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Is there a use of advanced Green's functions? If yes then when or in which context?
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In quantum field theory, why do we always use Feynman's prescription for finding the propagator and not the retarded one?
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Why does Feynman's prescription for the propagator not make sense in a classical field theory?
[Physics] Retarded and advanced Green’s function and Feynman propagator
classical-field-theoryelectromagnetismgreens-functionspropagatorquantum-field-theory
Related Solutions
The propagators of massless Bosons are only zero inside the lightcone if the number of spatial dimensions is odd and larger than 1.
You can determine the exact form of the propagator on the light-cone, even though it is divergent, and do so for any number of spatial dimensions, in the following way:
Use this expression to derive the propagator in d spatial dimensions from the 1 dimensional case with,
$P_d{(t,r)}\ =\ \frac{1}{2\pi^a}\ \frac{\partial^a }{\partial (s^2)^a}\ P_1{(t,r)} $
where $~P{(t,r)}~$ is the propagator, with $~~a=(d-1)/2~$ and $~~s^2=t^2-r^2$
For the real valued Klein Gordon propagator this becomes:
$P_d^{KG}{(t,r)}\ =\ \frac{1}{2\pi^a}\ \frac{\partial^a }{\partial (s^2)^a} \left\{\ {\cal H}(s^2) J_o(ms)\ \right\}$
Where ${\cal H}(s^2)$ is the Heaviside step function which is 1 inside the light-cone and 0 outside the light-cone. In the mass-less case this becomes:
$P_d^{KG}{(t,r)}\ =\ \frac{1}{2\pi^a}\ \frac{\partial^a }{\partial (s^2)^a} \left\{\ {\cal H}(s^2)\ \right\}$
The following graphs showing the mass-less cases are from numerical simulations:
In the case with 1 spatial dimension you see the Heaviside step function. The 3d case is the first order derivative of the step function. The 5d case is the 2nd order derivative of the step-function and so on.
The even dimension cases are non-zero inside the light-cone because of the 1/2 order derivatives.
The operator which derives the d-dimensional propagator from the 1-dimensional propagator is derived in my paper here: http://www.physics-quest.org/Higher_dimensional_EM_radiation.pdf in section V.
The convolution $G_{ret}*f$ of the retarded propagator $G_{ret}$ with a source term $f$ which vanishes sufficiently far in the past is the unique solution of the inhomogeneous Klein-Gordon equation with source term $f$ which vanishes in the far past. It is necessarily a nontrivial superposition of positive and negative energy solutions at all times when it is not identically zero, precisely because it vanishes outside the future light cone of the support of $f$.
The convolution $G_F*f$ of the Feynman propagator $G_F$ with a source term $f$ which vanishes outside a bounded space-time region, on its turn, is a solution of the inhomogeneous Klein-Gordon equation which is of positive energy outside the past light cone of the support of $f$ (i.e. in the far future) and of negative energy outside the future light cone of the support of $f$ (i.e. in the far past). Because of this, it cannot vanish in any nonvoid open region of space-time, unlike $G_{ret}*f$.
The relation between the position-space support and the momentum-space support of $G_{ret}$ and $G_F$ can be seen as a consequence of the uncertainty principle, which restricts the regions where a distribution can vanish in position and momentum space (the latter after a Fourier transform).
Best Answer
David Tong notes in his lecture notes that the advanced propagator "is useful if we know the end point of a field configuration and want to figure out where it came from".