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arXiv:gr-qc/0105021

Nonrelativistic field theoretic setting for gravitational self-interactions

Sergio De Filippo

Published May 7, 2001• 0 citations
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Abstract

It is shown that a recently proposed model for the gravitational interaction in non relativistic quantum mechanics is the instantaneous action at a distance limit of a field theoretic model containing a negative energy field. It reduces to the Schroedinger-Newton theory in a suitable mean field approximation. While both the exact model and its approximation lead to estimates for localization lengths, only the former gives rise to an explicit non unitary dynamics accounting for the emergence of the classical behavior of macroscopic bodies.

DOI
10.48550/arxiv.gr-qc/0105021

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It is shown that a recently proposed model for the gravitational interaction in non relativistic quantum mechanics is the instantaneous action at a distance limit of a field theoretic model containing a negative energy field. It reduces to the Schroedinger-Newton theory in a suitable mean field approximation. While both the exact model and its approximation lead to estimates for localization lengths, only the former gives rise to an explicit non unitary dynamics accounting for the emergence of the classical behavior of macroscopic bodies.
An explicit dynamical model for non relativistic quantum mechanics with an effective gravitational interaction is proposed, which, as being well defined, allows in principle for the evaluation of every physical quantity. Its non unitary dynamics results from a unitary one in a space with twice as many degrees of freedom as the ordinary ones. It exhibits a threshold for the emergence of classical behavior for bodies of ordinary density on rather long times or instantaneously, respectively at around 10^11 or 10^20 proton masses.
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 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.
The localization length for the center of mass motion of a matter lump, induced by gravitation, is obtained, without using any phenomenological constants. Its dependence from mass and volume is consistent both with unitary evolution of microscopic particles and with the classical behavior of macroscopic bodies required to account for wave function collapse in quantum measurements, the transition between the two regimes being rather sharp. The gravitational interaction of nonrelativistic matter is modelled by a Yukawa Hamiltonian with vanishing pion mass, no gravitational background is needed and the only hypothesis consists in assuming unentanglement between matter and the Yukawa field.
Abstract Isaac Newton’s theory of gravity seems to postulate action at a distance. Two distant massive bodies act on one another gravitationally instantaneously without a causal mechanism to mediate the action. This paper will argue that a strategy found in the Principia that allows Newton’s theory to be explanatory without providing a causal explanation can be extended to the quantum case. However, while causal action at a distance can be avoided, the explanatory cost of avoiding any form of nonlocality is higher in the quantum case than in the case of Newton’s theory of universal gravitation.