Dynamical gauge-fixing parameter ($\delta S/\delta \xi =-(\partial \cdot A{)}^2/(2{\xi }^2)$)

If a real nonzero field $\xi (x)$ appears in the Lorentzian density $-F_{ab}{F}^{ab}/4+({\partial }_a{A}^a{)}^2/(2\xi )$ and is varied, its algebraic Euler-Lagrange equation is $-({\partial }_a{A}^a{)}^2/(2{\xi }^2)=0$. Hence it imposes the Lorenz gauge condition on real classical fields. Varying $A_b$ gives ${\partial }_a{F}^{ab}-{\partial }^b[(\partial \cdot A)/\xi ]=0$, including derivatives of $\xi$. This constrained field theory differs from holding the gauge parameter fixed in a Gaussian functional integral.