The analog of unitary S-matrix from a curved K\"ahler geometry of the space of WCW spinor fields

The understanding of the unitarity of the S-matrix has remained a major challenge of TGD for 4 decades. It has become clear that some basic principle is missing. Assigning S-matrix to a unitary evolution works in non-relativistic theory but fails already in generic 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 Kähler metric of Hilbert space with a non-flat one allow to identify unitary S-matrix as a geometric property of Hilbert space?

An amazingly simple argument demonstrates that one can construct scattering probabilities from the matrix elements of Kähler metric and assign to the Kähler 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 Teichmueller matrix. Teichmueller space parameterizes the complex structures of space: could the allowed WCW K”ahler metrics- or rather the associated complex probability matrices – correspond to complex structures for some space? By the strong from of holography, the most natural candidate would be Cartesian product of Teichmueller spaces of partonic 2 surfaces with punctures and string world sheets.

Under some additional conditions one can assign to Kähler metric a unitary S-matrix but this does not seem necessary. The experience with loop spaces suggests that for infinite-D Hilbert spaces the existence of non-flat Kähler metric requires a maximal group of isometries. Hence one expects that the counterpart of S-matrix is highly unique.

In the TGD framework the world of classical worlds (WCW) has Kähler geometry allowing spinor structure. WCW spinors correspond to Fock states for second quantized spinors at space-time surface and induced from second quantized spinors of the imbedding space. Scattering amplitudes would correspond to the Kähler metric for the Hilbert space bundle of WCW spinor fields realized in zero energy ontology and satisfying Teichmueller condition guaranteeing non-negative probabilities.

Equivalence Principle generalizes to level of WCW and its spinor bundle. In ZEO one can assign also to the Kähler space of zero energy states spinor structure and this suggests strongly an infinite hierarchy of second quantizations starting from space-time level, continuing at the level of WCW, and continuing further at the level of the space of zero energy states. This would give an interpretation for an old idea about infinite primes asan infinite hierarchy of second quantizations of an arithmetic QFT.

See the article The analog of unitary S-matrix from a curved K”ahler geometry of the space of WCW spinor fields or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

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