The Measurement Problem within Quantum Mechanics on a Non-commutative Plane

This work addresses the measurement problem in quantum physics through the framework of quantum mechanics on a non-commutative plane. The study is motivated by the conceptual tension between unitary quantum evolution and the non-unitary collapse postulate, and by the incompatibility between general relativity and standard quantum mechanics at the Planck scale. Quantum mechanics on the non-commutative plane, characterized by the modified relation [ˆxi, ˆxj ] = iθεij , introduces a fundamental length scale that limits spatial resolution and embeds geometric uncertainty directly into the formal structure of the theory. The postulates of quantum mechanics on a non-commutative configuration space, where quantum states are represented by operator-valued functions acting on a Hilbert space of Hilbert-Schmidt operators, is formulated. This construction alters the structure of measurement, leading to an intrinsic coarse-graining of position and to the identification of position as an emergent preferred basis. The dynamics of such systems are analysed in detail through the double-slit and von Neumann measurement schemes, showing that interference suppression arises as geometric consequences of the intrinsic spatial non-commutativity rather than an external effect. The formalism is then extended to the path integral representation, where the noncommutative structure modifies the classical action and introduces damping of non-classical paths. In the macroscopic limit, this leads to the dominance of classical trajectories to the transition amplitude while suppressing those that deviate from such paths. In summary, this work demonstrates that non-commutative quantum mechanics provides both a consistent mathematical framework and proposes a natural physical mechanism for the apparent emergence of the classical from the quantum. This suggests that a non-commutative spatial structure could play a role in understanding the measurement problem.

Grantor dc:publisher

Stellenbosch : Stellenbosch University

Year dc:date.issued

2026

Author dc:creator

• Pittaway, Ian Bennett

Advisor dc:contributor.advisor

• Scholtz, Frederik G.

Language dc:language.iso

en

Repository record dc:identifier.uri

https://scholar.sun.ac.za/handle/10019.1/135935

OAI identifier oai:identifier

oai:scholar.sun.ac.za:10019.1/135935