Integrable Non-Holonomic Constraints and Gauge Fixing in Classical Field Theory

We re-examine the derivation of the equations of motion from an action principle for classical field theories with non-holonomic constraints, \textit{i.e.}, constraints involving derivatives of the fields. We find that the usual method for gauge fixing in classical and quantum field theories is highly non-trivial for non-holonomic gauge constraints, such as the Coulomb and Lorenz gauges. The subtlety appears at the use of the so-called transposition rule, $\delta(\partial_\nu A^\mu)=\partial_\nu(\delta A^\mu)$, which has been shown not to hold for general non-holonomic constraints in the point-particle context. We provide a sufficient definition of integrable non-holonomic constraints in classical field theory that allows us to prove that the transposition rule holds for all theories with these constraints; we are then able to recover the usual treatment of gauge fixing for gauges of this type.