# 14th Colloquium on Mathematics and Foundations of Quantum Theory

## February 4, 2022

## About

This event is organized alternately each seminar by the groups of Dirk - AndrÃ© Deckert (LMU), Wojciech Dybalski (U PoznaÅ„), Felix Finster (U Regensburg), and Peter Pickl (U TÃ¼bingen).**The event will take place on Zoom:**Please, join us on Zoom here.

## Program

Time
| Event |
Description |

13:20-13:30 | Opening | Welcome and gathering before the talks |

13:30-14:30 | Talk | Daniela Cadamuro: Fermionic integrable models and graded Borchers triples |

14:30-15:00 | Discussion break | BYO coffee and cookies |

15:00-16:00 | Talk | Miguel Ballesteros: Levinson Theorem for Matrix-Valued ShrÃ¶dinger Operators on the Discrete Line |

16:00-16:30 | Discussion break | BYO coffee and cookies |

16:30-17:30 | Talk | A. Shadi Tahvildar-Zadeh: Classical and Quantum Laws of Motion for Singularities of Spacetime |

17:30-18:00 | Discussion break | BYO coffee and cookies |

18:00 and possible open end | Closing | Final discussion, get-together, clsoing |

## Talks

### Fermionic integrable models and graded Borchers triples

**Speaker:** Daniela Cadamuro (U Leipzig)

**Abstract:** We present a construction of 2D quantum field
theories with asymptotic fermionic particles in the operator-algebraic
framework by using the notion of a "graded" Borchers triple. The
non-triviality of the graded local algebras is still implied by the usual
modular nuclearity condition, since this is unchanged under the grading. Only
the even part of such algebra is observable, even if it lacks Haag-duality. We
apply Haag-Ruelle scattering theory in the version that uses wedge-local
operators, and show that the asymptotic particles are indeed fermionic.
Finally, we provide an explicit example of such construction in the context of
quantum integrable models in 1+1 dimensions, and discuss the characterization
of the local observables in these models, and therefore also the connection to
the form factor programme.

### Levinson Theorem for Matrix-Valued ShrÃ¶dinger Operators on the Discrete Line.

**Speaker:** Miguel Ballesteros (UNAM Mexico City)

**Abstract: We study spectral and stationary scattering theories
for matrix-valued ShrÃ¶dinger operators on the discrete real line. We derive
formulas for the scattering matrix in the band edges in both: the exceptional
and the generic cases. In the exceptional case, the so-called half bound
states appear (they are generalized eigenvectors that are not square summable,
but they are bounded). We prove a formula that relates scattering data to
bound and half-bound states in the Levinson Theorem.**

### Classical and Quantum Laws of Motion for Singularities of Spacetime

**Speaker:** A. Shadi Tahvildar-Zadeh (U Rutgers)

**Abstract:** I will report on recent progress towards a fully
relativistic quantum-mechanical theory of motion for a fixed, finite number of
electrons, photons, and their anti-particles. I will briefly explain the
necessary conditions under which worldlines of charged particles can be
identified with time-like singularities of spacetime, and show examples of
classical as well as quantum theories of motion for them. I will present a
unifying framework for defining a quantum-mechanical wave function that guides
the motion of a particle, regardless of whether it is a boson or a fermion, and
use that to obtain a Lorenz-covariant system of multi-time wave equations for
an interacting two-body system in one space dimension, comprised of one
electron and one photon. I will demonstrate that the corresponding
initial-boundary-value problem is well-posed, and that the resulting electron
and photon trajectories behave in a way that is consistent with Compton
scattering. I will conclude with some future directions to pursue.