A note on the “Aharonov–Vaidman operator action representation theorem”

Abstract

The original Aharonov–Vaidman operator action theorem relates the action of an Hermitian operator \({\hat{A}} \) upon a state \(|\psi \rangle \) to its associated mean value \(\langle \psi |\hat{{A}}|\psi \rangle \), its uncertainty \(\Delta A\), and an orthogonal companion state \(|{\psi ^{\bot }} \rangle \). Here a slightly more general version of the original theorem and of the associated Aharonov–Vaidman gauge transformation is posited. It is shown that these more general versions are—and must clearly be—invariant under U(1) gauge transformations of \(|{\psi ^{\bot }}\rangle \). A new simple method for determining the nth moment of a Hermitian operator is also obtained using the operator action theorem for both the actions \({\hat{A}} |{\psi ^{\bot }} \rangle \) and \({\hat{A}} |\psi \rangle \).