A Quantum Field Theory View of Interaction Free Measurements


We propose a Quantum Field Theory description of beams on a Mach–Zehnder interferometer and apply the method to describe Interaction Free Measurements (IFMs), concluding that there is a change of momentum of the fields in IFMs. Analysing the factors involved in the probability of emission of low-energy photons, we argue that they do not yield meaningful contributions to the probabilities of the IFMs.

This is a preview of subscription content, log in to check access.

Fig. 1


  1. 1.

    Notice that if the detonation is detectable, these photons cannot be soft as soft photons have, by definition, such low energy that they evade detection. To avoid this semantic confusion, we shall refer to the detectable photons as low-energy photons.

  2. 2.

    The 4-momentum operator is

    $$P^{\mu } = \int_\Sigma {T^{{0\mu }} d^{3} x} ,$$

    where \(T^{\nu \mu }\) is the energy-momentum tensor [14].


  1. 1.

    Elitzur, A.C., Vaidman, L.: Quantum mechanical interaction-free measurements. Found. Phys. 23, 7 (1993)

    Article  Google Scholar 

  2. 2.

    Renninger, M.: Messungen ohne störung des meßobjekts. Z. Phys. 158, 417–421 (1960)

    ADS  Article  Google Scholar 

  3. 3.

    Paraoanu, G.S.: Interaction-free measurements with superconducting qubits. Phys. Rev. Lett. 97, 180406 (2006)

    ADS  Article  Google Scholar 

  4. 4.

    White, A.G., Mitchell, J.R., Nairz, O., Kwiat, P.G.: “Interaction-free imaging”. Phys. Rev. A 58, 605 (1998)

    ADS  Article  Google Scholar 

  5. 5.

    Paul, H., Pavic̆ić, M.: Nonclassical interaction-free detection of objects in a monolithic total-internal-reflection resonator. J. Opt. Soc. Am. B 14, 6 (1997)

    Article  Google Scholar 

  6. 6.

    Kwiat, P.G., White, A.G., Mitchell, J.R., Nairz, O., Weihs, G., Weinfurter, H., Zeilinger, A.: High-efficiency quantum interrogation measurements via the quantum Zeno effect. Phys. Rev. Lett. 83, 4725 (1999)

    ADS  Article  Google Scholar 

  7. 7.

    Namekata, N., Inoue, S.: High-efficiency interaction-free measurements using a stabilized Fabry–Perot cavity. J. Phys. B 39, 16 (2006)

    Article  Google Scholar 

  8. 8.

    Weinberg, S.: The Quantum Theory of Fields. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  9. 9.

    Weinberg, S.: Infrared photons and gravitons. Phys. Rev. 140, B516 (1965)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Bloch, F., Nordsieck, A.: Note on the radiation field of the electron. Phys. Rev 52, 54 (1937)

    ADS  Article  Google Scholar 

  11. 11.

    Simon, S.H., Platzman, P.M.: Fundamental limit on interaction-free measurements. Phys. Rev. A 61, 052103 (2000)

    ADS  Article  Google Scholar 

  12. 12.

    Vaidman, L.: The meaning of the interaction-free measurements. Found. Phys. 33, 491–510 (2003)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Karlsson, A., Björk, G., Forsberg, E.: “Interaction” (energy exchange) free and quantum nondemolition measurements. Phys. Rev. Lett. 80, 1198 (1998)

    ADS  Article  Google Scholar 

  14. 14.

    Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Perseus Books, New York (1995)

    Google Scholar 

  15. 15.

    Gerry, C., Knight, P.: Introductory Quantum Optics. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  16. 16.

    Cabrera, R., Strohecker, T., Rabitz, H.: The canonical coset decomposition of unitary matrices through Householder transformations. J. Math. Phys. 51, 082101 (2010)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Orfeu Bertolami.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

C. R. Barroso, F., Bertolami, O. A Quantum Field Theory View of Interaction Free Measurements. Found Phys (2020). https://doi.org/10.1007/s10701-020-00350-8

Download citation


  • Interaction free measurements
  • Weinberg’s soft photon theorem