Faculty of Mathematics, Physics
and Informatics
Comenius University Bratislava

Seminar of Department of Theoretical Physics - Peter Schupp (30.1.2024)

Tuesday 30.1.2024 at 14:00, Lecture room C

11. 01. 2024 22.43 hod.
By: Peter Maták

prof. Peter Schupp:
An algebraic formulation of nonassociative quantum mechanics

We present a suitably general formulation of quantum mechanics that can handle nonassociative quantum algebras. The approach is naturally probabilistic, and it reduces to standard quantum theory in the traditional associative setting. The main difference to the usual formulation of quantum mechanics is a careful distinction between the algebra of operators on Hilbert space and the more fundamental algebra of observables that describe the quantum system. We formulate properties of states together with notions of trace and use them to develop GNS constructions. Typical assumptions like power-associativity or alternativity are not needed, but the existence of a 3-cyclic trace is helpful. We describe Heisenberg and Schroedinger pictures of completely positive dynamics, and we illustrate our formalism on the explicit examples of finite-dimensional matrix Jordan algebras as well as the octonion algebra. Time-permitting we will give further examples of even more general quantum algebras.