Lonti Autumn 2021 Series: Lecture 3. Live Tutorial. Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom. CPT-symmetry is a fundamental symmetry of nature that is respected by all physical processes. It consists of a consecutive reversal of time (T), reflection of space at an arbitrary point (Parity P) and a charge conjugation that replaces particles by antiparticles. I briefly discuss the role P T -symmetry plays in quantum mechanics and how it may be utilised for a consistent formulation of non-Hermitian theories. For a relativistic quantum field theory the CPT-theorem provides the general framework for the validity of this symmetry to occur, by stating that the CPT-symmetry is equivalent to a strong reflection and a simultaneous Hermitian conjugation. In this lecture I will prove the theory in the framework of a Lagrangian quantum field theory for spin 0, 1 and spin 1/2 Dirac fields, by first identifying the separate transformation and a subsequent combination. Subsequently these transformations are used to identify the behaviour of various interaction terms under their action. I conclude by commenting on the experimental observations of CP-symmetry violation in the neutral K-meson decay.

Lonti Autumn 2021 Series: Lecture 3. Release of Recorded Lecture. Available at https://youtu.be/zABaRs1Ghmw CPT-symmetry is a fundamental symmetry of nature that is respected by all physical processes. It consists of a consecutive reversal of time (T), reflection of space at an arbitrary point (Parity P) and a charge conjugation that replaces particles by antiparticles. I briefly discuss the role P T -symmetry plays in quantum mechanics and how it may be utilised for a consistent formulation of non-Hermitian theories. For a relativistic quantum field theory the CPT-theorem provides the general framework for the validity of this symmetry to occur, by stating that the CPT-symmetry is equivalent to a strong reflection and a simultaneous Hermitian conjugation. In this lecture I will prove the theory in the framework of a Lagrangian quantum field theory for spin 0, 1 and spin 1/2 Dirac fields, by first identifying the separate transformation and a subsequent combination. Subsequently these transformations are used to identify the behaviour of various interaction terms under their action. I conclude by commenting on the experimental observations of CP-symmetry violation in the neutral K-meson decay.

I will provide a brief general introduction into non-Hermitian Hamiltonian systems with real eigenvalue spectra, arguing that they represent well defined self consistent physical systems. Such type of models possess usually an antilinear symmetry, as for instance PT-invariance (simultaneous parity and time reversal) and/or are quasi/pseudo Hermitian. Most crucial is that they allow for a consistent quantum mechanical framework possessing a unitary time evolution. The general framework will be applied to some integrable models, such a quantum spin chains, classical integrable systems associated to differential equations and Calogero-Moser-Sutherland models. I will present some recent results.

I will give a brief account of some examples of non-Hermitian Hamiltonian systems with real spectra, which have appeared in the literature over the last four decades. I will discuss the spectral properties of these type of Hamiltonians and explain how their reality results from PT-symmetry, quasi-Hermiticity, pseudo-Hermiticity or supersymmetry. Subsequently I review the general technicalities needed to formulate a consistent quantum mechanical system in this context by constructing an appropriate metric and domain. I will provide some Lie algebraic examples. Taking PT-symmetry as a guiding principle one may construct deformations of integrable models, such as Calogero-Moser-Sutherland models, the Korteweg-deVries or Burgers equation. It turns out that some of these deformations are supersymmetry preserving. Others even leave the integrabilty in tact and therefore lead to new types of integrable system.

We propose affine Toda field theories related to the non-crystallographic Coxeter groups H_2, H_3 and H_4. The classical mass spectrum, the classical three-point couplings and the one-loop corrections to the mass renormalisation are determined. The construction is carried out by means of a reduction procedure from crystallographic to non-crystallographic Coxeter groups. The embedding structure explains for various affine Toda field theories that their particles can be organised in pairs, such that their relative masses differ by the golden ratio.