PTsymmetry and related geometrical structures
PTsymmetry and related geometrical structures
Günther, U.
Abstract
In nonrelativistic quantum mechanics, the dynamics of closed quantum systems is described by Hamiltonians which are selfadjoint in appropriately chosen Hilbert spaces. For PTsymmetric quantum systems, the Hamiltonians are in general no longer selfadjoint in standard Hilbert spaces, but rather they are selfadjoint in Krein spaces, Hilbert spaces endowed with indefinite metric structures. Moreover, the spectra of PTsymmetric Hamiltonians are symmetric with regard to the real axis in the spectral plane. Apart from Hamiltonians with purely real spectra this includes also Hamiltonians whose spectra may contain sectors of pairwise complexconjugate eigenvalues. Considering families of parameterdependent Hamiltonians one can arrange for parameterinduced passages from sectors of purely real spectra to sectors of complexconjugate spectral branches. Corresponding passages can be regarded as PTphase transitions from sectors of exact PTsymmetry to sectors of spontaneously broken PTsymmetry. Approaching a PTphase transition point, the eigenvectors of the Hamiltonian tend toward their isotropic limit  an, in general, infinitedimensional (Kreinspace) generalization of the lightcone limit in Minkowski space. At a phase transition, the Hamiltonian is no longer diagonalizable, but similar to an arrangement of nontrivial Jordanblocks. The interplay of these structures is briefly reviewed with special emphasis on the related Liealgebraic and Liegroup aspects. With the help of Cartandecompositions associated hyperbolic structures and Lietriplesystems are discussed for finitedimensional setups as well as for their infinitedimensional generalizations (HilbertSchmidt (HS) Lie groups, HS Lie algebras, HS Grassmannians). The interconnection of Kreinspace structures and PTphase transitions is demonstrated on two exactly solvable models: PTsymmetric BoseHubbard models and PTsymmetric plaquette arrangements.
Keywords: PT symmetry; PT phase transitions; Krein spaces; Jordan blocks; Lie algebras; Lie triple systems; HilbertSchmidt Lie groups; HS Lie algebras; HS Grassmannians; PTsymmetric BoseHubbard models; plaquette arrangements

Invited lecture (Conferences)
Symmetry 2017  The First International Conference on Symmetry, 16.18.10.2017, Barcelona, Spain 
Abstract in refereed journal
Proceedings / MDPI AG 2(2018), 25
DOI: 10.3390/proceedings2010025
ISSN: 25043900
Permalink: https://www.hzdr.de/publications/Publ26910