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arXiv:1407.4370 · New Journal of Physics

The Schrödinger–Newton equation and its foundations

Mohammad Bahrami, André Großardt, Sandro Donadi, Angelo Bassi

Published Nov 7, 2014• 217 citations
arXivHTMLPDF

Abstract

The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was initially proposed for exactly this purpose. Together with the standard collapse postulate, fundamentally semi-classical gravity gives rise to superluminal signalling. A consistent fundamentally semi-classical theory of gravity can therefore only be achieved together with a suitable prescription of the wave-function collapse. We further discuss, how collapse models avoid such superluminal signalling and compare the nonlinearities appearing in these models with those in the Schrödinger-Newton equation.

DOI
10.1088/1367-2630/16/11/115007

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The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was initially proposed for exactly this purpose. Together with the standard collapse postulate, fundamentally semi-classical gravity gives rise to superluminal signalling. A consistent fundamentally semi-classical theory of gravity can therefore only be achieved together with a suitable prescription of the wave-function collapse. We further discuss, how collapse models avoid such superluminal signalling and compare the nonlinearities appearing in these models with those in the Schrödinger-Newton equation.
We examine the origin of the Newton–Schrödinger equations (NSEs) that play an important role in alternative quantum theories (AQT), macroscopic quantum mechanics and gravity-induced decoherence. We show that NSEs for individual particles do not follow from general relativity (GR) plus quantum field theory (QFT). Contrary to what is commonly assumed, the NSEs are not the weak-field (WF), non-relativistic (NR) limit of the semi-classical Einstein equation (SCE) (this nomenclature is preferred over the 'Moller–Rosenfeld equation') based on GR+QFT. The wave-function in the NSEs makes sense only as that for a mean field describing a system of N particles as , not that of a single or finite many particles. From GR+QFT the gravitational self-interaction leads to mass renormalization, not to a non-linear term in the evolution equations of some AQTs. The WF-NR limit of the gravitational interaction in GR+QFT involves no dynamics. To see the contrast, we give a derivation of the equation (i) governing the many-body wave function from GR+QFT and (ii) for the non-relativistic limit of quantum electrodynamics. They have the same structure, being linear, and very different from NSEs. Adding to this our earlier consideration that for gravitational decoherence the master equations based on GR+QFT lead to decoherence in the energy basis and not in the position basis, despite some AQTs desiring it for the 'collapse of the wave function', we conclude that the origins and consequences of NSEs are very different, and should be clearly demarcated from those of the SCE equation, the only legitimate representative of semiclassical gravity, based on GR+QFT.
The Schrödinger-Newton equation aims at describing the dynamics of massive quantum systems subject to the gravitational self-interaction. As a deterministic nonlinear quantum wave equation, it is generally believed to conflict with the relativistic no-signalling principle. Here we challenge this viewpoint and show that it is of key importance to study the quantitative and operational character of the superluminal effects. To this end we employ a rigorous formalism of probability measures on spacetime and quantify the probability of a successful superluminal bit transfer via the single-particle Schrödinger-Newton equation. We demonstrate that such a quantity decreases with the increasing size and mass of the system. Furthermore, we prove that the Einstein-Dirac system, which yields the Schrödinger-Newton equation in the non-relativistic limit, is perfectly compatible with the relativistic causal structure. Our study demonstrates that the Schrödinger-Newton equation, which is by construction non-relativistic, is in fact `more compatible' with the no-signalling principle than the ordinary free Schrödinger equation.
No experimental evidence exists, to date, whether or not the gravitational field must be quantised. Theoretical arguments in favour of quantisation are inconclusive. The most straightforward alternative to quantum gravity, a coupling between classical gravity and quantum matter according to the semi-classical Einstein equations, yields a nonlinear modification of the Schrödinger equation. Here, effects of this so-called Schrödinger-Newton equation are discussed, which allow for technologically feasible experimental tests.
The Schrödinger-Newton equation has gained attention in the recent past as a nonlinear modification of the Schrödinger equation due to a gravitational self-interaction. Such a modification is expected from a fundamentally semi-classical theory of gravity and can, therefore, be considered a test case for the necessity of the quantisation of the gravitational field. Here we provide a thorough study of the effects of the Schrödinger-Newton equation for a micron-sized sphere trapped in a harmonic oscillator potential. We discuss both the effect on the energy eigenstates and the dynamical behaviour of squeezed states, covering the experimentally relevant parameter regimes.
It has been suggested that the nonlinear Schrödinger–Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the Møller–Rosenfeld semiclassical theory. Numerical results for the spherically symmetric, time-dependent, single-particle case are presented, clarifying and extending previous work on the subject. It is found that, for a particle mass greater than 1.14(ℏ 2 /( G σ)) 1/3 , a wave packet of width σ partially 'collapses' to a groundstate solution found by Moroz, Penrose and Tod, with excess probability dispersing away. However, for a mass less than 1.14(ℏ 2 /( G σ)) 1/3 , the entire wave packet appears to spread like a free particle, albeit more slowly. It is argued that, on some scales (lower than the Planck scale), this theory predicts significant deviation from conventional (linear) quantum mechanics. However, owing to the difficulty of controlling quantum coherence on one hand, and the weakness of gravity on the other, definitive experimental falsification poses a technologically formidable challenge.
We incorporate non-local gravitational self-energy, motivated by string-inspired T-duality, into the Schrödinger-Newton equation. In this framework spacetime has an intrinsic non-locality, rendering the standard linear superposition principle only an approximation valid in the absence of gravitational effects. We then invert the logic by assuming the validity of linear superposition and demonstrate that such superpositions inevitably become unstable once gravity is included. The resulting wave-function collapse arises from a fundamental tension between the equivalence principle and the quantum superposition principle in a semiclassical spacetime background. We further show that wave functions computed in inertial and freely falling frames differ by a gravitationally induced phase shift containing linear and cubic time contributions along with a constant global term. These corrections produce a global phase change and lead to a spontaneous, model-independent collapse time inversely proportional to the mass of the system.