Separating operators into classical + quantum

[DrClaude]

In the paper http://link.aps.org/doi/10.1103/PhysRevA.85.062329, the authors separate the position and momentum operators into classical motion and quantum fluctuations:
$$\hat{X}_i \equiv \bar{X}_i + \hat{q}_i; \quad \hat{P}_i \equiv \bar{P}_i + \hat{\pi}_i$$
Can someone point me to a reference rigorously explaining why and how this can be done?

 

[Bill_K]

Can you give more details? There's no way to tell from what you have written what is going on.

 

[DrClaude]

There's no way to tell from what you have written what is going on.

That is exactly the problem I have.

Apart from quoting the article I linked to, there is not much more I can do. $\bar{X}_i$ is the "classical position" of the $i$th particle, but I do not understand how to express the operators $\bar{X}$, $\hat{q}$, etc.

 

[stevendaryl]

That is exactly the problem I have.

Apart from quoting the article I linked to, there is not much more I can do. $\bar{X}_i$ is the "classical position" of the $i$th particle, but I do not understand how to express the operators $\bar{X}$, $\hat{q}$, etc.

What I've seen some people do in the path-integral formulation of quantum mechanics is to split the path into a sum of the classical path + a quantum correction. I assume that you can do the same thing with operators in the Heisenberg picture (where instead of operators that are time-independent and states that are time-dependent, it's the other way around).