Quantum interference on the non-commutative plane and the quantum-to-classical transition

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2023-04, Vol.56 (16), p.165303
• Main Authors: Pittaway, I B, Scholtz, F G
• Format: Article
• Language: English
• Subjects: decoherence; foundational quantum mechanics; measurement problem; non-commutative quantum mechanics
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/acc3cd

We explore a possible link between the structure of space at short length scales and the emergence of classical phenomena at macroscopic scales. To this end we adopt the paradigm of non-commutative space at short length scales and explicitly compute the outcomes of a double slit experiment and a von Neumann measurement in the non-commutative plane. A very consistent picture of a continuous quantum-to-classical transition emerges. The mechanism driving this transition is standard decoherence, which arises quite naturally from the tensor product structure of the non-commutative quantum Hilbert space. The double slit calculation enables us to establish a lower bound on the non-commutative parameter for this transition to become effective at particle numbers of the order of Avogadro’s number. Similarly, the result of the von Neumann measurement establishes a criterion involving the non-commutative parameter, apparatus size and coupling between system and apparatus for classicality to emerge.