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arXiv:gr-qc/9511004 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum theory of nonrelativistic particles interacting with gravity

Charis Anastopoulos

Published Jul 15, 1996• 37 citations
arXivHTMLPDF

Abstract

We investigate the effects of the gravitational field on the quantum dynamics of nonrelativistic particles. We consider $N$ nonrelativistic particles, interacting with the linearized gravitational field. Using the Feynman-Vernon influence functional technique, we trace out the graviton field to obtain a master equation for the system of particles to first order in $G$. The effective interaction between the particles as well as the self-interaction is in general non-Markovian. We show that the gravitational self-interaction cannot be held responsible for decoherence of microscopic particles due to the fast vanishing of the diffusion function. For macroscopic particles though, it leads to diagonalization to the energy eigenstate basis, a desirable feature in gravity-induced collapse models. We finally comment on possible applications.

DOI
10.1103/physrevd.54.1600

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We investigate the effects of the gravitational field on the quantum dynamics of nonrelativistic particles. We consider NNN nonrelativistic particles, interacting with the linearized gravitational field. Using the Feynman-Vernon influence functional technique, we trace out the graviton field to obtain a master equation for the system of particles to first order in GGG. The effective interaction between the particles as well as the self-interaction is in general non-Markovian. We show that the gravitational self-interaction cannot be held responsible for decoherence of microscopic particles due to the fast vanishing of the diffusion function. For macroscopic particles though, it leads to diagonalization to the energy eigenstate basis, a desirable feature in gravity-induced collapse models. We finally comment on possible applications.
A quantum master equation describing the stochastic dynamics of a quantum massive system interacting with a quantum gravitational field is useful for the investigation of quantum gravitational and quantum informational issues such as the quantum nature of gravity, gravity-induced entanglement and gravitational decoherence. Studies of the decoherence of quantum systems by an electromagnetic field shows that a lower temperature environment is more conducive to successful quantum information processing experiments. Likewise, the quantum nature of (perturbative) gravity is far better revealed at lower temperatures than high, minimizing the corruptive effects of thermal noise. In this work, generalizing earlier results of the Markovian ABH master equation [1,2] which is valid only for high temperatures, we derive a non-Markovian quantum master equation for the reduced density matrix, and the associated Fokker-Planck equation for the Wigner distribution function, for the stochastic dynamics of two masses following quantum trajectories, interacting with a graviton field, including the effects of graviton noise, valid for all temperatures. We follow the influence functional approach exemplified in the derivation of the non-Markovian Hu-Paz-Zhang master equation [62,64] for quantum Brownian motion. We find that in the low temperature limit, the off-diagonal elements of the reduced density matrix decrease in time logarithmically for the zero temperature part and quadratically in time for the temperature-dependent part, which is distinctly different from the Markovian case. We end with a summary of our findings and a discussion on how this problem studied here is related to the quantum stochastic equation derived in [77] for gravitational self force studies, and to quantum optomechanics where experimental observation of gravitational decoherence and entanglement may be implemented.
Abstract Even though we have some proposals for the quantum theory of gravity like string theory or loop quantum gravity, we do not have any experimental evidence supporting any of these theories. Actually, we do not have empirical evidence pointing in the direction that we really need a quantum description of the gravitational field. In this scenario, several proposals for experimentally investigating quantum gravitational effects far from the Planck scale have recently appeared in literature, like gravitationally induced entanglement, for instance. An important issue of these approaches is the decoherence introduced by the quantum nature not only of the system under consideration but also from the gravitational field itself. Here, by means of the Feynman–Vernon influence functional, we study the decoherence of a quantum system induced by the quantized gravitational field—in the linearized gravity regime—and also by its own quantum nature. Our results may be significant in better understanding many phenomena like the decoherence induced by the gravitational time-dilation, the quantum reference frames, and the quantum equivalence principle.
We formulate the gravitationally induced quantum decoherence of a massive object prepared in a spatial superposition. Starting from linearized gravity coupled to a massive system particle and an environmental scalar field, we derive a closed-time-path influence functional governing the reduced system dynamics. In the nonrelativistic and quasi-static regime, the decoherence exponent can be written as a bilinear functional of the difference of the system stress-energy tensors and an effective noise kernel obtained by dressing the environmental stress-energy tensor correlator with graviton propagators. We then apply this framework to the Newtonian long-range gravitational interaction and evaluate the resulting decoherence function for a dilute nonrelativistic gas modeled by finite wave packets and coarse-grained in time and space. By performing controlled approximations, we obtain analytic expressions for the cumulative decoherence function and show that the dominant contribution is accumulated logarithmically over a broad range of distances, while remaining subdominant to conventional collisional decoherence under realistic conditions.
We derive a general quantum master equation for the dynamics of a scalar bosonic particle interacting with a weak, stochastic and classical external gravitational field. The dynamics predicts decoherence in position, momentum and energy. We show how our master equation reproduces the results present in the literature by taking appropriate limits, thus explaining the apparent contradiction in their dynamical description. Our result is relevant in light of the increasing interest in the low energy quantum-gravity regime.
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.
In a previous work [L. Asprea, G. Gasbarri, and A. Bassi, Gravitational decoherence: A general nonrelativistic model, Phys. Rev. D 103, 104041 (2021).] we derived a quantum master equation for the dynamics of a scalar bosonic particle interacting with a weak, stochastic and classical gravitational field. As standard matter is made of fermions, such an equation should be suitably extended to describe more relevant experimental situations. Here we derive a nonrelativistic model for the gravitational decoherence of spin 1/2 particles. We enrich the treatment by also considering a coupling with an external classical electromagnetic field. We comment on the differences with the scalar bosonic model and we describe the regimes in which they become negligible.