What is a "source term" & what is it physically in QFT?

["Don't panic!"]

Given an inhomogeneous ODE of the form $$a_{n}(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\cdots +a_{2}(x)y''(x)+a_{1}(x)y'(x)+a_{0}(x)y(x)=f(x)$$ where $y^{(n)}(x)\equiv \frac{d^{n}y(x)}{dx^{n}}$, why is the function $f(x)$ on the right hand side referred to as a "source term" ? In what way does it source left hand side (or the function $y(x)$)?
The reason I ask as in quantum field theory the Feynman propagator for a scalar field $\phi (x)$ can be derived by considering the Klein-Gordon equation with a so-called "source term" $J(x)$, i.e. $$\left(\Box +m^{2}\right)\phi (x)=J(x)$$ where $\Box\equiv\eta^{\mu\nu}\frac{\partial^{2}}{\partial x^{\mu}\partial x^{\nu}}$. I'm unsure as to how this should be interpreted physically, or whether it is just a useful mathematical trick that bears no physical interpretation?! (I understand the derivation of the Feynman propagator for a scalar field from this, but I'm unsure how to motivate the introduction of the source term $J(x)$ on the right hand side of the K-G equation?!)

N.B. I originally posted this in the differential equations forum, but it was suggested that perhaps it would be more suited in this forum, given what the question is relating to.

 

[A. Neumaier]

why is the function f(x) on the right hand side referred to as a "source term" ?

In many simple but paradigmatic situations it is a force switched on at time $t=0$. For $t<0$ the system is on its own, for $t>0$ it responds to the force. Thus the force is the source of the modified behavior. For example, you put a weight on a spring and it starts oscillating.

 

[dextercioby]

Did you study electrodynamics? I hope so, this comes before QM which comes before QFT (in a normal university syllabus), so this discipline should teach you about wave equations and field sources.

 

[&quot;Don&#039;t panic!&quot;]

Did you study electrodynamics? I hope so, this comes before QM which comes before QFT (in a normal university syllabus), so this discipline should teach you about wave equations and field sources.

Yes, I did (although it's been a while since I learned it), but I guess I was just viewing the scalar field differently for some reason :-\

For example, I get that in classical electrodynamics the electric current density $\mathbf{J}$ (along with a time varying electric field) sources a spatially varying magnetic field. Can one translate this over to QFT to say that $J(x)$ sources a propagating scalar field of mass $m$?

(Could one think of it classically in terms of a force, e.g. from Newton's 2nd law we could say that a force $\mathbf{f}$ sources an acceleration $\mathbf{a}$ of a particle of mass $m$, i.e. $\mathbf{f}=m\mathbf{a}$? Would be fair to say that this is the classical analogue of $\left(\Box +m^{2}\right)\phi (x)=J(x)$? [In this sense, $J(x)$ would be acting on the field $\phi$ sourcing a propagation in the field]).

 

[dextercioby]

Electrodynamics explains to you what a source term is. A QFT book emphasizing the relevance was written by David Bailin and Alexander Love. "Introduction to gauge Field theory", IOP, 1993.