What is the functional iW[J] for a system with 2 sources or 3 sources?

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Discussion Overview

The discussion revolves around the functional W(J) in the context of quantum field theory (QFT), particularly its interpretation and application when dealing with systems that have two or three sources. Participants explore the meaning of W(J) and its relationship to generating functionals and Green functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the nature of W(J) and its significance, referencing Zee's formulation of Z(J) and suggesting it involves summing over pairs of sources.
  • Another participant proposes that W(J) represents the sum of amplitudes for all possible interactions between sources, seeking validation of this interpretation.
  • A different participant identifies W(J) as the generating functional for unconnected Green functions and cites foundational QFT texts that explain the path integral approach.
  • One post references additional resources, including a link to a related thread and a recommendation for Srednicki's textbook on the topic.
  • Another participant mentions that iW[J], when multiplied by -kT, corresponds to the difference in Helmholtz free energy between the vacuum and a system with sources J, particularly under certain conditions.

Areas of Agreement / Disagreement

Participants express varying interpretations of W(J) and its implications, indicating that multiple competing views remain without a consensus on its precise meaning or application.

Contextual Notes

Some statements rely on specific definitions and assumptions about the nature of sources and interactions, which may not be universally accepted or clarified within the discussion.

Hymne
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Hello!

I have a hard time getting to know what this exponential W(J) really is about. What is it even called?

Zee writes:

Z(J) = Z(J=0) * e^(i W(J)), and I suppose that this is the term that should be evaluated by summing over all possible pair of sources?

What is W(J) for a system with 2 sources or 3 sources?

Thanks really much!
 
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Hmm, I just wrote in my notes:

"The interpretation of W(J) is that we sum the amplitudes for all possible interactions between our sources."

Could somebody with more insight maybe just give me a right or wrong here?
 
W(J) is called the generating functional for the unconnected Green functions. The approach to QFT based on path integrals is well explained in the QFT books written by P. Ramond, D. Bailin and A. Love, L.H. Ryder. It all can be traced back to the 1965 book by Feynman and Hibbs.
 
Hymne said:
Hello!

I have a hard time getting to know what this exponential W(J) really is about. What is it even called?

Thanks really much!

iW[J], when multiplied by -kT, is a functional that's equal to the difference in Helmholtz free energy of the vacuum and one with sources J (at least when using real values of time).
 

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