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

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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 \({\hat{A}} |\psi \rangle \).

Keywords

Aharonov–Vaidman operator action theorem Aharonov–Vaidman gauge transformation \(U\left( 1 \right) \) gauge invariance nth moments of Hermitian operators 

Notes

Acknowledgements

The first author would like to thank Yakir Aharonov whose talk “Finally making sense of Quantum Mechanics, part 3” at the Perimeter Institute’s conference: Concepts and Paradoxes in a Quantum Universe June 20–25, 2016 inspired the ideas which lead to this paper and PI for the invitation to participate in this conference. Funding was provided by Office of Naval Research (US) under the auspices of the NSWDD ILIR program.

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Copyright information

© Chapman University (outside the USA) 2017

Authors and Affiliations

  1. 1.Electromagnetic and Sensor Systems DepartmentNaval Surface Warfare Center Dahlgren DivisionDahlgrenUSA

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