Separating operators into classical + quantum

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Discussion Overview

The discussion revolves around the separation of position and momentum operators into classical and quantum components, as presented in a referenced paper. Participants seek clarification on the theoretical basis and implications of this separation, exploring its application in quantum mechanics, particularly in the context of path integrals and the Heisenberg picture.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant references a paper that proposes separating position and momentum operators into classical motion and quantum fluctuations, but seeks further explanation on the methodology.
  • Another participant requests more details to understand the context and implications of the initial claim.
  • Some participants express confusion regarding the definitions and expressions of the operators involved, specifically \bar{X}_i and \hat{q}_i.
  • There is mention of a common approach in path-integral formulation where paths are split into classical and quantum components, suggesting a potential parallel to operator separation in the Heisenberg picture.

Areas of Agreement / Disagreement

Participants generally express confusion and seek clarification, indicating that the discussion remains unresolved with no consensus on the explanation of the operator separation.

Contextual Notes

Participants note limitations in understanding the definitions and expressions of the operators, as well as the theoretical framework needed to rigorously explain the separation of classical and quantum components.

DrClaude
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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:
[tex]\hat{X}_i \equiv \bar{X}_i + \hat{q}_i; \quad \hat{P}_i \equiv \bar{P}_i + \hat{\pi}_i[/tex]
Can someone point me to a reference rigorously explaining why and how this can be done?
 
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Can you give more details? There's no way to tell from what you have written what is going on.
 
Bill_K said:
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. [itex]\bar{X}_i[/itex] is the "classical position" of the [itex]i[/itex]th particle, but I do not understand how to express the operators [itex]\bar{X}[/itex], [itex]\hat{q}[/itex], etc.
 
DrClaude said:
That is exactly the problem I have.

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

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