What does it mean to say that 2 fields are coupled? More generally, what does "coupling" mean?
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1$\begingroup$ This is probably too broad as it stands. Could you be a bit more precise in where you've seen this coupling, or what sort of answer (particularly in terms of mathematical detail) you're expecting? $\endgroup$– Emilio PisantyAug 9, 2013 at 14:15
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$\begingroup$ @EmilioPisanty: Thanks for commenting! I've been reading, and trying to understand, the last paragraph under "What is Quantum Field Theory?" on page 4 in these notes $\endgroup$– AlexAug 9, 2013 at 14:32
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$\begingroup$ @EmilioPisanty: P.S. any math detail is welcome. $\endgroup$– AlexAug 9, 2013 at 14:33
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$\begingroup$ Well, I would look at chapter three of those same notes. Equation (3.7) is a good example of two coupled fields. $\endgroup$– Emilio PisantyAug 9, 2013 at 14:43
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$\begingroup$ I don't understand why this was done to this question.. It seems like a valid question to me . It's essentially asking what coupling means in general. That isn't so broad . $\endgroup$– Abhimanyu Pallavi SudhirAug 10, 2013 at 11:57
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1 Answer
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"Coupling" is just a particle physicist's (but also a rather generic other physicist's) favorite word for an "interaction".
In a theory given by a Lagrangian, degrees of freedom (e.g. fields but not necessarily fields) $f,g$ are coupled if there exist terms in the Lagrangian that depend both on $f$ and $g$, typically (in field theory) on their product or a product of their derivatives. This term in the Lagrangian – e.g. $e\cdot \bar\Psi\gamma^\mu \Psi\cdot A_\mu$ in QED – is called a "coupling" by itself (in my example, a coupling between electrons and photons). The coefficient of such a term (e.g. $e$ in my example) is often called the "coupling constant", at least in certain conventions.
In the opposite case, $f,g$ are said to be "decoupled": they don't interact with each other. The action may be divided to an action that only depends on the $f$-degrees of freedom and those that only depend on the $g$-degrees of freedom which means that the minimization proceeds separately and these two degrees of freedom evolve independently of one another.
The word "coupling" became widespread long before quantum field theory. In its "toy model", a multi-dimensional harmonic oscillator, one may write terms such as $K x_1 x_2$ in the Lagrangian which will make the two springs with coordinates $x_1$ and $x_2$ interact (coupled): the energy will be transmitted from one to the other and back.
answered Aug 9, 2013 at 14:43
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1$\begingroup$ Today I was going to ask the very same question that the OP asked which was closed for lack of -- what?, specifity? However, this answer is just the kind of answer that satisfied my need and obviously the OP. I know this was asked 7 years ago but it should never have been closed in my opinion. Excellent answer to a question that someone else thought not worthy to be considered. $\endgroup$– K7PEHOct 10, 2020 at 23:54