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## MAXWELL EQUATION FROM FIELD THEORY LAGRANGIAN

$L = -\frac{1}{4} F^{ab} F_{ab}$

$F_{ab} = \partial_a A_b - \partial_b A_a$

The Euler-Lagrange equation reduces to $\partial_a \partial L /\partial( \partial_a A_b) = 0$ since the Lagrangian have no terms with $A_a$.  The trick is to rewrite

$F^{ab}F_{ab} = \eta^{ac} \eta^{bd} F_{cd} F_{ab}$ first and then apply the product rule

$\partial L/\partial(\partial_a A_b) = -1/4 \eta^{ac} \eta^{bd}( \frac{\partial F_{cd}}{\partial (\partial_a A_b)} + F_{cd} \frac{\partial F_{ab}}{\partial (\partial_a A_b)})$

This is

$-(1/4) \eta^{ac}\eta^{bd} (\delta_{ac}\delta_{bd} F_{cd} + F_{cd}) = -(1/4) \eta^{ac}\eta^{bd}(2 F_{cd}) = -(1/2) F^{ab}$

Therefore $\partial_a F^{ab} = 0$ which are Maxwell’s equations.  The nice thing about this trick is that we can use standard product rule without problems once we lower the indices of $F^{ab}$. This is pretty useful because deriving the Euler-Lagrange equations for Lagrangians is basic for all quantum field theory models.