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

1. Feb 1, 2016

### "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.

2. Feb 1, 2016

### A. Neumaier

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.

3. Feb 1, 2016

### 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.

4. Feb 2, 2016

### "Don't panic!"

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]).

Last edited: Feb 2, 2016
5. Feb 2, 2016

### 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.