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My understanding of QED is roughly that (please correct me if I have made a mistake, I am not very experienced in QFT):

  • Firstly, QED is not exact. At a certain energy scale, we have electroweak unification. Thus, many physicists don't care about predictions of pure QED at high energies, as it is not physical.
  • Regardless, we can study pure QED mathematically.
  • QED is perturbatively renormalizable to all orders due to the BPHZ theorem.
  • However, by a result of Dyson, perturbative expansions in QED are only asymptotic series. One expects that maximum accuracy is gotten by truncating the sum at $\sim \frac{1}{\alpha} \approx 137$ terms, and that adding more terms will lead the series to diverge.
  • In some cases, though, asymptotic expansions can be resummed to a finite value via methods like Borel summation. So naive divergence of the perturbation series does not necessarily mean the theory is ill-behaved.
  • There seems to be a Landau pole, but apparently this can't be trusted as the calculation leading to this prediction is perturbative.
  • Nevertheless, numerical calculations apparently suggest that QED is trivial for different reasons (i.e. see https://physics.stackexchange.com/a/359627/359416)

QED being trivial seems strange as we routinely make excellent theoretical predictions using the first few terms of the perturbative expansion. If the interacting theory really does not exist, then what theory are we "actually" using when we think we are using QED? Is it QED but with a UV cutoff?

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    $\begingroup$ "what theory are we "actually" using when we think we are using QED?" perturbative QED up to $n$ loops (depending on your level of sophistication insert values of $n$ from $0$ to $5$). Does that not answer your question? $\endgroup$ Feb 27, 2023 at 8:36
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    $\begingroup$ Every single theory we have in physics is only valid in its region of validity. The only theory that would fall outside of this would arguably be a theory of everything. While we may have seen glimpses of such a t.o.e. we sure haven't got it yet. $\endgroup$ Feb 27, 2023 at 13:10
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    $\begingroup$ @Prof.Legolasov Normally in physics, a perturbative expansion is an approximation to a true answer which we could in principle compute numerically to arbitrary precision. If QED is trivial, the true theory doesn't exist, so I don't understand what we are approximating in the first place. $\endgroup$
    – S.R.
    Feb 27, 2023 at 15:08
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    $\begingroup$ @S.R. I disagree. There are many "true answers" that can be approximated by the same perturbative expansion $\endgroup$ Feb 27, 2023 at 21:41
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    $\begingroup$ @Prof.Legolasov Thank you, that makes sense. I did not truly remember/grasp this until I saw a comment below by the user QCD_IS_GOOD. I think I understand the answer to my question, now. $\endgroup$
    – S.R.
    Feb 28, 2023 at 1:59

1 Answer 1

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  1. "Firstly, QED is not exact. At a certain energy scale, we have electroweak symmetry breaking. Thus, many physicists don't care about predictions of pure QED at high energies, as it is not physical."

Answer: The modern point of view is, to regard QED (to be concrete, let us think of the electromagnetic interaction of the electron/positron field) as an "effective quantum field theory", valid only below a certain energy scale, where contributions from other degrees of freedom (muon, hadrons, etc.) described by the standard model become relevant. By the way, this "philosophy" applies also to the standard model itself, which can be seen as a "low-energy effective theory" valid up to an (yet unknown) energy scale, where "new" physics might become relevant.

  1. "... we can study pure QED mathematically."

Answer: Sure, enjoy yourself!

  1. "QED is perturbatively renormalizable ..."

Answer: Yes, the QED Lagrangian is a gauge theory, containg only terms up to (operator) dimension $4$, being thus (perturbatively) renormalizable. This means, that only the electron mass $m_e$ and the fine-structure constant $\alpha \simeq 1/137$ are needed as experimental input for pure QED (of $e^\pm$ and the photon), all other observables (like the anomalous magnetic moment of the electron) can then be predicted.

  1. "... perturbative expansions in QED are only asymptotic series. One expects that maximum accuracy is gotten by truncating the sum at $\sim 1/ \alpha \sim 137$, and that adding more terms will lead the series to diverge."

Answer: Perturbation theory in QED leads indeed to an asymptotic expansion in the coupling parameter. The divergence of the perturbative series of pure QED has no physical relevance. In calculations of the contributions of pure QED to the anomalous magnetic moment of the electron and the muon, the perturbative expansion has been extended up to $5$-loop order, meeting the current experimental uncertainties of these observables.

  1. "... So naive divergence of the perturbation series does not necessarily mean the theory is ill-behaved."

Answer: This might be true or not, but this question is completely irrelevant from a physical point of view. The applicability of pure QED breaks down for purely physical reasons (presence of other particles) well below those energy scales, where this problem might appear (see also 6.).

  1. "There seems to be a Landau pole, but apperently this can't be trusted as the calculation leading to this prediction is perturbative."

Answer: The energy scale (the Landau pole in ancient terminology) $\Lambda_{\rm QED} \simeq m_e \exp(3 \pi/2 \alpha) \simeq 10^{277} \, \rm GeV$, where the perturbative expansion breaks down, is beyond good and evil. Compare it to the Planck mass $\simeq 10^{19} \, \rm GeV$, where gravitation becomes relevant! Seen as an effective low-energy theory, pure QED of the $e^\pm$ field is valid only up to an energy scale, where the presence of the muon ($m_\mu \sim 100 \, \rm MeV$), etc. becomes relevant. Seen as the unbroken $U(1)$ part of the standard model (taking into account all presently known particles), its applicability might be up to $\rm TeV$ energies.

  1. "QED being trivial..."

Answer: Triviality of a theory simply means, that taking such type of a theory seriously up to arbitrarily high energies (or, equivalently, down to arbitrarily small distances), i.e. requiring all the "axioms" of a relativistic quantum field theory, it only admits the free theory solution. Regarding QED as an effective theory, this "problem" is again irrelevant from a physical point of view.

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  • $\begingroup$ Shouldn't that $\ln$ be an $\exp$? $\endgroup$
    – J.G.
    Feb 27, 2023 at 15:41
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    $\begingroup$ Thank you, but I'm not sure whether this answers my question. Normally, an effective theory is a well-defined construct which gives us approximations to physically observable quantities. If I understand correctly, triviality implies lattice QED does not exist, so it cannot be used as a definition of QED. So my question is-- how do we still get well-behaved answers perturbatively, and what nonperturbative theory do those numbers actually correspond to? $\endgroup$
    – S.R.
    Feb 27, 2023 at 15:47
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    $\begingroup$ @Mankka Yes, but the main (theoretical) problem is not the "pure" QED contribution but the hadronic contribution and hadronic light-by-light scattering in muon g-2. $\endgroup$
    – Hyperon
    Feb 27, 2023 at 20:17
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    $\begingroup$ @S.R. As you say in your question, perturbative QED does approximate 'QED with a cutoff'. It also approximates the QED-part of any UV-completion of the standard model (something like a string theory). The point is that at low energies we are simply not sensitive to whatever is happening in the UV, so it's a bit of a moot question. You ask "if there's no nonperturbative theory, why does perturbation theory work", I feel like that's a bit backwards - it's all just made up models to begin with. $\endgroup$ Feb 27, 2023 at 20:25
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    $\begingroup$ @Tom You are probably mixing up the $n$-loop order with the order $2n$ of powers of $e$ in $a=(g-2)/2$. E.g., the famous Schwinger result $\alpha/2 \pi$ was the result of a one-loop vertex correction being of second order in $e$. The standard references for the pure QED contributions are: T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, Complete Tenth-Order QED Contribution to Muon $g-2$, Phys. Rev. Lett. 109 (2012) 111808, https//arxiv.org/pdf/12055370.pdf ; T. Aoyama, T. Kinoshita, M. Nio, Theory of the Anomalous Magnetic Moment of the Electron, Atoms 7 (2019) 28; ctd. $\endgroup$
    – Hyperon
    Feb 28, 2023 at 16:12

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