We provide a new ground state energy model, which ensures convergent quantum oscillator energy series. This enables the definition of a truly infinitesimal geometry. The corresponding inner product with its induced norm gives the appropriate metric. The domains of related self-adjoint, positive definite operators to build appropriate eigenpair structures are built on Cartan´s differential forms. By this, H. Weyl´s "truly" infinitesimal (affine connexions, parallel displacements, differentiable manifolds based) geometry is replaced by a truly infinitesimal (rotation group based) geometry.
The today´s well accepted zero energy formula of the quantum oscillator is (just (!)) a divergent series. Nobody seems to be concerned about his. Sophisticated renormalization techniques were developed to overcome this homemade "issue", when trying to build a quantum gravity theory, which failed until today. ("Superstrings" have not yet reached a status to be called a "theory", ... at least, to the author's opinion, from a mathematical perspective; the "loop quantum gravity" builds a Hilbert space framework, but puts the whole complexity in a sum of a series of Hamiltonian operators with corresponding (Hilbert space) domains ). The free energy of a system of interacting oscillators to model the Planck blackbody radiation law contains same divergent series as the quantum oscillator (Feynman R., P. Hibbs A. R., "Quantum Mechanics and Path Integrals", (10.85)).
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.
The new concept has a direct relationship and impact to the idea of H. Weyl of a (to-be-built) truly infinitesimal geometry:we claim, that in case of space dimension n=1 the proposed inner (!) product for 1-forms is the enabler of this. It also provides an alternative to the concept of S. Lie about "contact transformation", which was developed to allow an analysis (enabling the concept of co-variant derivatives) on manifolds. A contact transform is a point transform, which also transforms the Pfaff problem dz-pdx-qdy=0 into itself, i.e. two contacting areas are transformed into again two contacting areas (only in an infinitesimal small area, of cause). This concept is required to "bridge" the gap between the mathematical concept of affine connexion on manifolds with a (still missing) truly infinitely geometry of the continuum, which overcomes the today's particle-field dualism paradox.
Our proposed new mathematical framework replaces the manifold concept by a Hilbert space concept:
the manifold concept requires additionally the concept of gauge theory to ensure group properties in relation to the "affine built" vector spaces, just because those properties are not provided by the manifold itself.
Our proposed alternative Hilbert space framework H(-a) provides “geometrical space” properties per definition, i.e. a Hilbert space anticipates appropriate “geometry structure” properties per definition by its inner product with corresponding norm, which ultimately builds a metric. The tool to build this inner product is based on an appropriately defined alternative "contact" transformation for infinitesimal small entities. In case of space dimension n=1 it is built on ("nomen est omen") the Hilbert transform (which also plays a key role in conformal mapping theory), with its remarkable properties, especially in the context of H(0) Hilbert space theory. It is applied to 1-forms, which is basically a Riemann-Stieltjes (singular) convolution integral, which enables a linkage to Hilbert scale theory.
The Hilbert transformation is a PDO of order "0" with in our case chosen domain of 1-forms. It can be reformulated as singular Calderon-Zygmund (convolution) PDO of order "1".
With respect to F. Klein's algebraic approach to classify a geometry, we note: „the entirety of all properties, which do not change by the transformations of a group defines the geometry".
By this principle there is the relationship between:
- the Euclidean geometry and the group of movements (not truly covering infinitesimal displacements)
- the affine geometry and the affine group (not truly covering all kinds of infinitesimal displacements)
- the projective geometry and the projective group (modeling the “infinity” by managing straight lines, which clip at infinity (but not truly covering all kinds of infinitesimal displacements)).
With respect to our proposition above we propose and claim the following relationship:
"A truly infinitesimal geometry can only be defined by an infinitesimal rotation group":
a "truly" infinitesimal affine (i.e. parallel) only geometry requires uniquely to-be-defined (measurement) directions of the required displacements (which relates one-to-one to the underlying space (-time) dimension). "All" remaining potentially other "out-of-scope "displacements" are of same cardinality as the unit interval, i.e. same cardinality as the field of the real numbers (i.e. the same cardinality as the field of the Non-standard numbers)!!
Therefore, the affine geometry should not be accepted as a "truly" infinitesimal model.