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Regularizing (Away) Vacuum Energy

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Abstract

In this paper I formulate Minimal Requirements for Candidate Predictions in quantum field theories, inspired by viewing the standard model as an effective field theory. I then survey standard effective field theory regularization procedures, to see if the vacuum expectation value of energy density (\(\langle \rho \rangle\)) is a quantity that meets these requirements. The verdict is negative, leading to the conclusion that \(\langle \rho \rangle\) is not a physically significant quantity in the standard model. Rigorous extensions of flat space quantum field theory eliminate \(\langle \rho \rangle\) from their conceptual framework, indicating that it lacks physical significance in the framework of quantum field theory more broadly. This result has consequences for problems in cosmology and quantum gravity, as it suggests that the correct solution to the cosmological constant problem involves a revision of the vacuum concept within quantum field theory.

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Notes

  1. For recent philosophical discussions of EFTs, see ([3,4,5,6,7]).

  2. Koberinski and Smeenk [4] provide a more sustained argument that the cosmological constant problem signals a failure of naturalness for vacuum energy, in QFT and in general relativity as an EFT. The solution proposed there is to embrace new heuristics in theory construction, and to accept the limitations of the EFT framework for understanding fundamental physics.

  3. Using precision tests of the standard model, one may find deviations from the predictions made using only the renormalizable terms. Examples of possible experimental tests include the anomalous magnetic moment of the electron or muon [8,9,10] as well as the fine structure of positronium and muonium [11]. In all of these cases, small deviations from the predictions made using the renormalizable standard model may be accounted for with higher-order couplings, suppressed by the physical cutoff scale.

  4. By physical significance of vacuum energy density, I mean the inference from a vacuum expectation value of an energy density term within a model of QFT to a real physical quantity onto which that value maps. One can believe that there is some real physical quantity of a suitably averaged value of vacuum energy density, to which our best physical theories don’t accurately map (cf. [12]). The arguments in this paper undermine taking values from QFT to map onto the world; they say nothing about whether vacuum energy density exists. Undermining the physical significance of vacuum energy density for QFTs means that we should not trust that our best QFTs to accurately capture the relevant physics. Continuing the process discussed in Saunders [13], a further revision of the vacuum concept in QFT may be required, or perhaps even a full theory of quantum gravity.

  5. The lower bound may be interpreted as encoding the fact that QFTs are only used in local regions of spacetime. Imposing some set of boundary conditions for long distances just means that we don’t expect the model to apply in all of spacetime.

  6. For a more detailed analysis of the differences between the two approaches to renormalization, see ([7, 18]). The latter argues that EFTs are best understood strictly under cutoff regularization. However, as I show below for the vacuum energy density, many features of QFTs are most easily understood under dimensional regularization.

  7. The fact that Lorentz invariance is lost if the lattice structure of effective field theories is taken literally should have observable consequences. Incredibly sensitive tests have failed to detect violation of Lorentz invariance at small scales [20]. Though outside the scope of this paper, one might argue that a literal interpretation of the lattice is therefore unmotivated from the point of view of both QFTs and general relativity.

  8. This is only a disadvantage if one expects a regulator to be physically significant. If regularization is treated simply as a procedure for taming divergences, then the regulators need not have a physical significance. Further, if the analogy between lattice regularization in condensed matter physics and particle physics is misleading, then the physical interpretation that lattice regularization provides may actually lead to an unjustified physical interpretation (cf. ([22, 23])).

  9. I use \(\mu\) as an arbitrary scale factor here because it appears in the formal expression for \(\langle {\tilde{\rho }}_{dim}\rangle\) in the same way that the (arbitrary) momentum cutoff appears in the lattice regularized expression. The fact that these scales have different meanings supports my argument that these terms differ significantly. The same term for the regulator is used simply to aid algebraic comparison.

  10. This is a first-order renormalized calculation. As Martin [19, Sect. VI] highlights, this prediction is largely unchanged under a Gaussian approximation to an interaction term (i.e., to one loop). Since the expression remains the same, I refer to Eq. (15) as a one-loop renormalized term.

  11. This example is discussed in more detail in Sect. 4.

  12. The treatment of point-splitting in this section follows the presentation in Scharf [26, Ch. 3].

  13. It is possible that \(\omega\) will not be an integer for some distributions, though this does not occur in QED. When \(\omega\) is not an integer, the polynomial will be rank \(\omega '\), the largest integer that is less than \(\omega\).

  14. Technically, old demands of renormalizability were imposed on the S-matrix of a model of QFT, believed to encode all physically meaningful content of scattering amplitudes and other dynamics [21, 31]. The QFTs comprising the standard model of particle physics are all renormalizable, despite the fact that the vacuum energy for each is nonrenormalizable. If one demands renormalizability of a model in terms of its S-matrix, additional nonrenormalizable structure that can be extracted from the action should be thought of as ill-defined surplus structure, about which the theory remains silent.

  15. The other, of course, being the Higgs mass. In that case the physical significance is undeniable, since the Higgs boson has been discovered, and has mass about 125 GeV [32]. The physical significance of vacuum energy is a bit less direct, and is subject to criticism. Aside from the criticism raised in this paper, see Bianchi and Rovelli [33].

  16. Cf. Koberinski [2] for an argument that the Casimir effect and Lamb shift do not license the inference to a constant vacuum expectation value of energy.

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Acknowledgements

The author is grateful to Chris Smeenk, Robert Brandenberger, Doreen Fraser, and the UCI Philosophy of Physics Research group for helpful feedback on early drafts of this paper, as well as the comments from two anonymous reviewers. This work was supported by the Social Sciences and Humanities Research Council of Canada (756-2020-0198), and the John Templeton Foundation Grant 61048, New Directions in Philosophy of Cosmology. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation.

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    Adam Koberinski

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Koberinski, A. Regularizing (Away) Vacuum Energy. Found Phys 51, 20 (2021). https://doi.org/10.1007/s10701-021-00442-z

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