A few structural remarks on matrix PTQM and beyond


A few structural remarks on matrix PTQM and beyond

Günther, U.

In the first part of the talk, the historical and structural origin of PT-symmetric ix3 quantum models is briefly sketched: the Yang-Lee edge singularities for the distribution of the zeros of the partition function of the 2D Ising model in the complex plane, the close relation to criticality in the complex extended Landau-Ginzburg model for 2nd-order phase transitions, Fisher's infra-red (IR) approximation near criticality by a quantum field theory with ix3 coupling. Recent conceptually puzzling results from operator theoretic investigations of related quantum mechanical toy models with PT-symmetric ix3 couplings are reinterpreted in this phase transition context.
In the second part of the talk, the specific structural features of PT-symmetric matrix models are discussed: hidden group theoretical aspects, Lie triple systems following from Cartan decompositions of the corresponding Lie algebras, projectivization embeddings to resolve singularities at PT phase transitions. Starting from these structural findings for finite-dimensional PT-symmetric matrix setups, possible technically feasible extensions toward infinite-dimensional Hilbert-Schmidt Lie groups, Fredholm groups and PT-symmetry related Hilbert-Schmidt Grassmannians are sketched.

Keywords: PT symmetric quantum systems; phase transitions; 2D Ising model; Landau-Ginzburg model; infra-red limit; criticality; operator theory; group theory; Lie triple systems; projectivization embeddings; Hilbert-Schmidt Lie groups; Fredholm groups; Hilbert-Schmidt Grassmannians

  • Invited lecture (Conferences)
    PHHQP16: Progress in Quantum Physics with Non-Hermitian Operators, 08.-12.08.2016, Kyoto, Japan

Permalink: https://www.hzdr.de/publications/Publ-24936
Publ.-Id: 24936