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If I try to quantize the vector field in the Coulomb gauge (radiation gauge)

[tex] A_0(x)=0,\quad \vec\nabla\cdot\vec A=0. [/tex]

by imposing the equal-time commutation relation

[tex] [A_i(x),E_j(y)]=-i\delta_{ij}\delta(\vec x-\vec y)[/tex]

then I should find

[tex]\partial_i[A_i,E_j]=[\vec\nabla\cdot\vec A,E_j]=0, [/tex]

since [itex] \vec\nabla\cdot\vec A=0, [/itex] which is inconsistent with [itex] \partial_i\delta_{ij}\delta(\vec x-\vec y)\neq 0[/itex].

My question is simply how to take this divergence

[tex]\partial_i[A_i,E_j]=[\vec\nabla\cdot\vec A,E_j] [/tex]

I'm getting

[tex] \partial_i[A_i,E_j]=[\vec\nabla\cdot\vec A,E_j]+A_i\partial_i E_j-(\partial_i E_j)A_i .[/tex]

I must be missing something in the math here. Can anyone help me?