Zero Energy Ontology, Hierarchy of Planck Constants, and Kahler Metric Replacing Unitary S-matrix: Three Pillars of New Quantum Theory

Abstract

The understanding of the unitarity of the S-matrix has remained a major challenge of Topological Geometrodynamics (TGD) for 4 decades. It has become clear that some basic principle is still lacking. Assigning S-matrix to a unitary evolution works in non-relativistic theory but fails already in the generic quantum field theory (QFT). The solution of the problem turned out to be extremely simple. Einstein's great vision was to geometrize gravitation by reducing it to the curvature of space-time. Could the same recipe work for quantum theory? Could the replacement of the flat Kahler metric of Hilbert space with a non-flat one allow the identification of the analog of unitary S-matrix as a geometric property of Hilbert space? Kahler metric is required to geometrize hermitian conjugation. It turns out that the Kahler metric of a Hilbert bundle determined by the Kahler metric of its base space would replace the unitary S-matrix. An amazingly simple argument demonstrates that one can construct scattering probabilities from the matrix elements of Kahler metric and assign to the Kahler metric a unitary S-matrix assuming that some additional conditions guaranteeing that the probabilities are real and non-negative are satisfied. If the probabilities correspond to the real part of the complex analogs of probabilities, it is enough to require that they are non-negative: complex analogs of probabilities would define the analog of a Teichmueller matrix. Teichmueller space parameterizes the complex structures of Riemann surface: could the allowed WCW Kahler metrics - or rather the associated complex probability matrices - correspond to complex structures for some space? By the strong form of holography (SH), the most natural candidate would be a Cartesian product of Teichmueller spaces of partonic 2 surfaces with punctures and string world sheets.