13593 - Applied Algebraic Quantum Theory Modulübersicht

Module Number: 13593
Module Title:Applied Algebraic Quantum Theory
  Angewandte algebraische Quantentheorie
Department: Faculty 1 - Mathematics, Computer Science, Physics, Electrical Engineering and Information Technology
Responsible Staff Member:
  • Prof. Dr.-Ing. habil. Wolff, Matthias
Language of Teaching / Examination:English
Duration:1 semester
Frequency of Offer: On special announcement
Credits: 6
Learning Outcome:After successfully completing the module, students will have a in-depth knowledge of algebraic quantum theory. They know its application in different areas. They can assess the importance of algebraic quantum theory in the context of artificial intelligence.
Contents:Algebraic quantum theory, i.e. the theory of operator algebras and their representations is an important branch of modern functional analysis, connecting non-commutative algebra with topology, measure theory and lattice theory. In the past, many applications in statistical mechanics and quantum field theory have been developed. Yet most recently, algebraic quantum theory appears as a powerful framework for artificial intelligence and cognitive dynamical systems as well. The lecture elucidates the basic concepts of algebraic quantum theory, such as observable algebras, representation theory, and contextual emergence in the light of present and future applications.

Outline
  1. The ideas of pioneer quantum theory as motivation: Systems, state preparation, measurement, dynamics. Operator algebras on Hilbert space.
  2. Coordinate-free (Dirac) and representation-free (von Neumann) descriptions: C*-algebras, W*-algebras, GNS-construction and representation theory.
  3. Classical dynamical systems: symbolic dynamics, neural automata and vector symbolic architectures.
  4. Contextual emergence: context states, weak topologies, singular perturbations, emergent descriptions.
  5. Projector lattices: ontology inference for cognitive dynamical systems.
Recommended Prerequisites:None
Mandatory Prerequisites:None
Forms of Teaching and Proportion:
  • Lecture / 2 Hours per Week per Semester
  • Self organised studies / 150 Hours
Teaching Materials and Literature:
  • Primas, H. (1981). Chemistry, Quantum Mechanics and Reductionism. Lecture Notes in Chemistry. Springer, Berlin.
  • Haag, R. (1992). Local Quantum Physics: Fields, Particles, Algebras. Texts and Monographs in Physics. Springer, Berlin.
  • Sakai, S. (1971). C*-Algebras and W*-Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin.
  • beim Graben, P. & Atmanspacher, H. (2006). Complementarity in classical dynamical systems. Foundations of Physics, 36, 291 – 306.
  • beim Graben, P.; Barrett, A. & Atmanspacher, H. (2009). Stability criteria for the contextual emergence of macrostates in neural networks. Network: Computation in Neural Systems,  20, 178 – 196.
  • Carmantini, G. S.; beim Graben, P.; Desroches, M. & Rodrigues, S. (2017). A modular architecture for transparent computation in recurrent neural networks. Neural Networks, 85, 85 – 105.
  • beim Graben, P.; Huber, M.; Meyer, W.; Römer, R. & Wolff, M. (2021). Vector symbolic architectures for context-free grammars. Cognitive Computation, 10.1007/s12559-021-09974-y.
  • Huber-Liebl, M.; Römer, R.; Wirsching, G.; Schmitt, I.; beim Graben, P. & Wolff, M. (subm.). Quantum-inspired cognitive agents. Frontiers in Applied Mathematics and Statistics.


Module Examination:Final Module Examination (MAP)
Assessment Mode for Module Examination:
  • Written examination, 120 min. OR
  • Oral examination, 30-40 min. (with small number of participants)
In the first lecture it will introduced, if the examination will organized in written or oral form.
Evaluation of Module Examination:Performance Verification – graded
Limited Number of Participants:None
Part of the Study Programme:
  • Master (research-oriented) / Angewandte Mathematik / PO 2019
  • Master (research-oriented) / Artificial Intelligence / PO 2022
  • Master (research-oriented) / Informatik / PO 2008 - 2. SÄ 2017
Remarks:
  • Study programme Artificial Intelligence M.Sc.: Compulsory elective module in complex  „Acquisition, Representation, and Processing“
  • Study programme Artificial Intelligence Engineering M.Sc.: Compulsory elective module in complex „Cognitive Science and Neuroscience“
  • Study programme Computer Science M.Sc.: Compulsory elective module in „Mathematics“ or in field of application „Mathematics“
  • Study programme Applied Mathematics M.Sc.: Compulsory elective module in complex „Analysis / Algebra / Combinatorics“
The module is offered as a block course.
Module Components:
  • Lecture: Applied Algebraic Quantum Theory
  • Related examination
Components to be offered in the Current Semester: