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On Haag’s Theorem and Renormalization Ambiguities

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Abstract

We revisit the implications of Haag’s theorem in the light of the renormalization group. There is still some lack of discussion in the literature about the possible impact of the theorem on the standard (as opposite of axiomatic) quantum field theory, and we try to shed light in this direction. Our discussion then deals with the interplay between Haag’s theorem and renormalization. While we clarify how perturbative renormalization (for the sub-class of interactions that are renormalizable) marginalizes its impact when the coupling is formally small, we argue that a non-perturbative and non-ambiguous renormalization cannot be built if there is any reference to the interaction picture with free fields. In other words, Haag’s theorem should be regarded as a no-go theorem for the existence of a non-ambiguous analytic continuation from perturbative to non-perturbative QFT.

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The authors declare that no data have been taken during the completion of this work.

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Notes

  1. In Eqs. (11) and (13), one can also put coefficient a, b, with \(|a|=|b|=1\), but this does not affect Eq. (16).

  2. Recall also that the unitarity of U can be rigorously proven in quantum mechanics [15].

  3. The drawback is that, at a given order in the expansion, all previous terms have to be computed as well, perhaps making it less useful in practical computations.

  4. This arbitrary constant makes the result obtained using the Borel–Ecalle resummation ambiguous.

  5. In principle, one might also consider power correction in \(\rho\) into the coefficient of L, but this does not change the conclusions here presented.

  6. We should stress again that the approach to RGE, leading to Eq. (29), is an approximation and some assumptions are required, such as the non-linearity in \(\rho\) are small, in agreement with the theory in [30].

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Acknowledgements

We thank Oleg Antipin and Jahmall Bersini for the comments on the manuscript.

Funding

Alessio Maiezza was partially supported by the Croatian Science Foundation Project Number 4418. Juan Carlos Vasquez was supported in part under the U.S. Department of Energy Contract DE-SC0015376.

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  1. Ruder Bošković Institute, Bijenička cesta 54, 10000, Zagreb, Croatia

    Alessio Maiezza

  2. Department of Physics, Amherst Center for Fundamental Interactions, University of Massachusetts, Amherst, MA, 01003, USA

    Juan Carlos Vasquez

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Maiezza, A., Vasquez, J.C. On Haag’s Theorem and Renormalization Ambiguities. Found Phys 51, 80 (2021). https://doi.org/10.1007/s10701-021-00484-3

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