However, one can take the point of view that whatever the theorists think is plausible or not should still be experimentally tested. Maybe the semi-classical theory does in fact correctly describe the way a quantum wave-function creates a gravitational field; maybe gravity really is classical and the semi-classical limit exact, we just don't understand the measurement process. So what effects would such a funny coupling between the classical and the quantum theory have?
Luckily, to find out it isn't really necessary to work with full general relativity, one can instead work with Newtonian gravity. That simplifies the issue dramatically. In this limit, the equation of interest is known as the Schrödinger-Newton equation. It is the Schrödinger-equation with a potential term, and the potential term is the gravitational field of a mass distributed according to the probability density of the wave-function. This looks like this
However, the gravitational interaction is very weak, and so is the effect. For typical systems in which we study quantum effects, either the mass is not large enough for a collapse, or the typical time for it to take place is too long. Or so you are lead to think if you make some analytical estimates.
The details are left to a numerical study though because the non-linearity of the Schrödinger-Newton equation spoils the attempt to find analytical solutions. And so, in 2006 Carlip and Salzmann surprised the world by claiming that according to their numerical results, the contraction caused by the Schrödinger-Newton equation might be possible to observe in molecule interferometry, many orders of magnitude off the analytical estimate.
It took five years until a check of their numerical results came out, and then two papers were published almost simultaneously:
- Schrödinger-Newton "collapse" of the wave function
J. R. van Meter
- Gravitationally induced inhibitions of dispersion according to the Schrödinger-Newton Equation
Domenico Giulini and André Großardt
It is interesting what this equation does, so let me summarize the findings from the new numerical investigation. These studies, I should add, have been done by looking at the spread of a spherical symmetric Gaussian wave-packet. The most interesting features are:
- For masses smaller than some critical value, m less than ~ (ℏ2/(G σ))1/3, where σ is the width of the initial wave-packet, the entire wave-packet expands indefinitely.
- For masses larger than that critical value, the wave-packet fragments and a fraction of the probability propagates outwards to infinity, while the rest remains localized in a finite region.
- From the cases that eventually collapse, the lighter ones expand initially and then contract, the heavier ones contract immediately.
- The remnant wave function approaches a stationary state, about which it performs dampened oscillations.