Master` Equation for the Penrose-Diosi wavefunction collapse

[Agrippa]

TL;DR

Hope you'll help me understand specific aspects of the equation better

I'm trying to understand equation 209 on page 81 here: https://arxiv.org/pdf/1204.4325.pdf

Here's what I understand so far:
- We are to imagine a system in a superposition of |X> and |X'>, where these describe distinct particle configurations.
- On the RHS of the eqn, the first term is Schoredinger evolution, the second term represents effects of objective collapse.
- The term [tau-d(X, X')] gives a number, which represents the average time for a collapse of this superposition to occur. This is calculated in terms of the gravitational energy required to create system S1 in |X> and system S2 in |X'> starting with S1 and S2 both being in state |X>.

Here's what I don't yet understand?
- What is the exact meaning of the LHS term, in particular, what does it mean to sandwich the density operator of the system between the two eigenstates of the superposition like this?
- Why does the first rho on the RHS of the eqn not have a hat on it, whereas the other two rhos do?
- Why do we take [tau-d(X, X')] to the power of -1?
- What is the physical significance of the two subtraction signs? e.g. what happens if the first is deleted, or the second is changed to addition?

Hope you can help! :-)

 

[Agrippa]

p.s. this thread concerns the master equation for Penrose-Diosi collapse (as opposed to, for example, the state vector evolution equation for Penrose-Diosi collapse). It is PhysForum editors that keep changing the title from "Master equation for..." to the slightly less sexy "Mastering this equation for..."

 

[PeterDonis]

Moderator's note: Thread title restored.