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

TL;DR

This work identifies a subtle breakdown of the transposition rule $\delta(\partial_\nu A^\mu)=\partial_\nu(\delta A^\mu)$ for general non-holonomic constraints in classical field theory. It introduces integrable non-holonomic constraints, proving the transposition rule holds by mapping to holonomic form via $g_i=\partial_\nu f_i^\nu$ and using a constrained variational principle with Lagrange multipliers. The authors illustrate the framework with a $U(1)$ gauge field in Coulomb and Lorenz gauges, arguing these derivative-dependent gauges are integrable and compatible with standard action variation. The result provides a rigorous foundation for gauge fixing in classical and quantum field theories that employ derivative-based constraints, clarifying subtleties in gauge constraints.

Abstract

We show how the usual derivation of the equations of motion for a classical field theory with non-holonomic constraints, constraints that depend on the derivatives of the field, fails. As a result, the usual method for gauge fixing in classical and quantum field theories fails for general gauges that depend on the derivatives of the fields, such as the Coulomb and Lorenz gauges. The point of failure occurs at the use of the transposition rule, $δ(\partial_νA^μ)=\partial_ν(δA^μ)$, in the derivation of the equations of motion from the extremization of the action; we show that the transposition rule does not hold for general non-holonomic constraints. We define the concept of integrability of non-holonomic constraints in field theory and prove that the usual transposition rule holds for theories with these constraints. We are thus able to recover the usual treatment of gauge fixing for gauges of this type. We apply our formalism to the specific examples of the Coulomb and Lorenz gauges.