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Schwinger's variational principle

From Biocrawler, the free encyclopedia.

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

δ < A | B > = i < A | δ S A B | B >

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