I've heard of UV divergences appearing in perturbative calculations as both a mathematical artifact and as having a physical origin (the latter understanding due to Wilson). Could one clarify what this means? This sounds contradictory at first glance.


This is a very deep question. The full answer (which I do hope someone will attempt or link to a post which answers it) involves quite a long and technical story and involves path integrals, the renormalization group, regularization and renormalization schemes, and on and on. There are many nice expositions in books and online; here is an article by Burgess on arxiv that I like: https://arxiv.org/abs/hep-th/0701053

I am going to attempt a shorter but necessarily incomplete answer to get across the main idea without assuming too much background knowledge of quantum mechanics and quantum field theory.

Observationally, we only ever probe a physical theory over some finite range of energies. The exact lower end of this range is not so important for this question, but as a lower bound we can say that the cosmological constant sets an energy scale of $\sim 10^{-33}\ {\rm eV}$ below the Compton wavelength of a particle is larger than the horizon length of the Universe. The upper end is set by the most energetic experiments performed; the LHC probes processes up to 1 TeV with excellent precision. Cosmic ray experiments have detected particles with energies as large as $5 \times 10^{19} \ {\rm eV}$ (although these events are very rare and not well understood).

Theoretically, processes from arbitrarily large energies contribute to a given observable. One nice way to see this is to look at the expression for the second order perturbative corrections to the energy of the $n$-th state of a quantum system (the details of this equation are less important than the general idea I'll get into below): \begin{equation} E^{(2)}_n = \sum_{m \neq n} \frac{\langle n| \delta H | m \rangle \langle m | \delta H | n \rangle}{E_n - E_m} \end{equation} where $\delta H$ is the perturbation Hamiltonian, and where $|m \rangle$ are eigenstates of the unperturbed Hamiltonian. The key thing about this equation I want to draw your attention to is the sum over all states (except $m=n$), $\sum_{m\neq n}$. In particular, this implies that states with energies much larger than $E_n$ contribute to the observed energy level of the state $n$. You may have heard words along the lines "particles pop into and out of existence in the quantum vacuum, and these vacuum fluctuations dress physical observables" -- this equation is a rigorous manifestation of those words. Quantum mechanics tells us that there is a probability amplitude for every possible way a process can occur and we need to sum the probability amplitudes, so there is some contribution from ways where very high energy states are excited.

Now, naively, combining the above two paragraphs, we might think that we are in a very powerful situation. By making observations over a finite range of energies, we can measure effects that include contributions to arbitrarily high energies. So, with enough precision, it seems we can make observations at 1 TeV, and probe quantum gravity physics at the Planck scale.

This general idea has some truth to it. The reason the recent (as of spring 2021) measurement of the muon $g-2$ is exciting is that one can make a very precise calculation of the observable from theory, and a comparably precise measurement, and by comparing them we check if the theory is right to a very precise level. The disagreement between experiment and theory may point to a higher energy correction not included in the theory. (But -- don't get too excited in this specific case; this is a difficult calculation and a competing group has done the calculation a different way and gets a result that agrees with experiment).

But even though we can probe high energies with precise low energy experiments, the situation is not as simple as it naively looks. The deep reason for this is known as the decoupling theorem. Loosely stated, the contribution of processes from very high energies (very far above the energy scale we are probing), can be parameterized by coefficients of local operators in the low energy Hamiltonian. To give a specific example, which hopefully illustrates the basic idea: you can think of the electron as a point particle moving through a quantum vacuum in which virtual photons, electrons, and positrons (and other particles) pop into and out of existence. The combined motion of the electron and this cloud of virtual particles is what we actually observe as an electron in the lab. We can parameterize the effect of the cloud of virtual particles, as a correction to the electron's "intrinsic" mass. To restate this, the observed mass of the electron, at the low energy scales we can probe in a lab, is a combination of the "bare mass" of the point-like electron, and the "dressing" due to the interactions with quantum fluctuations. The exact way we split the contributions between a "bare" mass and the "dressing" due to interactions with virtual particles depends somewhat on our conventions, but the combination of the two is an unambiguous physical observable. So to bring it back around: even though in principle there are contributions due to arbitrarily high energy processes, the effect of these processes gets wrapped up into a finite number of parameters describing the low energy theory, and so in the end the observable effects of these high energy processes are very small.

Now with all this background, we get to your question.

It turns out that in typical perturbative schemes, the contribution of virtual particles to the mass of the electron (or generically, many low energy observables) is infinite. The reason for this infinity is that there is a sum over states with arbitrarily high energy, and this sum diverges. Of course we can't actually be sure what the states are at energies above the scale of the process we are probing. So it may be (and probably is!)the case that the infinity is an artifact of extrapolating our theory to arbitrarily high energies, and the sum in the "real" theory is finite. But by the magic of the decoupling theorem, we don't need to know the correct theory to arbitrarily high energies, to be able to get sensible predictions. We can simply absorb these corrections into our definition of the bare mass of the electron.

Mathematically, this means we add an infinite constant to the bare mass of the electron, which is chosen to cancel the infinite contribution of the virtual particles (at least, the contribution of the virtual particles is infinite if we extrapolate our theories to infinitely large energies). The magic is that we only need to do this process a finite number of times to cancel all infinities that occur in any process (at least for a renormalizable theory; this can also be done for a non-renormalizable theory if we only work to a finite order in energy -- this is the approach of "effective field theory").

Physically, the actual numerical value of the contribution due to virtual particles (which turns out to be infinite, if we extrapolate our current theories to infinitely large energies) is not particularly important. The key physical idea is that the virtual particles have some contribution, which "dresses" the "intrinsic" properties of the particle. All we can observe is the combined effect of dressed particle.

This whole story in fact has observational consequences, through the renormalization group. Taking again the example of the electron mass, we can ask what happens if we probe the electron at higher energies, such that we "penetrate" the cloud of electrons. We would expect physical properties, such as the mass and charge, to change. In fact, this does happen. For instance, the fine structure constant (the squared charge of the electron in dimensionless units) is approximately $1/137$ at "normal" every day energies, but in fact takes a value of $\sim 1/127$ at energies of about 90 GeV.

To summarize: This is obviously quite an involved story, and even my attempt to give a short answer has turned into many paragraphs. So let me summarize the logic: Any observable such as a mass or charge contains an "intrinsic" contribution and a "dressing" due to the interactions with virtual particles. The exact split between these is arbitrary, but the combination is physically observable. If we take our current theories and extrapolate them to high energies, the contribution of the dressing is infinite. This is probably not right (our current theories are not guaranteed to work at energies above what we can probe experimentally). But, it is also irrelevant, because we can absorb these infinities into the definition of the "intrinsic" or "bare" parameters, so that the final combined result (which is all we can actually observe) is finite. If this sounds like a convoluted song and dance, well, you aren't wrong, but this scheme does lead to observable effects, such as the running of the fine structure constant (and other observables) with energy.

  • $\begingroup$ This is a mainstream "explanation". It implicitely uses a hypothesis that the theory construction is unique, whereas it is not! Those "absorbings" of harmful corrections by "bare" parameters is nothing but "doctoring" the calculation results. There may be theories with different initial approximations, with different interaction terms that give physical results directly. $\endgroup$ May 29 at 8:44
  • 1
    $\begingroup$ @VladimirKalitvianski (a) The corrections are not harmful, and counterterms are not doctoring the calculation, for all the reasons I said in my answer. (b) Sure, of course if we knew the correct theory and didn't need to start with a free + perturbative split, it's possible that the counterterms may not need to be infinite (I also said this in my answer). (c) Still any approach needs to account for the renormalization group. $\endgroup$
    – Andrew
    May 29 at 12:18
  • $\begingroup$ Also it's not 100% clear to me exactly what you are criticizing in my answer, but an important property of the "mainstream" viewpoint is that it can be used to successfully calculate observables that agree with experiment, such as the running of the fine structure constant. It sounds like you have some alternate scheme that disagrees with the standard one; can you point me to a reference where you calculate the running of the fine structure constant? $\endgroup$
    – Andrew
    May 29 at 13:00
  • $\begingroup$ I clearly stated the origin of UV difficulties - it is due to our mistake in constructing a theory. As you pointed out, it is "a very deep question"; that's why I proposed a simplified toy model in the link, not in the "short answer". If you read my papers, you will find answers to your questions posed in your comments. $\endgroup$ May 29 at 16:56
  • $\begingroup$ @VladimirKalitvianski Pretend I'm a busy person who doesn't have time to read every paper. Can you point me to a specific section where you derive the running of the fine structure constant? If not can you sketch out in detail how you could derive a prediction for the value of the fine structure constant at 90 GeV using your methods? $\endgroup$
    – Andrew
    May 29 at 22:43

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