QFT Nonlinearity: Interactions & Maxwell's Eqns

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Discussion Overview

The discussion revolves around the relationship between nonlinearity in quantum field theory (QFT) and interactions, particularly in the context of Maxwell's equations, which are linear. Participants explore whether linear equations can predict interactions and the implications of nonlinearity in QFT.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions if the linearity of Maxwell's equations implies they do not predict interactions.
  • Another participant explains that while free field theories are linear, adding a nonlinear term (like in φ^4 theory) accounts for interactions, suggesting that interactions can exist even in linear frameworks.
  • A participant asserts that Maxwell's equations do not lead to self-interaction of electromagnetic fields, indicating that interactions in QFT do not necessarily stem from nonlinearity.
  • There is a request for examples of linear interactions, which remains unanswered, with one participant expressing skepticism about the existence of such examples.
  • Another participant confirms that Maxwell's equations predict no interactions, stating that classical electromagnetic waves and quantum photons pass through each other without interaction.

Areas of Agreement / Disagreement

Participants express disagreement regarding the implications of linearity and nonlinearity in relation to interactions. There is no consensus on whether linear interactions exist or how they relate to the predictions of Maxwell's equations.

Contextual Notes

The discussion highlights the complexity of defining interactions in the context of linear versus nonlinear theories, with participants acknowledging the limitations of their examples and reasoning.

gentsagree
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Nonlinearlity of QFT produces interactions, or so I was told today. Maxwell's eqns, though, are perfectly linear. Does that mean that Maxwell's eqns don't predict interactions?

Thanks
 
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Free field theories in which no interaction is present(of course) are linear,you add a nonlinear term to account for interaction like in [itex]\phi^4[/itex] theory in which a self interacting term is present.But that does not mean that interaction term is always nonlinear but nature does not chose one.Maxwell's eqns are linear but they still have interaction.
 
The Maxwell equations don't lead to self-interaction. That is perfectly fine, as there is no direct (without other fields involved) self-interaction of electromagnetic fields.

In addition, "Nonlinearlity of QFT produces interactions" does not mean the reverse ("Interactions in QFT have to come from nonlinearity") has to be true.
 
mfb said:
In addition, "Nonlinearlity of QFT produces interactions" does not mean the reverse ("Interactions in QFT have to come from nonlinearity") has to be true.
Please, can you give an example of an interaction which is linear?
 
I cannot, and I don't think there is one, but the statement "A=>B does not imply B=>A" is a mathematical fact.
 
gentsagree said:
Does that mean that Maxwell's eqns don't predict interactions?

Correct. Two classical electromagnetic waves will simply pass through each other without affecting each other at all. This is what we mean by a "noninteracting" or "free" field theory.
 
Maxwell's equations with no source have no interactions. In the classical theory we have waves which pass straight through each other and in the quantum theory we have photons passing right through each other.
 

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