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arXiv:1401.0176

A clock containing a massive object in a superposition of states; what makes Penrosian wavefunction collapse tick?

Tjerk H. Oosterkamp, Jan Zaanen

Published Dec 31, 2013• 1 citation

Abstract

Penrose has been advocating the view that the collapse of the wave function is rooted in the incompatibility between general relativity and quantum mechanics. On the basis of conceptual analysis, he arrived at an estimate for the collapse time. To better understand his estimate, in this paper we present a thought experiment, which singles out the role of time-dilations in massive superpositions. First we investigate the behavior of a hypothetical clock containing a component which can be in a superposition of states. The clock contains a massive object, whose only purpose is to introduce a curvature of space time into the problem. We find that a state of this massive object with a smaller radius, but with the same mass, experiences a larger time dilation. Considering a coherent superposition of the large and small object, introduces an ambiguity in the definition of a common time for both states. We assert that this time ambiguity can be thought to affect the time evolution of a state in different ways and that the relative phase difference between these different interpretations can be calculated. We postulate that the wave function collapse will occur when this phase difference becomes of order unity. An absolute energy scale enters this equation and we recover Penrose's estimate for the collapse time by equating the absolute energy scale to the rest mass of the object.

DOI: 10.48550/arxiv.1401.0176

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According to Penrose, the fundamental conflict between the superposition principle of quantum mechanics and the general covariance principle of general relativity entails the existence of wavefunction collapse, e.g. a quantum superposition of two different space-time geometries will collapse to one of them due to the ill-definedness of the time-translation operator for the superposition. In this paper, we argue that Penrose's conjecture on gravity's role in wavefunction collapse is debatable. First of all, it is still a controversial issue what the exact nature of the conflict is and how to resolve it. Secondly, Penrose's argument by analogy is too weak to establish a necessary connection between wavefunction collapse and the conflict as understood by him. Thirdly, the conflict does not necessarily lead to wavefunction collapse. For the conflict or the problem of ill-definedness for a superposition of different space-time geometries also needs to be solved before the collapse of the superposition finishes, and once the conflict has been resolved, the wavefunction collapse will lose its physical basis relating to the conflict. In addition, we argue that Penrose's suggestions for the collapse time formula and collapse states are also problematic.

According to Penrose, the fundamental conflict between the superposition principle of quantum mechanics and the principle of general covariance of general relativity entails the existence of wavefunction collapse, e.g. a quantum superposition of two different space-time geometries will collapse to one of them due to the ill-definedness of the time-translation operator for the superposition. In this paper, we argue that Penrose's conjecture on gravity's role in wavefunction collapse is debatable. First of all, it is still a controversial issue what the exact nature of the conflict is and how to resolve it. Secondly, Penrose's argument by analogy is too weak to establish a necessary connection between wavefunction collapse and the conflict as understood by him. Thirdly, the conflict does not necessarily lead to wavefunction collapse. For the conflict or the problem of ill-definedness for a superposition of different space-time geometries also needs to be solved before the collapse of the superposition finishes, and once the conflict has been resolved, the wavefunction collapse will lose its physical basis relating to the conflict. In addition, we argue that Penrose's suggestions for the collapse time formula and the preferred basis are also problematic.

On the conjectured gravity-related collapse rate

E_\Delta/\hbar

of massive quantum superpositions.

Roger Penrose and the author share the proposal that the spatial superposition

|x_1\rangle+|x_2\rangle

of a massive object collapses into its localized components

|x_1\rangle

or

|x_2\rangle

with the characteristic time

\hbar/E_\Delta

where

E_\Delta

is the gravitational self-energy excess of the superposition versus the localized states. Underlying arguments of such radical departure from standard quantum mechanics and different derivations of the rate equation are briefly recapitulated and discussed.

The Diósi-Penrose model states that the wave function collapse ending a quantum superposition occurs due to the instability of coexisting gravitational potentials created by distinct geometric conformations of the system in different states. The Heisenberg time-energy principle can be invoked to estimate the collapse time for the energy associated with this instability, the gravitational self-energy. This paper develops atomistic models to calculate the Diósi-Penrose collapse time. It applies them to a range of systems, from small molecules to large biological structures and macroscopic systems. An experiment is suggested to test the Diósi-Penrose hypothesis, and we critically examine the model, highlighting challenges from an atomistic perspective, such as gravitational self-energy saturation and limited extensivity.

For nearly 100 years, the paradox of Schrödinger's Cat has remained unresolved. Why does the world we live in appear classical despite being composed of quantum particles governed by the Schrödinger wave equation? Lajos Diósi and Roger Penrose propose the wavefunction collapses because it describes two incompatible space-times, demonstrating an inconsistency between quantum mechanics and general relativity. To avoid this paradox, collapse must occur within Heisenberg's time-energy uncertainty limit. Subatomic particles with low mass, and correspondingly low energy, collapse in years, while superposed cats would collapse almost instantaneously. We propose a table-top experiment to put two small mirrors into superposition and observe them collapse in a time consistent with the Diósi-Penrose model. We employ two techniques to perform this experiment in ambient laboratory conditions. Most experiments separate a small mass by a large distance. In contrast, we displace a large mass by a small distance where the self-energy follows an inverse square law with correspondingly high collapse times. We further use a symmetrical apparatus, where a break in symmetry indicates collapse independent of decoherence.