I'm wondering what exactly happens if a QFT is regularized with a smooth cutoff, but it is not sent to infinity at the end of the day. I'm certainly thinking about string theory's exponential smoothness around the string scale.

Polchinski says in his introduction to String Theroy vol. 1

In quantum field theory it is not easy to smear out interactions in a way that preserves the consistency of the theory. We know that Lorentz invariance holds to very good approximation, and this means that if we spread the interaction in space we spread it in time as well, with consequent loss of causality or unitarity.

The first thing that came to my mind was the Ostrogradski instability, but the theorem applies to a finite number of higher derivatives. A smooth cutoff however needs an infinite number of them. Another possible issue that I thought of is that the cutoff is normally enforced in Euclidean signature. But this affects loop momenta only, which are being integrated over.

I also found this blog post (paragraph about discretized theories) by Jacques Distler saying only that it requires fine-tuning due the cosmological constant.

So how exactly does a smooth finite cutoff violates unitarity?


One cheap way to see this maybe (as in: mathematical, but that doesn't connect immediately to Polchinki) is that a smooth cut-off in momentum space will necessarily alter the analytic structure of the answer and it's hard to see how this would not affect immediately the unitarity cuts. For instance, add a $\exp(-\ell^2 \alpha')$ to a scalar box integral, where $\ell$ is the loop momentum (one-loop) and $\alpha'$ some inverse mass scale. This shuts off high energy modes smoothly, but it stays in any unitarity cut as the exponential of some on-shell momenta and is not a function you expect from a tree-level graph in your theory, at least not if it's just conventional field theory with finitely many particles. Unitarity tells you that at tree-level you might have only poles. This exponential has a bad divergence at infinity on the complex plane. I'm not sure how this relates directly to Polchinski's "wrapping in time" argument though.

Also, note that in string theory, the exponential smoothness of the amplitudes is not a loop effect, it's the result of delicate cancellations between the infinite tower of states. These guys http://arxiv.org/abs/1607.04253 proved that it's universal to any theory of weakly interacting higher spin particles. In string theory, even at loop-level, you don't have a sharp cut-off in momentum space, it is in Schwinger proper-time space that the integration is truncated (as is explained somewhere in Polchinski).

  • $\begingroup$ I implicitly assumed that the cutoff function is part of the propagator, i.e. the theory is regularized already at the level of the (Wick rotated) Lagrangian. So, it should show up in tree-level diagrams, right? However, the bad divergence you mentioned prevents constructing unitarity cuts altogether, if I understand section 7.3 of Peskin/Schroeder correctly. What happens if the exponential is replaced by a more tame function on the complex plane, at the risk of introducing additional zeros and/or poles? $\endgroup$ – and008 Mar 1 at 11:34
  • $\begingroup$ The paper looks deep, but I need more time to digest it. Thanks! About loops: When performing the sewing construction (Polchinksi chapter 9.4) to reduce the genus 1 closed bosonic string amplitude to a genus 0 amplitude, I got (after a sketchy calculation) a factor of $4^{-\alpha' \ell^2}$ for the tachyon propagator from the integration over the moduli. The momentum cutoff translates to a truncation of the Schwinger proper time. This is actually how I started to think about QFTs with a finite cutoff. It would be interesting to see the connection to the infinite tower of states you mentioned. $\endgroup$ – and008 Mar 1 at 13:33
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    $\begingroup$ @and008 hmm I'm not sure that this is a factor that you would get from that computation. You might have things like $\exp(\alpha' \tau (\ell^2-4/\alpha'))$ where $\tau$ has the usual UV cutoff at the lower side of the fundamental domain, but it's not really in connection with the loop-momentum part. Concerning your first comment, I'm not sure what to say. If the analytic structure at tree-level is not the one correpsonding to unitarity, you're in trouble... If you insert another function with a heaviside step for instance, you'll change the analytic structure of your answer and that's also bad $\endgroup$ – picop Mar 2 at 14:18
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    $\begingroup$ You were right, in doing a more thorough calculation, I got for the tachyon partition function (in the $\alpha' \rightarrow 0$ limit) $Z=-\int d^Dp \Gamma(0,p^2+m^2)$, where $m^2=-4/\alpha'$. Interestingly, this corresponds exactly to the Lagrangian $(1)$ and $(8)$ in 1803.08827 (up to constants). But they didn't write where they got it from. $\endgroup$ – and008 Mar 5 at 11:10
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    $\begingroup$ Apparently I asked the question at the right time. This paper appeared the day after I asked the question: arxiv.org/abs/2103.00353 . The same authors show that smearing out interaction terms indeed doesn't make sense in Minkowski signature (perturbatively), only in Euclidean signature and analytically continuing afterwards. $\endgroup$ – and008 Mar 5 at 11:40

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