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MP: Fachverband Theoretische und Mathematische Grundlagen der Physik

MP 11: Many-body Theory II

MP 11.3: Talk

Thursday, March 21, 2024, 10:10–10:30, HL 102

On the applicability of Kolmogorov’s theory of probability for the description of quantum phenomena — •Maik Reddiger — Anhalt University of Applied Sciences, Germany

It is a common view that with his axiomatization of quantum mechanics von Neumann laid the foundations of a “non-commutative probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a mathematical theory of probability at the time.

Taking the latter view, I motivate an approach to the foundations of non-relativistic quantum theory that is based on Kolmogorov’s axioms. It relies on the Born rule for particle position probability and employs Madelung’s reformulation of the Schrödinger equation for the introduction of physically natural random variables. While an acceptable mathematical theory of Madelung’s equations remains to be developed, one may nonetheless formulate a mathematically rigorous “hybrid theory”, which is empirically almost equivalent to the quantum-mechanical Schrödinger theory. A major advantage of this approach is its conceptual coherence, in particular with regards to the question of measurement.

The talk is based on Reddiger, Found. Phys. 47, 1317 (2017) and Reddiger & Poirier, J. Phys. A: Math. Theor. 56, 193001 (2023).

Keywords: Madelung equations; Classical limit; Probability theory; de Broglie-Bohm theory; Stochastic mechanics