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 Incoming links: Coulomb-gauge photon propagator

Coulomb-gauge propagator between conserved currents 2026-10-07

For

q \cdot J = q \cdot K = 0

, the instantaneous term in the Coulomb-gauge photon propagator combines with its spatial longitudinal subtraction:

J^{μ} D_{μν}^{C} (q) K^{ν} = - \frac{i}{q ^{2} + i 0} J \cdot K .

The identity follows from

q \cdot J = q^{0} J^{0}

and

q^{2} = (q^{0})^{2} - ∣ q ∣^{2}

. It is equality after contraction with conserved currents, not equality of the uncontracted propagator tensors. It is especially useful in showing that tree scattering amplitudes computed in Coulomb gauge are Lorentz invariant.

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Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 50 / 3 / d / Solution 2026-10-14

At order

e^{2}

, scalar particle–antiparticle scattering has a

t

-channel photon exchange and an

s

-channel annihilation and recreation diagram. They are shown below, with incoming momenta

p, q

and outgoing momenta

p^{'}, q^{'}

. The seagull vertex has only two scalar legs and does not yield a four-scalar tree diagram at this order.

Figure 1.
Leading scalar particle–antiparticle scattering diagrams with exchanged and annihilation-channel photons
. Details.

For the exchange graph, let

k = p - p^{'} = q^{'} - q

. Apart from overall charge signs, the two scalar currents are

J = p + p^{'}

and

K = q + q^{'}

. The external mass shell conditions give

k \cdot J = p^{2} - p^{'2} = 0, k \cdot K = q^{'2} - q^{2} = 0.

For the annihilation graph,

k = p + q = p^{'} + q^{'}

, with currents

J = p - q

and

K = p^{'} - q^{'}

, again up to vertex signs. Their contractions are

p^{2} - q^{2} = 0

and

p^{'2} - q^{'2} = 0

. Thus each graph couples the photon to conserved currents, the relevant scalar quantum electrodynamics Ward identity.

Write

k = (ω, k)

and

κ^{2} = ∣ k ∣^{2}

. conserved current means

k \cdot J = ω J^{0}

and

k \cdot K = ω K^{0}

. For

κ \neq = 0

, contraction with the Coulomb-gauge photon propagator gives, displaying the rational identity before the pole boundary limit,

J^{μ} D_{μν}^{C} K^{ν} = \frac{i J ^{0} K ^{0}}{κ ^{2}} + \frac{i}{k ^{2}} (J \cdot K - \frac{( J \cdot k ) ( K \cdot k )}{κ ^{2}}) = \frac{i J \cdot K}{k ^{2}} + \frac{i J ^{0} K ^{0}}{κ ^{2}} (1 - \frac{ω ^{2}}{k ^{2}}) = \frac{i}{k ^{2}} (J \cdot K - J^{0} K^{0}) = - \frac{i}{k ^{2}} J \cdot K .

We used

k^{2} = ω^{2} - κ^{2}

. Restoring the common Feynman i-epsilon prescription yields the Coulomb-gauge propagator between conserved currents result

J^{μ} D_{μν}^{C} K^{ν} = J^{μ} (\frac{- i η _{μν}}{k ^{2} + i 0}) K^{ν} .

Hence each on-shell tree amplitude can use the Lorentz-invariant propagator. This is equality after contraction, not equality of the gauge-dependent tensors. At

k = 0

, the separate Coulomb-gauge terms are ill-defined; combine them first and take the conserved-current limit. The combined expression has only the physical photon pole.

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