10.08.01 · em-sr / advanced-electrodynamics

Green's function method for the scalar and vector potentials: retarded, advanced, and Feynman propagators

shipped3 tiersLean: none

Anchor (Master): Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6; Zangwill, Modern Electrodynamics (2013), Ch. 19-20

Intuition Beginner

The Green's function is the solution to a differential equation with a point source. For the wave equation, the Green's function tells us how a disturbance at one point and time propagates to another point at a later time.

For electromagnetic waves, the key result is the retarded Green's function: a flash of light at position $r^{'}$ at time $t^{'}$ arrives at position $r$ at time $t = t^{'} + ∣ r - r^{'} ∣/ c$ . The signal travels at the speed of light, delayed by the travel time. This is the causal Green's function: the effect (observation at $r$ ) comes after the cause (source at $r^{'}$ ).

There is also an advanced Green's function where the signal arrives before it is sent: $t = t^{'} - ∣ r - r^{'} ∣/ c$ . This violates causality and is unphysical, but it is mathematically valid as a solution. The physical solution uses only the retarded Green's function.

The Feynman propagator is the average of the retarded and advanced Green's functions (divided by 2). It plays a central role in quantum electrodynamics, where time ordering of events is handled by the propagator rather than by explicit causal selection.

Visual Beginner

Green's function	Causality	Physical?
Retarded: $G_{ret}$	Effect after cause	Yes (used in classical EM)
Advanced: $G_{adv}$	Effect before cause	No (unphysical, used in theory)
Feynman: $(G_{ret} + G_{adv}) /2$	Time-symmetric	Only in quantum field theory

Worked example Beginner

A point charge $q$ at the origin suddenly appears at $t = 0$ . The retarded potential at distance $r$ is:

V (r, t) = \frac{q}{4 π ϵ _{0} r} Θ (t - r / c)

where $Θ$ is the Heaviside step function. Before time $t = r / c$ , the signal has not arrived yet, so $V = 0$ . At exactly $t = r / c$ , the potential jumps to $q / (4 π ϵ_{0} r)$ . The information that the charge exists propagates outward at speed $c$ .

The advanced potential would be $V (r, t) = q Θ (t + r / c) / (4 π ϵ_{0} r)$ , which is nonzero for $t > - r / c$ — the potential exists before the charge appears. This is unphysical.

Check your understanding Beginner

Formal definition Intermediate+

The Green's function $G (r, t; r^{'}, t^{'})$ for the wave equation satisfies:

(\nabla^{2} - \frac{1}{c ^{2}} \frac{\partial ^{2}}{\partial t ^{2}}) G (r, t; r^{'}, t^{'}) = - 4 π δ^{3} (r - r^{'}) δ (t - t^{'})

Retarded Green's function.

G_{ret} (r, t; r^{'}, t^{'}) = \frac{δ ( t - t ^{'} - ∣ r - r ^{'} ∣/ c )}{∣ r - r ^{'} ∣}

This is nonzero only on the forward light cone $t - t^{'} = ∣ r - r^{'} ∣/ c$ (the signal travels at speed $c$ ).

Advanced Green's function.

G_{adv} (r, t; r^{'}, t^{'}) = \frac{δ ( t - t ^{'} + ∣ r - r ^{'} ∣/ c )}{∣ r - r ^{'} ∣}

This is nonzero on the backward light cone $t - t^{'} = - ∣ r - r^{'} ∣/ c$ .

Retarded potentials. Using $G_{ret}$ , the solutions for $V$ and $A$ are:

V (r, t) = \frac{1}{4 π ϵ _{0}} \int \frac{ρ ( r ^{'} , t _{r} )}{∣ r - r ^{'} ∣} d^{3} r^{'}

A (r, t) = \frac{μ _{0}}{4 π} \int \frac{J ( r ^{'} , t _{r} )}{∣ r - r ^{'} ∣} d^{3} r^{'}

where $t_{r} = t - ∣ r - r^{'} ∣/ c$ is the retarded time. The field at $(r, t)$ depends on the source at the earlier time $t_{r}$ .

Kirchhoff integral. The field at any point can be expressed as an integral over a closed surface (the Kirchhoff representation):

V (r, t) = - \frac{1}{4 π} \oint_{S} [\frac{1}{R} \frac{\partial V}{\partial n} - V \frac{\partial}{\partial n} \frac{1}{R} + \frac{1}{c R} \frac{\partial V}{\partial t} \frac{\partial R}{\partial n}]_{t_{r}} d a

This is the basis for diffraction theory 10.08.03.

Key derivation Intermediate+

Derivation (The retarded Green's function for the wave equation).

Theorem. The retarded Green's function $G_{ret} (r, t) = δ (t - r / c) / (4 π r)$ satisfies the inhomogeneous wave equation with a point source.

Proof. We need to show $□ G_{ret} = - 4 π δ^{3} (r) δ (t)$ where $□ = \nabla^{2} - c^{- 2} \partial_{t}^{2}$ .

Away from the origin ( $r \neq = 0$ ), $G_{ret}$ is a distribution supported on the sphere $t = r / c$ (the forward light cone). For $r > 0$ and $t \neq = r / c$ : $G_{ret} = 0$ , and the delta function vanishes, so the equation is satisfied.

Near the light cone, integrate $□ G_{ret}$ over a small 4-volume enclosing the origin. By Gauss's theorem in 4D:

\int_{V} □ G_{ret} d^{4} x = \oint_{\partial V} (\nabla G_{ret} \cdot d a - \frac{1}{c ^{2}} \frac{\partial G _{ret}}{\partial t} c d t d A)

The integral of $G_{ret}$ over the light cone picks up the delta function, and the $1/ r$ factor provides the correct normalisation. Evaluating the integral on a small cylinder around the origin: the radial and time derivatives of $G_{ret}$ contribute surface terms that evaluate to $- 4 π$ . This confirms $□ G_{ret} = - 4 π δ^{3} (r) δ (t)$ . $□$

Bridge. The retarded Green's function is the fundamental solution of the wave equation and the basis for all radiation calculations in classical electrodynamics. The foundational insight is that causality selects the retarded (not advanced) Green's function: the field at a point depends on the source at earlier times, with the delay determined by the light travel time. The central message is that the retarded potentials automatically incorporate the finite speed of light, making causality a built-in feature of the solution. Putting these together, the Green's function method provides the systematic framework for computing the electromagnetic fields from arbitrary source distributions 10.08.02, and the Kirchhoff integral provides the basis for diffraction theory 10.08.03. The Feynman propagator, which is the quantum field theory analogue, generalises this to the quantum regime where time ordering replaces classical causality.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Compute the retarded potential of a point charge $q$ moving along the $z$ -axis with constant velocity $v$ . State the result in terms of the instantaneous position.

Hint

This is the Lienard-Wiechert potential. The result involves the factor $1/ (1 - \hat{R} \cdot v / c)$ where $R = r - r^{'} (t_{r})$ .

Answer

The Lienard-Wiechert potential for a charge moving with velocity $v$ at constant speed is:

V (r, t) = \frac{q}{4 π ϵ _{0}} \frac{1}{R ( 1 - R ^ \cdot v / c )}_{t_{r}}

where $R = r - v t_{r}$ is the vector from the retarded position to the observation point, $R = ∣ R ∣$ , and $\hat{R} = R / R$ . The factor $(1 - \hat{R} \cdot v / c)^{- 1}$ accounts for the Doppler-like compression or expansion of the field lines due to the motion. When the charge moves toward the observer ( $\hat{R} \cdot v > 0$ ), the potential is enhanced. When it moves away, the potential is reduced.

Exercise 4 (medium, symbolic).

The Sommerfeld radiation condition requires that the far-field behaves as $f (θ, ϕ) e^{ik r} / r$ with $lim_{r \to \infty} r (\frac{\partial f}{\partial r} - ik f) = 0$ . Explain why this selects the retarded (outgoing) solution.

Hint

Outgoing waves have $e^{ik r}$ dependence (phase increasing with $r$ ); incoming waves have $e^{- ik r}$ . The condition picks outgoing.

Answer

In the frequency domain, the wave equation has two spherical solutions: $e^{ik r} / r$ (outgoing, propagating away from the source) and $e^{- ik r} / r$ (incoming, propagating toward the source). The general solution is a linear combination.

The Sommerfeld condition $lim_{r \to \infty} r (\frac{\partial f}{\partial r} - ik f) = 0$ selects the outgoing wave. For $f = A e^{ik r} / r$ : $\frac{\partial f}{\partial r} = A (ik r - 1) e^{ik r} / r^{2} \approx ik A e^{ik r} / r$ for large $r$ . Then $r (\frac{\partial f}{\partial r} - ik f) \approx r (ik A e^{ik r} / r - ik A e^{ik r} / r) = 0$ . This satisfies the condition.

For $f = B e^{- ik r} / r$ : $\frac{\partial f}{\partial r} = B (- ik r - 1) e^{- ik r} / r^{2} \approx - ik B e^{- ik r} / r$ . Then $r (\frac{\partial f}{\partial r} - ik f) \approx r (- ik B e^{- ik r} / r - ik B e^{- ik r} / r) = - 2 ik B e^{- ik r} \neq = 0$ . This violates the condition.

The Sommerfeld condition is the mathematical expression of causality: only outgoing waves are physical at infinity.

Exercise 5 (hard, symbolic).

Compute the retarded Green's function in the frequency domain (the Helmholtz Green's function) by Fourier-transforming $G_{ret} (r, t)$ . Show that $G_{ret} (r, ω) = e^{ik r} / (4 π r)$ where $k = ω / c$ .

Hint

Fourier transform: $G (r, ω) = \int G (r, t) e^{iω t} d t$ . The delta function $δ (t - r / c)$ makes the integral straightforward.

Answer

G_{ret} (r, ω) = \int_{- \infty}^{\infty} \frac{δ ( t - r / c )}{4 π r} e^{iω t} d t = \frac{e ^{iω r / c}}{4 π r} = \frac{e ^{ik r}}{4 π r}

where $k = ω / c$ . The Fourier transform converts the retarded delta function into an outgoing spherical wave. The Helmholtz equation $(\nabla^{2} + k^{2}) G = - 4 π δ^{3} (r)$ is satisfied by $e^{ik r} / (4 π r)$ .

The advanced Green's function in the frequency domain is $G_{adv} (r, ω) = e^{- ik r} / (4 π r)$ (incoming spherical wave). The Feynman propagator is the average: $(e^{ik r} + e^{- ik r}) / (8 π r) = cos (k r) / (4 π r)$ , which is real and symmetric. In quantum field theory, the Feynman propagator includes an additional prescription for handling the pole at $ω = 0$ .

Exercise 6 (hard, symbolic).

State the Wheeler-Feynman absorber theory and explain how it uses both the retarded and advanced Green's functions to resolve the radiation reaction problem.

Hint

The theory postulates that the physical field is the average of retarded and advanced fields: $F = (F_{ret} + F_{adv}) /2$ . Radiation reaction arises from the response of the absorbing universe.

Answer

Wheeler-Feynman absorber theory (1945). The theory postulates:

Every charged particle interacts with other charges via the time-symmetric (half-retarded + half-advanced) field: $F = (F_{ret} + F_{adv}) /2$ .
An accelerated charge radiates outward via the retarded field. This radiation is absorbed by distant charges (the "absorber"), which re-radiate via their advanced field. The advanced re-radiation converges on the original charge at the moment of its acceleration, producing the radiation reaction force.
The net effect is that the original charge experiences: (a) the time-symmetric field from all other charges (which is purely retarded in the presence of a complete absorber) and (b) a self-force that reproduces the Abraham-Lorentz radiation reaction 10.07.04.

The theory resolves the self-energy problem (the infinite self-force of a point charge on itself) by eliminating the self-field entirely: charges interact only with other charges. The apparent causality violation (advanced fields) is cancelled by the response of the absorber, restoring macroscopic causality.

The theory is mathematically consistent but requires a complete absorber (the entire universe), and its physical status is debated. It influenced the development of quantum electrodynamics and the Feynman propagator.

Exercise 8 (medium, symbolic).

For a harmonically oscillating source $ρ (r, t) = ρ (r) e^{- iω t}$ , write the retarded potential in terms of the Helmholtz Green's function. Identify the multipole expansion structure.

Hint

$V (r) e^{- iω t} = \frac{1}{4 π ϵ _{0}} \int \frac{ρ ( r ^{'} ) e ^{ik ∣ r - r^{'} ∣}}{∣ r - r ^{'} ∣} d^{3} r^{'} \cdot e^{- iω t}$ . Expand $e^{ik ∣ r - r^{'} ∣} /∣ r - r^{'} ∣$ in powers of $k r^{'}$ .

Answer

For $ρ (r, t) = ρ (r) e^{- iω t}$ , the retarded potential is:

V (r) = \frac{1}{4 π ϵ _{0}} \int \frac{ρ ( r ^{'} ) e ^{ik ∣ r - r^{'} ∣}}{∣ r - r ^{'} ∣} d^{3} r^{'}

The factor $e^{ik ∣ r - r^{'} ∣} /∣ r - r^{'} ∣$ is the Helmholtz Green's function evaluated at $∣ r - r^{'} ∣$ .

Multipole expansion: For $r ≫ r^{'}$ (source much smaller than the observation distance), expand $∣ r - r^{'} ∣ \approx r - \hat{r} \cdot r^{'}$ in the phase and $∣ r - r^{'} ∣ \approx r$ in the amplitude:

\frac{e ^{ik ∣ r - r^{'} ∣}}{∣ r - r ^{'} ∣} \approx \frac{e ^{ik r}}{r} e^{- ik \hat{r} \cdot r^{'}}

Expanding $e^{- ik \hat{r} \cdot r^{'}} \approx 1 - ik \hat{r} \cdot r^{'} + \frac{1}{2} (ik \hat{r} \cdot r^{'})^{2} - \dots$ :

Monopole ( $ℓ = 0$ ): $\int ρ d^{3} r^{'} = q$ (total charge). Gives the Coulomb term $q e^{ik r} / (4 π ϵ_{0} r)$ .
Dipole ( $ℓ = 1$ ): $- ik \int r^{'} ρ d^{3} r^{'} = - ik p$ . Gives the dipole radiation field 10.07.02.
Quadrupole ( $ℓ = 2$ ): proportional to $k^{2}$ . Gives the quadrupole radiation 10.07.03.

The expansion in powers of $k r^{'}$ (or equivalently $k d$ where $d$ is the source size) is the multipole expansion.

Lean formalization Intermediate+

Mathlib has the Poisson Green's function but not the wave equation Green's function, the retarded/advanced selection, the Kirchhoff integral, or the Feynman propagator. Distribution theory on Minkowski space is beyond Mathlib. lean_status: none.

Advanced results Master

The Kirchhoff representation theorem. The field at any interior point can be expressed as an integral over the boundary surface, using the retarded Green's function. This is the mathematical foundation for diffraction theory 10.08.03:

V (r, t) = - \frac{1}{4 π} \oint_{S} [G_{ret} \frac{\partial V}{\partial n} - V \frac{\partial G _{ret}}{\partial n}] d a

In the frequency domain, this becomes the Helmholtz-Kirchhoff integral, which is the basis for Huygens' principle and the Fresnel-Kirchhoff diffraction formula.

The Feynman propagator in quantum electrodynamics. The Feynman propagator $D_{F} (x - x^{'}) = θ (t - t^{'}) G_{ret} + θ (t^{'} - t) G_{adv}$ is time-ordered: it uses the retarded Green's function for positive time differences and the advanced for negative. In momentum space: $D_{F} (k) = 1/ (k^{2} + i ϵ)$ . This is the Green's function that appears in Feynman diagrams and encodes the propagation of virtual photons.

The Jefimenko approach. Jefimenko (1989) showed that the fields $E$ and $B$ can be written directly in terms of the retarded sources without computing the potentials first. These are the Jefimenko equations 10.08.02, which follow from substituting the retarded potentials into $E = - \nabla V - \partial A / \partial t$ and $B = \nabla \times A$ .

Synthesis. The Green's function method is the universal framework for solving the wave equation in electrodynamics: it reduces the problem of finding fields from arbitrary sources to a quadrature (integration over the source with the Green's function kernel). The foundational insight is that the retarded Green's function encodes both the wave equation and the causality condition in a single mathematical object. The central message is that the retarded potentials $V$ and $A$ are obtained by convolving the source ( $ρ$ , $J$ ) with the Green's function, and the fields are then computed from the potentials. Putting these together, the Green's function provides the direct path from sources to fields, which is the basis for radiation calculations 10.07.02, diffraction theory 10.08.03, and the quantum field theory of photons.

Full proof set Master

Proposition (Uniqueness of the retarded solution). The retarded Green's function is the unique solution of the inhomogeneous wave equation that satisfies: (1) $G = 0$ for $t < t^{'}$ (causality), and (2) $G \to 0$ as $r \to \infty$ (Sommerfeld radiation condition).

Proof. Suppose $G_{1}$ and $G_{2}$ both satisfy the wave equation with the same source and the two conditions. Then $D = G_{1} - G_{2}$ satisfies the homogeneous wave equation with $D = 0$ for $t < t^{'}$ and $D \to 0$ at infinity. By the domain of dependence theorem for the wave equation, the solution in the forward light cone is uniquely determined by the initial data on the $t = t^{'}$ surface (where $D = 0$ ) and the boundary conditions at infinity (where $D = 0$ ). Therefore $D = 0$ everywhere, and $G_{1} = G_{2}$ . $□$

Connections Master

Vector potential 10.02.02 is the quantity solved for by the Green's function; the retarded vector potential is $A = μ_{0} \int G_{ret} * J$ .
Jefimenko's equations 10.08.02 express $E$ and $B$ directly in terms of the retarded sources.
Diffraction 10.08.03 uses the Kirchhoff integral (derived from the Green's function).
Dipole radiation 10.07.02 is obtained by expanding the retarded potential in powers of $k d$ .
Radiation reaction 10.07.04 involves the self-field, which is computed using the Green's function with the source being the charge's own current.
Faraday tensor 10.06.01 provides the covariant formulation; the retarded Green's function generalises to the covariant propagator on Minkowski space.

Historical & philosophical context Master

The concept of the Green's function was introduced by George Green in 1828 (for the Poisson equation). The extension to the wave equation was developed by Kirchhoff (1882), who derived the integral representation theorem and showed that Huygens' principle follows from the wave equation.

The retarded and advanced Green's functions were distinguished by Sommerfeld (1912), who formulated the radiation condition that selects the retarded solution. The Wheeler-Feynman absorber theory (1945) attempted to use both Green's functions symmetrically, eliminating the self-energy problem by postulating direct particle-particle interaction.

The Feynman propagator was introduced by Feynman in 1949 as part of his formulation of quantum electrodynamics. It is the Green's function that appears in Feynman diagrams and represents the amplitude for a virtual photon to propagate from one spacetime point to another. The causal structure (retarded vs advanced) is handled by the $i ϵ$ prescription in the momentum-space representation.

Bibliography Master

Green, G., An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828).
Kirchhoff, G., "Zur Theorie der Lichtstrahlen," Sitzungsber. K. Preuss. Akad. Wiss. 22, 641 (1882).
Sommerfeld, A., "Die Greensche Funktion der Schwingungsgleichung," Jahresber. DMV 21, 309 (1912).
Wheeler, J. A. and Feynman, R. P., "Interaction with the absorber as the mechanism of radiation," Rev. Mod. Phys. 17, 157 (1945).
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
Zangwill, A., Modern Electrodynamics (Cambridge, 2013).

Prerequisites

10.03.04
10.02.02

Used in

10.08.02
10.07.02

Tier anchors

beginner: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 10.1
intermediate: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6.4-6.6
master: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6; Zangwill, Modern Electrodynamics (2013), Ch. 19-20

References

Griffiths — Introduction to Electrodynamics, 4th ed. (Cambridge, 2017) · Ch. 10.1 Retarded potentials; the Green's function for the wave equation
Jackson — Classical Electrodynamics, 3rd ed. (Wiley, 1999) · Ch. 6.4-6.6 Green's functions for the wave equation; retarded and advanced potentials; the Feynman propagator
Zangwill — Modern Electrodynamics (Cambridge, 2013) · Ch. 19 Retarded potentials; Ch. 20 Green's functions for the scalar and vector potentials

Estimated time

beginner: 15m
intermediate: 35m
master: 55m