Schwinger's variational principle

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In Schwinger's variational approach to quantum field theory, introduced by Julian Schwinger, the quantum action is an operator. This is unlike the functional integral (path integral) approach where the action is a classical functional.

Suppose we have a complete set of commuting operators (or anticommuting for fermions) $\hat{A}$ and another set $\hat{B}$ . Let |A> be the eigenstate of $\hat{A}$ with eigenvalue A and similarly for |B>. There is some ambiguity in the phase, but that can be taken care of in the quantum action S_AB associated with $\hat{A}$ and $\hat{B}$ .

Suppose also we have not just one model of quantum mechanics or quantum field theory but a whole family of them, varying smoothly. So, |A> and |B> are "different" for each model in the family. S_AB also varies smoothly. Schwinger's variational principle tells us