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

  1. Feb 1, 2016 #1
    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.
     
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  3. Feb 1, 2016 #2

    A. Neumaier

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    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.
     
  4. Feb 1, 2016 #3

    dextercioby

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    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.
     
  5. Feb 2, 2016 #4
    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
  6. Feb 2, 2016 #5

    dextercioby

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    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.
     
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