period mapping via quantum calculus and the H(1 2) space on the circle.

We propose a quantum gravity model

- building on Hilbert space, alternatively to manifolds (metric space, only)

- enabling an infinitesimal small geometry model with an inner product defined by "rotating differential forms", alternatively to exterior derivatives based on differentiable (!) manifolds

- enabling a truly infinitesimal geometry, alternatively to the affine connexions (affine, parallel infinitesimal displacements, only)

- 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.


Here we are:

    

Braun K., A quantum gravity and ground state energy Hilbert space model