Substituting fields within the action: consistency issues and some applications

TL;DR

This paper examines the consistency issues arising from substituting fields within the action in field theory and mechanics, highlighting non-commutativity of operations, the special role of auxiliary variables, and implications for symmetries and gauge fixing.

Contribution

It provides a detailed analysis of the conditions under which field substitutions preserve equations of motion and symmetries, clarifying the differences between substitution at the Lagrangian level and at the equations of motion.

Findings

Substitutions generally do not commute with deriving equations of motion.

Auxiliary variables allow consistent substitutions under certain conditions.

Symmetry preservation depends on the nature of the substitution and reduction process.

Abstract

In field theory, as well as in mechanics, the substitution of some fields in terms of other fields at the level of the action raises an issue of consistency with respect to the equations of motion. We discuss this issue and give an expression which neatly displays the difference between doing the substitution at the level of the Lagrangian or at the level of the equations of motion. Both operations do not commute in general. A very relevant exception is the case of auxiliary variables, which are discussed in detail together with some of their relevant applications. We discuss the conditions for the preservation of symmetries - Noether as well as non-Noether - under the reduction of degrees of freedom provided by the mechanism of substitution. We also examine how the gauge fixing procedures fit in our framework and give simple examples on the issue of consistency in this case.