Single-Particle Electrodynamics - Assassination Science

It is clear that, to carry out the partial derivatives present in (5.58), we shall
need to compute derivatives such as
∂ α ∂ β g[f(x, τ)] , (5.59)
where g[f] is either a Heaviside step function or Dirac delta function. Clearly,
the derivatives of the step function will yield terms that are non-vanishing
only on the worldline, which we are again ignoring in this section. Now, from
(5.17) we have
∂ u ∂ v g[f(u, v, τ)] = ∂ v
{
∂u f · d f g[f(u, v, τ)] } .
Carrying out the v differentiation by the product rule, we obtain
∂ u ∂ v f · d f g[f] + ∂ u f · ∂ v
(
df g[f] ) .
To compute this last term, one can consider d f g[f] as some new function,
h[f], upon which we perform the same process as carried out in (5.17). One
then finds that
∂ u ∂ v g[f(u, v, τ)] = ∂ u ∂ v f · d f g[f] + ∂ u f · ∂ v f · d 2
f g[f]. (5.60)
We now proceed to replace the d f derivatives in (5.60) by derivatives with
respect to τ. We again use the chain rule, namely
From the above, it is also clear that
d f ≡ d f τ · d τ ,
d 2
f ≡ d f d f ≡ d 2
f τ · d τ + (d f τ) 2 · d 2 τ .
d 2
f τ ≡ −d 2 τ f · (d τ f) −3 .
Using these identities in (5.59), we finally obtain the desired relation:
∂ α ∂ β g[f] = ∂ α ∂ β f · (d τ f) −1 · d τ g[f]
+ ∂ α f · ∂ β f · (d τ f) −2 · d 2 τ g[f]
− ∂ α f · ∂ β f · d 2 τ f · (d τ f) −3 · d τ g[f]. (5.61)
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