A few remarks on the structure of PT quantum mechanics

In the first part of the talk, starting from a historical discussion of the 2-dimensional Ising model, the Yang-Lee analysis of the zeros of the corresponding partition function and the occurrence of the Yang-Lee edge singularities the structural origin of the quantum mechanical toy model Hamiltonian with ix^3 potential is elucidated. The close relationship of this Hamiltonian to the Landau theory of phase transitions and conformal field theories (CFTs) is sketched what provides an intuitive explanation for the operator-theoretic difficulties in treating a conjectured Hermitian structure of the ix^3 model in full depth.
In the second part of the talk, the Krein space and Hilbert space metric structures of quasi-Hermitian PT-symmetric matrix models are discussed with emphasis on the underlying general Lie group structures of these metric operators. The Cartan decomposition into compact and noncompact metric components is used to show the existence of an underlying Lie triple system and its relation to the curvature of homogeneous coset spaces.
Finally, several extension schemes from finite-dimensional Lie groups toward ∞−dimensional Lie groups and Hilbert-Schmidt Lie groups are sketched.