Quantum action - Definition and Overview

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 <math>\hat{A}<math> and another set <math>\hat{B}<math>. Let |A> be the eigenstate of <math>\hat{A}<math> 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 <math>\hat{A}<math> and <math>\hat{B}<math>.

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

<math>\delta=i<math>.