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arXiv:2101.06108 · Journal of Physics A Mathematical and Theoretical

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

I B Pittaway, Frederik G. Scholtz

Published Mar 13, 2023• 6 citations
arXivHTMLPDF

Abstract

Abstract 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.

DOI
10.1088/1751-8121/acc3cd

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Abstract 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.
We investigate a quantum-to-classical transition which arises naturally within the fuzzy sphere formalism for three-dimensional non-commutative quantum mechanics. This transition may be understood as the mechanism of decoherence, but without requiring an additional external heat bath. We focus on treating a two-pinhole interference configuration within this formalism, as it provides an illustrative toy model for which this transition is readily observed and quantified. Specifically, we demonstrate a suppression of the quantum interference effects for objects passing through the pinholes with sufficiently-high energies or numbers of constituent particles. Our work extends a similar treatment of the double slit experiment by Pittaway and Scholtz (2021) within the two-dimensional Moyal plane, only it addresses two key shortcomings that arise in that context. These are, firstly that the interference pattern in the Moyal plane lacks the expected reflection symmetry present in the pinhole setup, and secondly that the quantum-to-classical transition manifested in the Moyal plane occurs only at unrealistically high velocities and/or particle numbers. Both of these issues are solved in the fuzzy sphere framework.
We present the standard double slit experiment with non-relativistic particles in the Heisenberg Picture of quantum mechanics. Our motivation is threefold. First and foremost, and contrary to some claims in the literature, we show that there is no need to talk about non-locality when explaining the interference fringes. Secondly, we emphasise the fact that even in the non-relativistic regime, and in order to preserve locality, we should define the position and momentum observables of a particle as functions of both space and time (and not just time). Thirdly, our presentation compares the projective measurements in the Heisenberg picture with the "Church of the Larger Hilbert Space", the latter of which is seldom discussed in the Heisenberg picture of quantum mechanics.
In previous papers we have explained how a sequence of theorems by John von Neumann on infinite tensor products (ITP) can be understood as providing elements to support both sectorisation of the Hilbert space of large quantum systems, and a mechanism of self decoherence thereof. These two effects may help understanding the articulation of the classical and quantum realms. However, as they involve considering an infinite number of quantum degrees of freedom, legitimate concerns can be raised on their applicability. In this paper, we address explicitly the interface between both realms through the example of a simplified model of a photon polarisation measurement device. Guided by the fact that there is von Neumann sectorisation at infinity, and by the necessity of classical contexts to perform measurements, we show that this limit can be under control, and that although the full force of the sectorisation theorems requires taking the infinite limit, early signs of the macroscopic behaviour appear before infinity. In our example, this shows up in photodiodes through diverging electron avalanches that simultaneously make the system classical, localise it randomly in a macroscopic sector and provide a macroscopic signal. This lays the grounds for justifying the inclusion in quantum physics of the ITP formalism, which involves non-separable Hilbert spaces and potentially type-III von Neumann algebras. Such an approach could make sense of the quantum-classical transition as a primarily algebraic one.
The double slit experiment provides a classic example of both interference and the effect of observation in quantum physics. When particles are sent individually through a pair of slits, a wave-like interference pattern develops, but no such interference is found when one observes which "path" the particles take. We present a model of interference, dephasing, and measurement-induced decoherence in a one-dimensional version of the double-slit experiment. Using this model, we demonstrate how the loss of interference in the system is correlated with the information gain by the measuring apparatus/observer. In doing so, we give a modern account of measurement in this paradigmatic example of quantum physics that is accessible to students taking quantum mechanics at the graduate or senior undergraduate levels.
This paper presents a theory of quantum interference based on Time-Scale Dynamics, aiming to explain the essence of phenomena such as the double-slit experiment from first principles. The core physical picture is that interference is not an inherent property of waves, nor an effect of consciousness, but a structural consequence of whether different evolutionary paths still share the same intrinsic time scale τ. The theory is built upon the fundamental "stability-activity" contradiction (S \times A \geq C_d) and the resulting spontaneous time-scale differentiation of systems (\tau-clustering). We demonstrate that when two (or more) evolutionary paths have equal \tau values, their statistical weights form a superimposable structure, manifesting as interference fringes. Measurement (or any structural coupling) essentially introduces \tau-splitting, destroying time-structural equivalence and causing interference to vanish. The theory naturally derives the Born rule, explains the necessity of three-dimensional space, and unifies concepts such as quantum interference, wavefunction collapse, and decoherence as statistical manifestations of time-scale structures. Complete Python implementations are provided, supporting computational verification of all conclusions. Keywords: time scale; quantum interference; \tau-structure; Born rule; three-dimensional space; emergent symmetry