From Landau-Ginzburg to PTQM: a few structural remarks

The historical and structural origin of PT symmetric ix³ quantum models is briefly sketched. Open questions are discussed and related possible lines for future research are suggested.

Starting from the Yang-Lee result on the distribution of complex zeros of the partition function for the 2D Ising model the structure and origin of the so called Yang-Lee edge singularities is discussed. The close structural relation to complex extended criticality in the Landau-Ginzburg model of 2nd-order phase transitions is shown and Fisher's result is recalled on the field theoretical infra-red (IR) limit of the fluctuating fields as an effective quantum field theory (QFT) with ix³ coupling. Based on this specific phase-transition related conceptual background of the ix³ model recent operator-theoretic findings and still open puzzles of the corresponding quantum mechanical toy model approximations are interpreted as having their origin in such still operator-theoretically unexplored phase-transition related model peculiarities. Possible strategies for future research are briefly sketched to fill corresponding mathematical and technical gaps.