schwinger's variational principle

In Schwinger's variational approach to quantum field theory, 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 (or anticommuting for fermions) operators

\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

\delta =i

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