Applied Algebraic Quantum Theory
Angewandte algebraische Quantentheorie
6 LPunregelmäßigEN
Inhalte
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 • The ideas of pioneer quantum theory as motivation: Systems, state preparation, measurement, dynamics. Operator algebras on Hilbert space. • Coordinate-free (Dirac) and representation-free (von Neumann) descriptions: C*-algebras, W*-algebras, GNS-construction and representation theory. • Classical dynamical systems: symbolic dynamics, neural automata and vector symbolic architectures. • Contextual emergence: context states, weak topologies, singular perturbations, emergent descriptions. • Projector lattices: ontology inference for cognitive dynamical systems.
Lernziele
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.
Prüfungsleistung
• 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.
Literatur (8)
- 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.
Bemerkungen
• Study programme Artificial Intelligence M.Sc.: Compulsory elective module in complex „Knowledge 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.
Quelle: Modulbeschreibung der BTU · abgerufen 29.09.2026Original