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I stumbled across the Wikipedia article on coupling constants [1] and didn't quite unterstand, what the paragraph on running couplings is trying to express.

It relates the virtual particles taking part in some scattering process and the apparent violation of energy conservation to the uncertainty principle of energy and time and concludes: “The previous remark only applies to some formulations of quantum field theory, in particular, canonical quantization in the interaction picture. In other formulations, the same event is described by ‘virtual’ particles going off the mass shell. Such processes renormalize the coupling and make it dependent on the energy scale, $\mu$, at which one observes the coupling.“

Now there are two things, that I don't understand:

  1. How can canonical quantization in the interaction picture circumvent the introduction of a running coupling?
  2. How does the connection to the uncertainty principle arise (at least heuristically, as it says)?

[1] https://en.wikipedia.org/wiki/Coupling_constant#Running_coupling

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The two are alternative (complementary) "pictures" of variability with scale.

The second one, the running coupling covariant perturbation theory picture relies on no metaphors or intuition: it tells you to follow the math and apply the resulting running coupling to the process you probe at a given energy, the "modern" RG picture.

The first, alternative, earlier, more primitive, picture reminds you of the elementary QM intuition connecting wavelengths with inverse momenta (and frequencies/energies with inverse times), the central theme in estimating the resolving power of such probes, universal in high energy physics.

It emphasizes the virtuality of states, hence "violation" of eigenenergies of intermediate states at short times (inverse to the probing frequencies), the "slop" circumscribed and summarized by the uncertainty principle. That is, as you use shorter wavelengths to probe shorter distances, you employ higher momentum and frequency probes (~ shorter times) of heavier intermediate states, as the reader is assumed to be familiar with in the context of atomic physics transitions. With higher resolving power, you observe more and "heavier" virtual states to get your effective amplitude, here the effective coupling.

This is a shared visualization/mnemonic/intuition that most practitioners carry in the back of their mind, but it is understood you need not take too much of it too literally and descend into rigid consistency vortices. You may ignore it altogether, if it fails to speak to you, but it would be a pity.

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