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I'm studying the generating functional of correlation functions. It takes the form \begin{equation} Z[J]=\int D\phi e^{i(S[\phi)]+J_i\phi^i)} \end{equation}

My question is: what is $J_i$ physically? In studying Klein-Gordon equation, for example, $J(x)$ is the source of the Klein-Gordon field, but I don't think this is the case, as it seems I have multiple sources (as much as my indices "i", which is just a discretization of a continuous index as far as I understood).

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  • $\begingroup$ What is the meaning of the index here? For Klein-Gordon, $J$ should be a scalar function. $\endgroup$ – Aaron Mar 8 '17 at 18:39
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    $\begingroup$ It's a condensed notation. $J_i \phi^i$ is equivalent to $\int d^4x J(x)\phi(x)$ $\endgroup$ – Luthien Mar 8 '17 at 20:46
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    $\begingroup$ Doesn't that answer the question then? $J$ is then just the source of the KG field, which depends on space-time. $\endgroup$ – Aaron Mar 8 '17 at 21:55
  • $\begingroup$ Nope because in the KG equation my source is fixed. Without knowing the source I cannot solve the equation. Here I'm trying to find "probability amplitudes", let's say, of a particle travelling from point A to point B. No sources involved, it seems. Then at some point we introduce sources and then we take the functional derivative of the correlation functions in J=0. So, do they even have a physical meaning or are they here just for calculus reasons? $\endgroup$ – Luthien Mar 8 '17 at 22:02
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    $\begingroup$ I think $J$ is auxiliary thing and must be set to zero after the calculations if there are no source. $\endgroup$ – user145902 Mar 8 '17 at 23:29
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There are two perspectives of how to interpret $J$. The first perspective is that $J$ is merely introduced to obtain a generating functional; it is simply a mathematical tool to obtain correlation functions. This is in the general spirit of how generating functionals are found.

However, one can also interpret $J$ as a source of the KG field. To see this, write down the classical equation of motion of this action. You will get \begin{equation} -\partial_\mu \partial^\mu \phi - m^2 \phi + J(x) = 0 \end{equation} As is clear from this, this is simply the KG equation with a source term $J(x)$.

How are these two perspectives related? Roughly speaking, imagine introducing small fluctuations to your system by adding $\delta J$. This will induce changes $\delta\phi$, and these fluctuations are captured in the correlations. Hence, by looking at small variations in the source $J$ one can extract correlations.

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