Spacetime Fluctuations and a Stochastic Schrödinger-Newton Equation

Spacetime Fluctuations and a Stochastic Schrödinger-Newton Equation We propose a stochastic modification of the Schrödinger-Newton equation which takes into account the effect of extrinsic spacetime fluctuations. We use this equation to demonstrate gravitationally induced decoherence of two gaussian wave-packets, and obtain a decoherence criterion similar to those obtained in the earlier literature in the context of effects of gravity on the Schrödinger equation. Is the apparent collapse of the wave-function during a quantum measurement caused by a dynamical physical process which results from possible modification of the Schrödinger equation? Or can it be explained within the framework of standard quantum theory via environmental decoherence and the many-worlds interpretation, or through a reformulation such as Bohmian mechanics? In the coming years it might become possible to decisively answer this question experimentally, thanks to advances in technology, and new innovative ideas for experiments based on optomechanics and interferometry [1]. The focus of such experiments and ideas for experiments is to test dynamical collapse theories such as Continuous Spontaneous Localisation [CSL] which involve a stochastic nonlinear modification of the Schrödinger equation. CSL is a phenomenological theory with two free parameters, designed to solve the measurement problem, explain the Born probability rule, and to explain the apparent absence of superpositions of macroscopic states [2, 3]. How- 1 ever, at the present state of understanding it is unclear as to what is the fundamental origin of CSL: why should there be a stochastic modification of the Schrödinger equation? Possible explanations include the existence of a fundamental stochastic field in nature, which couples nonlinearly to matter fields and results in an anti-Hermitean modification to the Hamiltonian. Alternatively, quantum theory maybe a coarse-grained approximation to a deeper theory such as Trace Dynamics, and stochastic modifications arise when one goes beyond the leading order approximation. A third possible explanation is that gravity plays a role in bringing about collapse of the wave-function [1, 4, 5]. The present paper is concerned with a specific, modest aspect concerning the possible role of gravity. The idea that gravity plays a role in collapse of the wave-function has been around for the last fifty years, and has been pursued by many investigators starting with the works [6–16], and also pursued by Diosi and collaborators [17–20]. The basic principle behind the idea is easy to state and understand. Gravitational fields are produced by material bodies; and largely by macroscopic material bodies. However even macroscopic bodies are not exactly classical, and their position and momenta are subject to the uncertainty principle. It is plausible then [unless one invokes semiclassical gravity] that the gravitational field produced by these bodies is also subject to intrinsic fluctuations, which induce stochasticity in the space-time geometry, which cannot be ignored. Thus when one is studying the Schrödinger evolution of a quantum system on a background spacetime (even a flat Minkowski spacetime), one can in principle not ignore these spacetime fluctuations. When one makes models to see how these fluctuations affect the standard Schrödinger evolution, it is found [as should be the case] that microscopic objects are not affected by the gravitational fluctuations, so that the conventional picture of quantum theory and the linear superposition principle continues to hold for them. However, the Schrödinger evolution of a macroscopic object is significantly affected, leading to gravitationally induced decoherence, thus providing at least a partial resolution of the measurement problem. While it has not been shown that collapse of the wave-function can be achieved through gravity, models strongly suggest that fundamental decoherence [loss of interference without loss of superposition] can be achieved through gravity [without the need for an environment]. It is hoped that when properly understood, gravity might be able to provide an underlying explanation for CSL. One of the earliest pioneering works investigating gravity induced decoherence is due to Karolyhazy, who proposed that the quantum nature of objects imposes a minimum uncertainty [different from Planck length] on the accuracy with which length and time intervals can be measured. This is interpreted as an intrinsic property of spacetime, which is modelled 2 as resulting from a stochastic metric perturbation having a [non-white] gaussian two-point correlation. The Schrödinger evolution of a quantum object is modified to include the effect of this stochastic potential, and it is shown that gravitational decoherence can be achieved for macroscopic objects. This model has been studied further by Karolyhazy and collaborators. In a different model, Diosi has modelled the intrinsic quantum uncertainty of the Newtonian gravitational potential [resulting from the quantum nature of the probe] by a [white-noise] gaussian correlation, and again demonstrated gravitational decoherence. This model has also been studied further by various authors. The Karolyhazy model and the Diosi model have been recently compared in [21]. Models such as those of Karolyhazy and Diosi study the effect of extrinsic space-time fluctuations on the Schrödinger equation. A different gravitational effect is due to the selfgravity of the quantum object: how does the Schrödinger equation get modified by the gravity of the very particle for which this equation is being written? One possible way to describe this effect is to propose that to leading order the particle produces a classical potential satisfying the Poisson equation, whose source is a density proportional to the quantum probability density. The Schrödinger equation is then modified to include this potential [a kind of back-reaction] and the modified equation is known as the Schrödinger-Newton [SN] equation [22–25]. The SN equation has been studied extensively in many papers, for its properties and possible limitations [26–37]. One important feature of the SN equation is a gravitationally induced inhibition of dispersion of a wave-packet [27]. However, the SN equation is not intended to explain gravitationally induced decoherence or collapse of the wave-function. It cannot achieve that because it lacks a stochastic feature, unlike the Karolyhazy and Diosi models, which employ a stochastic gravitational field in the Schrödinger equation. The SN equation only incorporates the deterministic back-reaction of self-gravity in a semiclassical fashion, and one worrisome outcome of this deterministic nonlinearity is superluminal signalling. It is desirable to modify the SN equation into a stochastic equation, possibly by including higher order corrections to self-gravity, or otherwise. This brings home the possibility that the SN equation can take into account self-gravity as well as perhaps produce gravitational decoherence, though it remains to be seen whether the superluminal feature can be gotten rid of by including stochasticity. Another interesting aspect which seems worth considering, and which is the subject of the present paper, is to simultaneously take into account the effect of self-gravity and of extrinsic spacetime fluctuations. After all, that seems to be a rather natural and wholesome way of accounting for the role of gravity in Schrödinger evolution. In this spirit, we write down, in 3 the next section, a modified SN equation which includes a stochastic potential representing extrinsic spacetime uncertainty and having Diosi’s white noise correlation. In Section III, we use this stochastic equation to demonstrate gravitational decoherence of two gaussian states of a free particle, and obtain decoherence criteria similar to those obtained by Diosi. In Section IV we discuss the implications of our results, and compare them with earlier work. Details of some of the integrals that appear in Section III, are given in Appendix I.