The toolbox of the proposed unified plasma, quantum und gravity field theory covers among other things the mathematical areas
- orthogonal decompositions of Hilbert spaces - potential operators defined by indefinite inner products - Krein spaces in a quarternionic setting - variational (approximation) methods in Hilbert scales - the complex Lorentz group and its two connected components - classical & quantum tensor theory - quantum (two-component) spinor theory - two-component Magnetohydrodynamics.
The Hilbert scale decompositions model appropriately defined (quantum element and quantum energy) Hilbert spaces. More specifically, there is an energy Hilbert space H(1/2), which is decomposed into a kinematical sub-Hilbert space H(1) and an orthogonal closed "ground state energy" space H(1,ortho). The kinematical Hilbert space H(1) is governed by a countable orthogonal Riesz basis enabled by the eigen-pairs of a (kinematical) selfadjoint, positive definite operator with domain H(1). Geometrically speaking, the kinematical Hilbert space H(1) can be interpreted as a coarse-grained embedded Hilbert sub-spaces of H(1/2). The related closed (ground state energy) sub-space H(1,ortho) is governed by a continuous spectrum.
We note that the above proposed new „conceptual elements“ match to the "unconventional features of the mathematical formalism" in (HeW).