 Solution

There is a sign inconsistency in the original PDF: varying its action gives a plus sign for the spatially projected magnetic term, whereas its displayed target equation has a minus sign. The following Lagrange multiplier for antisymmetry calculation derives the result of the action as printed.

Temporarily vary all sixteen entries $M^{\mu \nu }$ independently and addVariation of the symmetric Lagrange multiplier imposes $M^{\mu \nu }+M^{\nu \mu }=0$. DefineOn this constraint surface, $C_{\nu \mu }=-C_{\mu \nu }$. Differentiating the original contractions before imposing the constraint givesandIn particular, the reversed order $\sigma \rho$ in the last factor of the action matters. The unrestricted Euler-Lagrange field equation, evaluated on the antisymmetry constraint, isSubtract its index-transposed equation to eliminate the symmetric multiplier:A direct check is to restrict the action to a rest-frame magnetic component $M^{12}=m$, $M^{21}=-m$: its quadratic contribution is $+m^2/(2\chi )$, whose derivative has a positive sign. To obtain the PDF's target equation instead, one must reverse the sign of that quadratic term in the action, or replace its last $M^{\sigma \rho }$ by $M^{\rho \sigma }$. These are alternative repairs; changing conventions cannot make the printed action and printed equation agree.