Schrödinger meets Newton

In January, we discussed semi-classical gravity: Classical general relativity coupled to the expectation value of quantum fields. This theory is widely considered to be only an approximation to the still looked-for fundamental theory of quantum gravity, most importantly because the measurement process messes with energy conservation if one were to take it seriously, see earlier post for details.

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

Inserting a potential that depends on the expectation value of the wave-function makes the Schrödinger-equation non-linear and changes its properties. The gravitational interaction is always attractive and thus tends to contract pressureless matter distributions. One expects this effect to show up here by contracting the wave-packet. Now the usual non-relativistic Schrödinger equation results in a dispersion for massive particles, so that an initially focused wave-function spreads with time. The gravitational self-coupling in the Schrödinger-Newton equation acts against this spread. Which one wins, the spread from the dispersion or the gravitational attraction, depends on the initial values.

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
  arXiv:1105.1579 [quant-ph]
• Gravitationally induced inhibitions of dispersion according to the Schrödinger-Newton Equation
  Domenico Giulini and André Großardt
  arXiv:1105.1921 [gr-qc]

They showed independently that Carlip and Salzmann's earlier numerical study was flawed and the accurate numerical result fits with the analytical estimate very well. Thus, the good news is one understands what's going on. The bad news is, it's about 5 orders of magnitude off today's experimental possibilities. But that's in an area of physics were progress is presently rapid, so it's not hopeless!

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.

That the Schrödinger-Newton equation leads to a continuous collapse might lead one to think it could play a role for the collapse of the wave-function, an idea that has been suggested already in 1984 by Lajos Diosi. However, this interpretation is questionable because it became clear later that the gravitational collapse that one finds here isn't suitable to be interpreted as a wave-function collapse to an eigenstate. For example, in this 2002 paper, it was found that two bumps of probability density, separated by some distance, will fall towards each other and meet in the middle, rather than focus on one of the two initial positions as one would expect for a wave-function collapse.