The famous Einstein field equations give the relationship between space-time tensor G=G(i,k) and the corresponding energy-matter tensor T=T(I,k). One of the challenges of this great system of partial differential equations is the fact, that the space-time tensor describes the universe structure and the energy-matter tensor describes the “dynamics” within this universe, i.e. the space-time tensor is a modelling element of the “stage”, while the energy-matter tensor is a modelling element of “events resp. actors” happening/acting on this stage.
The equations say that one of the elements determines the other. From a model design perspective this is the root cause, while the field equations do not have, and cannot have, initial value functions or boundary value functions, which is a mathematical problem per se. In the corresponding Hilbert-Einstein functional minimizing (the action) description this issue is reflected by not mathematical adequately defined domain of the underlying operator equation. With respect to a quantum gravity theory this is seen by the author as first opportunity for a more generalized, but then appropriate Hilbert-Einstein characterization of the Einstein field equations. As all those kinds of mathematical models are anyway describing transcendental areas, there is no loss of “truth”, but the chance to get a consistent model, which fits also for quantum theory.
The conceptual mathematical elements of quantum theory are functional analysis, Hilbert space framework with INNER product and spectral theory.
The conceptual mathematical elements of gravity theory are PDE, manifolds framework with metric and exteriorderivatives, and affine connexions.
From a properly designed (mathematical) model of quantum gravity all elements from the above need to be deduced, if the current (gravity and quantum) models should be kept valid for their specific areas. How this can be achieved, when there is no possibility
- to derive a Hilbert space framework from a purely metric space framework (the other way around would be possible, as any Hilbert space is also a metric space)
- to derive a manifolds/affine connexion concept based on exterior differential forms from a (quantum theory) Hilbert space framework ?
The only way out, based on the constraint to keep the Hilbert space framework, is, to build an alternative Hilbert space (which is basically about the definition of an appropriate inner product), which is able to define a geometry for infinitesimal small differentials. This is the linkage to the section “ground state energy”.
The above mentioned issues with initial value and boundary functions of the Einstein field equations then turn over to adequate definition of the domains of operators, acting on those domains. Of course, the regularity of such a Hilbert space needs to be less regular than current quantum theory framework. To derive current quantum theory from the new quantum gravity Hilbert space framework can then be achieved by standard orthogonal projection, enabling also the full power of spectral theory; Hilbert space approximation theory then can even quantify the “approximation error”: the “truly” quantum gravity model is given by operator norm minimization formulation in the less regular HS framework (which is equivalent to corresponding variational equation in appropriate Hilbert space energy inner product), while the “approximating model” gives the today’s “observation model”, which is basically the probability Lebesgue L(2) Hilbert space with its orthogonal (Hermite) polynomials.The practical utility of the field equations is pure. There are only a few metrics/solutions derived out of it. The most prominent and applied one is the Schwarzschild metric. The additional mathematical assumptions to define a well posed problem to enable its calculation, are very strong (Trefftz E., "Das statische Gravitationsfeld zweier Massenpunkte in der Einsteinschen Theorie"). To the author's best knowledge this metric is in most of the time the preferred metric, when analysing black wholes, big bang and related singularities scenarios. The outcome/consequence of the model seems to "generate" necessarily singularities, which then becomes the starting point for philosophical discussion about space-time structure (expanding universe, the very first moments of the universe, etc.). Why not challenging the mathematical assumption of the model itself, which is basically the metric (affine connexions enabling) space, with its missing capability to capture non-affine manifolds relationships/derivatives. An alternative Hilbert space framework would very likely provide alternative interpretations of "time arrow" and "entropy".