How can we expand product of two currents into series?

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Discussion Overview

The discussion revolves around the expansion of the product of two currents in quantum field theory (QFT), specifically referencing a formulation found in the book by Peskin & Schroeder. Participants are exploring how to derive a series expansion for this product, including the roles of operators and coefficients in the expansion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks guidance on how to expand the product of two currents as presented in a QFT text.
  • Another participant suggests considering operators that transform similarly to the product of the two currents as a way to identify those that will appear in the expansion.
  • A different participant mentions that coefficients in the series can be calculated using Feynman diagrams but expresses uncertainty about how to derive these diagrams for the coefficients.
  • One participant questions why the coefficient of the first operator is derived from diagrams that do not include external legs aside from the current insertions.
  • Another participant notes that there seems to be some arbitrariness in the operator product expansion (OPE), suggesting that coefficients can be adjusted based on the situation.
  • One participant proposes that the product of operators could be expanded using Wick's theorem and Taylor series, indicating a possible method for the expansion.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and propose different methods for approaching the expansion, indicating that multiple competing views remain without a consensus on the best approach.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the operators and the definitions of the coefficients, as well as unresolved details about the application of Feynman diagrams and the specific conditions under which the expansion is valid.

Who May Find This Useful

This discussion may be of interest to those studying quantum field theory, particularly students or researchers looking to understand operator product expansions and their applications in theoretical physics.

ndung200790
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Please teach me this:
In QFT book of Peskin&Schroeder write:
J_{\mu}(x)J_{\nu}(0)~C_{\mu\nu}^{1}(x).1+C_{\mu\nu}^{q^{-}q}q^{-}q(0)+C_{\mu\nu}^{F^{2}}(x)(F_{\alpha\beta}^{a})^{2}(0)...
I do not know how to expand the product into this series.
Thank you very much for your kind helping.
 
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Oh dear... I would love to know the answer to this one.

The best answer I have been given to date is: think of all the operators that transforms in the same way as the product of two currents, and those will be the ones appearing in the expansion.

It would be so awesome if someone could give a constructive algorithm for generating the series expansion for the product of two currents (or for that matter, for the non-local product of any two local operators).
 
Given the series,they say that the coefficients in the series can be calculate by Feynman diagrams.Then I do not know how to give the Feynman diagrams for the coefficients.
 
Why is the coefficient of operator 1 being the sum of diagrams with no external legs other than the current insertions?
 
It seems that there is ''some arbitrary'' in OPE.We can ''adjust'' the coefficients in the series according with ''situation''.Is that correct?
 
It seem to me that we can expand product of operators by Wick theorem and Taylor series?
 

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