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arXiv:2111.04604 · AVS Quantum Science

On the conjectured gravity-related collapse rate EΔ/ℏ of massive quantum superpositions

Lajos Diósi

Published Feb 7, 2022• 8 citations
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Abstract

Roger Penrose and the author share the proposal that the spatial superposition |x1⟩+|x2⟩ of a massive object collapses into its localized components |x1⟩ or |x2⟩ with the characteristic time ℏ/EΔ, where EΔ 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.

DOI
10.1116/5.0077919

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arXiv:1401.0176•Dec 31, 2013•1 citation
Roger Penrose and the author share the proposal that the spatial superposition |x1⟩+|x2⟩ of a massive object collapses into its localized components |x1⟩ or |x2⟩ with the characteristic time ℏ/EΔ, where EΔ 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.
On the conjectured gravity-related collapse rate EΔ/ℏE_\Delta/\hbarEΔ​/ℏ of massive quantum superpositions.
Roger Penrose and the author share the proposal that the spatial superposition ∣x1⟩+∣x2⟩|x_1\rangle+|x_2\rangle∣x1​⟩+∣x2​⟩ of a massive object collapses into its localized components ∣x1⟩|x_1\rangle∣x1​⟩ or ∣x2⟩|x_2\rangle∣x2​⟩ with the characteristic time ℏ/EΔ\hbar/E_\Deltaℏ/EΔ​ where EΔE_\DeltaEΔ​ 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.
Inspired by an ontic view of the wavefunction in quantum mechanics and motivated by the universal interaction of gravity, we discuss a possible gravity implication in the state collapse mechanism. Concretely, we investigate the stability of the spatial superposition of a massive quantum state under the gravity effect. In this context, we argue that the stability of the spatially superposed state depends on its gravitational self-energy originating from the effective mass density distribution through the spatially localized eigenstates. We reveal that the gravitational self-interaction between the different spacetime curvatures created by the eigenstate effective masses leads to the reduction of the superposed state to one of the possible localized states. Among others, we discuss such a gravity-driven state reduction. Then, we approach the corresponding collapse time and the induced effective electric current in the case of a charged state, as well as the possible detection aspects.
After providing an extensive overview of the conceptual elements – such as Einstein’s ‘hole argument ’ – that underpin Penrose’s proposal for gravitationally induced quantum state reduction, the proposal is constructively criticised. Penrose has suggested a mechanism for objective reduction of quantum states with postulated collapse time τ = ¯h/∆E, where ∆E is an ill-definedness in the gravitational self-energy stemming from the profound conflict between the principles of superposition and general covariance. Here it is argued that, even if Penrose’s overall conceptual scheme for the breakdown of quantum mechanics is unreservedly accepted, his formula for the collapse time of superpositions reduces to τ → ∞ (∆E → 0) in the strictly Newtonian regime, which is the domain of his proposed experiment to corroborate the effect. A suggestion is made to rectify this situation. In particular, recognising the cogency of Penrose’s reasoning in the domain of full ‘quantum gravity’, it is demonstrated that an appropriate experiment which could in principle corroborate his argued ‘macroscopic ’ breakdown of superpositions is not the one involving non-rotating mass distributions as he has suggested, but a Leggett-type SQUID or BEC
Abstract Inspired by an ontic view of the wave function in quantum mechanics and motivated by the universal interaction of gravity, we discuss a possible gravity implication in the state collapse mechanism. Concretely, we investigate the stability of the spatial superposition of a massive quantum state under the gravity effect. In this context, we argue that the stability of the spatially superposed state , depends on its gravitational self-energy originating from the effective mass density distribution through the spatially localized eigenstates . We reveal that the gravitational self-interaction between the different spacetime curvatures created by the eigenstate effective masses leads to the reduction of the superposed state to one of the possible localized states . Among others, we discuss such a gravity-driven state reduction. Then, we approach the corresponding collapse time and the induced effective electric current in the case of a charged state, as well as the possible detection aspects.
Diosi and Penrose have suggested a criterion for spontaneous wave function collapse. According to Penrose the profound conflict between the principle of superposition and general covariance entails the existence of reduction of quantum states, e.g. a quantum superposition of two different space–time geometries will collapse onto one of them. In his proposal, collapse time has an inverse relationship with ill-definedness in the gravitational self-energy, for the static gravitational fields. Anandan obtained the same result using the fluctuations of the connection of gravitational field. We show that in Newtonian limit of the superposition of a static and non-static gravitational field the results of the methods of Anandan and Penrose are different, but the numerical value of the extra term in Anandan’s approach, involving angular velocity, is not even considered in a case of practical interest. Then, we investigate the collapse time of the superposition of a static and non-static gravitational field.
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.