Commutativity of Substitution and Variation in Actions of Quantum Field Theory

Physics – High Energy Physics – High Energy Physics - Theory

Scientific paper

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11 pages

Scientific paper

10.1103/PhysRevD.80.105001

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.

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