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

Regular Paper


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


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



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.


  1. 1.
    Aharonov, Y., Vaidman, L.: Properties of a quantum system during the time interval between two measurements. Phys. Rev. A 41, 11–20 (1990). doi: 10.1103/PhysRevA.41.11 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Goldenberg, L., Vaidman, L.: Applications of a simple quantum mechanical formula. Am. J. Phys. 64, 1059–1060 (1996). doi: 10.1119/1.18307 CrossRefGoogle Scholar
  3. 3.
    Parks, A.: Weak values and the Aharonov-Vaidman gauge. J. Phys. A 43(13), 035305 (2010). doi: 10.1088/1751-8113/43/3/035305 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Isham, C.: Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press, London (1995)CrossRefMATHGoogle Scholar
  5. 5.
    Chen, C.: Linear System Theory and Design. Holt, Rinehart and Winston, New York (1970)Google Scholar
  6. 6.
    Parks, A., Cullin, D., Stoudt, D.: Observation and measurement of an optical Aharonov-Albert-Vaidman effect. Proc. R. Soc. Lond. A 454, 2997–3008 (1998)CrossRefMATHGoogle Scholar
  7. 7.
    Papoulis, A.: Probability, Random Variables, and Stochastic Processes, 3rd edn. McGraw-Hill, New York (1991)MATHGoogle Scholar

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© 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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