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Suppose we have an ultraviolet-complete (UV-complete) quantum field theory (QFT), e.g., quantum chromodynamics (QCD) or some other asymptotically free theory. Let's say we are trying to calculate some observable, which, for concreteness, we will assume is a scattering amplitude, using perturbation theory. Then, it is known that perturbation series for the amplitude diverges. In many QFTs, the divergence of perturbation theory occurs at such high order that it doesn't matter for practical purposes/calculations; however, I was wondering what the sources (either mathematical or physical) of these divergences are since, for a UV-complete theory, they cannot come from the theory being incomplete. Specifically, I have two questions:

  1. Do we know the mathematical reason the divergence occurs? For example, is it different from the reason perturbation theory doesn't converge in quantum mechanics? (I also don't know the mathematical reason perturbation theory doesn't converge in quantum mechanics, so I would be interested in hearing about this too).
  2. Do we know, or have at least some idea of, what all the non-perturbative contributions to the scattering amplitude are? I know what instantons are and how they work, but do people think that instantons are the only source of non-perturbative contributions, or do people think there are other non-perturbative effects that contribute, and if so, what are they?

There are posts, e.g., [1], that ask similar questions; however, they do not really address the physical meaning of UV divergences for UV-complete theories in particular and also do not comment on if there are non-perturbative corrections beyond effects due to instantons.

zeroknowledgeprover
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1 Answers1

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This is only a partial answer. But, I know that for Quantum field theories that are rigorously defined e.g. $\phi^4$ in 2 dimensions (see Constructive Quantum Field Theory I am specifically referencing the functional integral approach of Glimm and Jaffe), the non-convergence of the perturbation series comes from the unjustified commutation of limits. The theory is defined by the measure of the euclidean path integral $\mu(\phi) = e^{-S[\phi]} \mathcal{D}\phi$, which is quite complicated to construct mathematically.

The Euclidean correlation functions (Schwinger functions) can then be found by taking the expectation of the field variables in this measure e.g. $\langle \phi(x_1) \phi(x_2) \rangle = \mathbb{E}[\phi(x_1) \phi(x_2)]$. The perturbation expansion comes from expanding out the taylor series of the interaction exponential. $$ \mathbb{E}[\phi(x_1) \phi(x_2)] = \int \phi(x_1) \phi(x_2) e^{-S[\phi]} \mathcal{D}\phi = \int \phi(x_1) \phi(x_2) \lim_{N\to\infty} \left(\sum_{n=0}^N \frac{\lambda^n (-1)^n\phi^{4n}}{n!(4!)^n} \right)\mu_0(\phi) $$ Where $\mu_0(\phi) = \exp\{-S_0[\phi]\}\mathcal{D}\phi$ is a free field (gaussian) measure. Now, if we pull the limit out of the integral we get the standard perturbation series and Feynman rules. The problem is we cannot commute these limits. The perturbation series is an Asymptotic series which in practical terms means that it is a series which does not converge but does give a good approximation with a small number of terms.

The divergence of the perturbation series is not connected to the renormalization as in order to define the measure $\mu(\phi)$ we have already had to do renormalization. Here we see that the divergence of the perturbation series is just comes from the fact that we have switched the order of limits in a way that is not allowed mathematically.

Here are some more references: Divergence of perturbation theory for bosons and An SPDE approach to perturbation theory of $\Phi^4_2$: asymptoticity and short distance behavior.

Josh Newey
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