We formulate an “action principle” for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings “In memoriam Rudolf Haag” in September 2016 and the “Wolfhart Zimmermann memorial symposium” in May 2017.
The principle of compensation of dangerous diagrams has previously been the only general criterion for determining the coefficients in the Bogoliubov canonical transformation to quasiparticles. In this paper it is shown that the criterion of maximum overlap of the true ground state with the BCS ground state gives the same condition for determining the coefficients. The criterion of diagonalizing the quadratic part of the reaction operator, which is a generalization of the Brillouin-Brueckner condition, is also equivalent to the above criteria. Finally these criteria are evaluated from the standpoint of "least labor for the most accuracy."