Okay, hello.

We've just been discussing some extensions of quantum mechanics that try to introduce

some fundamental new decoherence sources.

The main motivation apparently is that you want to avoid these embarrassing to some microscopic

superposition states.

And so last time we discussed this pathogenetic Givaldi-Rimini-Weber example of some fundamental

decoherence process where the main idea is that you can arrange parameters such that

only the microscopic superposition states gets destroyed.

They didn't really say where this comes from microscopically, and so it's interesting

to learn about other such theories.

And the one we started discussing was first developed by Penrose and D'Ossi.

The idea is for some relatively mysterious reason there is some fundamental decoherence

associated with gravity.

So it's about gravitationally induced collapse of the wave front.

But other than that, the theory is still formulated within the context of usual quantum mechanics.

So we can write down a master equation for the density matrix, for example.

Okay so let me remind you.

The idea is that if we have some massive particle in a superposition of two places, we want

to know the decoherence rate that would depend on these positions or more precisely on the

difference of these positions.

And the speculation by Penrose and others now is that this is connected to the gravitational

self-energy of such a configuration.

More precisely it's connected to the gravitational self-energy that you would obtain if you insert

the difference in densities for these two positions.

So delta O, that is position one and position two.

Delta O is O1 minus O2 and then you would write down an integral, a double integral

over delta O of r, delta O of r prime over the distance and if you multiply this by the

gravitational constant then you get an energy, something similar to the self-energy of a

mass density distribution.

And if you get an energy you divide by h bar in order to get a rate.

Of course since this is highly speculative there's no guarantee as to what the numerical

pre-factor should be or whether there's anything to do with reality in the first place.

Okay now I also plotted this decoherence rate as a function of distance in this superposition

and what you see is that it starts out at zero of course, if they just coincide then

the difference is zero.

It saturates to some finite value when you take the distance to infinity because then

in this formula only those parts of the integrative will contribute where r and r prime are located

inside the same configuration and not in different configurations.

And then there are two regimes, there's the regime where the distance is still much larger

than the radius of these objects.

In that case there's the first correction to this constant which just goes by one over

the distance.

So these are the first contributions from considering r and r prime actually in different

configurations and then it quickly goes to zero once the distance falls below the radius

of the object.

Now let us first just manipulate this formally just by rewriting it in a fashion so that

we see it's exactly in the same scheme as for example the RDP mini-baba or the decoherence

of thermal molecules.

So we can write the time evolution of the density matrix due to this decoherence mechanism

as the following.