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I noticed that the Lorenz gauge term is represented by partial derivatives acting on the four-potential. Is it possible that the Lorenz gauge term could somehow be a similar object that belongs to the electromagnetic fields?

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Related: –  Qmechanic Mar 15 '14 at 17:07

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The Lorenz gauge term $\partial_\mu A^{\mu}$, or equivalently the expression (in $c=\hbar=1$)

$$\nabla \cdot \vec{A} + \frac{\partial \phi}{\partial t} $$

as a scalar quantity in its own right has no physical significance. Furthermore, the $4$-potential is not an object "that belongs to the electromagnetic fields," as the OP stated. The field $A_\mu$ is a quantity from which the electromagnetic fields may be derived; for example

$$\vec{E} = - \left( \nabla \phi + \frac{\partial \vec{A}}{\partial t}\right).$$

On the other hand, the quantity $\partial_\mu A^\mu$ is used for gauge fixing, we do not derive from it the electromagnetic fields.

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