In the 1999 paper published in Nature by Markus Arndt et al entitled "Wave-particle duality of C60 molecules" he states:

The fullerenes are of course in this funny state of superposition as they are 'falling'.

Couple questions:
1. Is this drop of 0.7 mm identical to what any object would experience?
2. If there were 2 of these fullerenes in a superposition out in space (ie: isolated), would they experience a gravitational attraction to one another?
3. If the 2 fullerenes are attracted to each other, how in principal (please omit any math) would one calculate this attractive force? The point being that these two particles don't have a specified location, so there would seem to be a logical problem with the two particles being able to interact through gravitational attraction.

Sorry, I don't understand your equation. Don't know what the variables are, nor what the meaning is.

Are you saying the gravitational force one would calculate between the two is a function of the probability of the particles (fullerenes) being at any location in space? Does this mean their gravitational attraction is 'smeared out' (so to speak) as well? Can you put it into terms such as Scientific American might use?

This sounds pretty accurate! The gravitational force is inversely proportional to the distance between the particles, and the position of the particles is determined by the probability distribution.

But the real role of the gravitational force in quantum mechanics is in determining what will this probability distribution be. Contrarily to classical mechanics, it becomes very uninteresting to know what the gravitational force between two particles is at some given moment since the force no longer equals acceleration so force no longer determines path in space.

So to speak, that's as good a description of it as any I guess.

I think I'm missing something important here. Are you saying the gravitational force between two particles in superposition has no affect on the future evolution of the particles? In other words, imagine calculating the 'probability' of locating a particle at some given location. Does that particle's location depend at all on some gravitational force exerted on it by another particle which is also in a superposition? That's really the crux of the OP. I'd like to understand if there is any gravitational force that affects two such particles. Obviously, for one particle in a superposition (the C60 molecule) and the Earth being in a decoherent state, Arndt considers the affect to be real, but I'd like to know if the affect is real between two particles in superposition.

Absolutely. The probability of finding each particle at each point in space is what is affected by the existence of a mutual gravitational attraction between them. Namely, each particle will have a stronger probability of being observed in the vicinity of the other than if no grav. force existed.

My point was simply that since we know the mathematical formula for the gravitational force (inversely proportional to distance between them), if we know the distance between the particles at a given instant, then we can calculate the gravitation attraction between them at that instant. But it is uninteresting to do so because it won't enable us to do any prediction whatever about what the future state of the system will be (contrary () to classical mechanics)

Playing that back to you - you're saying the probability of locating these particles (that are in superposition) in space is dependant on this 'smeared out' gravity. So gravity does indeed influence two particles (that are in superposition) with respect to each other.

I wonder what affect this has (if any) on massive clouds of intersteller gas. I wonder if these clouds of gas have sufficiently few interactions such that the molecules are in superposition long enough to make any difference in the overall gravitational field. Any thoughts?

Assuming a decoherent particle is at some specific location, it has all it's mass at that location and we can treat it as a classical particle. Any gravitational attraction associated with this particle affects the local gravitational field as if the particle were at a single point. Is that correct so far?

If on the other hand, the particle doesn't interact with it's environment very frequently, then just like the fullerenes in Arndt's experiment, there is no single point at which the mass can be 'found'. I guess this is where I'm still a bit confused.

A single particle, regardless of whether it's in a superposition or not, only has some mass m. Therefore, its affect on the local field is different than if it were in a decoherent state, isn't it? If such a particle can't act as a classical particle, I'm thinking this would affect the gravitational field in something like a large, intersteller gas cloud where interactions may be relatively infrequent in comparison to the distance a particle could move between interactions. The particle can't act as if it has mass m at point p, it has to act as if it has some small percentage of mass m at point p, and some percentage of mass at point p1, p2, p4,... etc.. so I'm imagining this affect being similar to a geography map where there might be lines of equal probability drawn around the last known location of the particle. But the total mass of this particle (m) is 'smeared out' over this entire map, not at some point.

In essence, I'm wondering if such a phenomena has any affect on the evolution of the universe.

Thanks for the comments, quasar. I'll be away the rest of the day so I'll have to respond tonight.

In the paper How to complete Quantum-Mechanical Description? http://xxx.lanl.gov/abs/quant-ph/0212139
describe the wave-function of quantum particles as a summered effect of classical gravity waves of classical test particles. It is mean that in the classical experiment on electron's diffraction an electron pass throw single hole but wave-function (Psi-function) throws both. It is mean that Psi-function is the property of the space and classical test particles will be have quantum properties in this space.

Q, if the particle acts as if it's mass is spread out over a larger area then it's gravitational field will be spread out as well and so will any forces acting on it. Also, gravity is dependent on mass as welll as distance so the gravity of a cloud would be limited to the mass of the cloud.

Hi catuz,
I read through the paper you suggested but found it rather difficult to read primarily because I have little formal background in QM. What is meant by "stochastic gravitation"? To me that might imply we can't assign gravitational force to a single particle in superposition, though I suspect I've misinterpreted this. Would you agree or disagree that the gravity of any particle actually exists, and is 'smeared out' over some volume of space when this particle is in superposition?

Just out of curiosity, do you know this author (Kamalov)?

Hi baryon,
There seems to be a problem with this concept of smeared out gravity that I'm trying to get a handle on. If something can be isolated, it's gravity must be isolated as well, but I can't see any reasonable way of accomplishing this. Perhaps the intersteller cloud idea wasn't the best example. Take instead Schroedinger's cat. Let's put him in a box with the vile of poison and radioactive decay aparatus. But instead, let's put the box on a set of weight scales that will tell where the center of gravity of the entire box is and thus the location of the cat. If the cat is walking around the box, we know it's alive and thus gravity must prevent this cat from being in this state of dead/alive. If on the other hand, if the cat drops dead we might detect the cat falling to the floor by using these weight sensors which might registar a momentary decrease in weight as the cat's legs buckle, then increase in weight as the cat hits the floor. Gravity alone is all that's required to distinguish a live/dead cat.

In Arndt's paper he discusses which type of interactions are necessary for decoherence and I've heard the exact same list given by others, but it doesn't include gravity. I'll copy it here for reference:
Arndt's paper continues:

Do you see any problem with the concept of 'smeared out gravity'?

Hi, Q_Goest. Excuse me for delay.
The answers for your questions.
I hope you are know what is the stochastic force. Particular it is probabilistic force. If we to add here gravitational it is mean the nature of the force. The example of this is the background of gravitational waves. It may be relict gravitational forces. May be others.
About the author of the paper is http://www.timkamalov.narod.ru/.
Thank you.