Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation:
L′=−12∂μAν∂μAν+12∂μAν∂νAμ−12∂μAμ∂νAν
=−12∂μAν∂μAν+12∂μ[Aν(∂νAμ)−(∂νAν)Aμ]The last term is a four-divergence which has no influence on the field equations. Thus the dynamics of the electromagnetic field (in the Lorentz gauge) can be described by the simple Lagrangian
L′′=−12∂μAν∂μAν
Yes, if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field, indeed according to the variational principle this four-divergence will not influence the field equation.
And actually I calculated the time-derivative dependence of 0-component of [Aν(∂νAμ)−(∂νAν)Aμ], in which only [A0(∂0A0)−(∂0A0)A0] could possibly contain time-derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case.
But how can he seem to claim that it holds for a general four-divergence term,The last term is a four-divergence which has no influence on the field equations
?
EDIT:
I only assumed the boundary condition to be Aμ=0 at spatial infinity, not at time infinity. And the variation of the action S=∫t2t1Ldt is due to the variation of fields which vanish at time, δAμ(x,t1)=δAμ(x,t2)=0, not having the knowledge of δ˙Aμ(x,t1) and δ˙Aμ(x,t2), which don't vanish generally, so the four-divergence term will in general contribute to the action, δSj=δ∫t2t1dt∫d3x∂μj(A(x),∇A(x),˙A(x))μ=δ∫t2t1dt∫d3x˙j0=∫d3x[δj(x,t2)0−δj(x,t1)0]
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