Commutativity of Substitution and Variation in Actions of Quantum Field Theory

TL;DR

This paper investigates a paradox in quantum field theory related to substitution and variation of field configurations, illustrating it with specific instanton examples and emphasizing the importance of boundary conditions and Legendre terms.

Contribution

It clarifies the conditions under which substitution and variation commute in quantum field theory, highlighting the role of boundary conditions and Legendre terms in resolving the paradox.

Findings

Substitution and variation are not always interchangeable in quantum field theory.

Matching boundary conditions requires adding Legendre terms to the action.

Examples include $S^4$ and Freund-Rubin-like instantons.

Abstract

There exists a paradox in quantum field theory: substituting a field configuration which solves a subset of the field equations into the action and varying it is not necessarily equivalent to substituting that configuration into the remaining field equations. We take the $S^4$ and Freund-Rubin-like instantons as two examples to clarify the paradox. One must match the specialized configuration field variables with the corresponding boundary conditions by adding appropriate Legendre terms to the action. Some comments are made regarding exceptional degenerate cases.