Applied Algebraic Quantum Theory

DATES: 17 - 21 October 2022 (5 sessions)
TIME: 11:00-13:00 (a total of 10 hours)
LOCATION: BCAM Seminar Room and Online

PLEASE NOTE THAT THE COURSE WILL BE HELD ALSO ONLINE

ABSTRACT:
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.


CONTENTS:
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.




REFERENCES:
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.



*Registration is free, but mandatory before 12 October 2022. To sign-up go to https://forms.gle/pS6roZ71zimafdcC9 and fill the registration form.

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