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What is the modern understanding of why UV infinities arise in Quantum Field Theories? While nowadays there is clearly a consensus about how to remove them, I'm interested in the reason why they show up in the first place.

Below are a few thoughts of mine but please feel free to ignore them. And please ignore the so-far only answer by Lubos Motl.


So... UV divergences arise and thus we need to renormalize, because:

  1. We have infinite number of degrees of freedom ín a field theory. (From this perspective, the infinites seem inevitable.)
  2. We multiply fields to describe interactions, fields are distributions and the product of distributions is ill-defined.
  3. We neglect the detailed short-wavelength structure of scattering processes, and the infinites are a result of our approximations with delta potentials. (From this point of view, the UV divergences aren't something fundamental, but merely a result of our approximation method. )
  4. We are dealing with non-self-adjoint Hamiltonians. (This is closely related to the 3. bullet point. From this perspective an alternative to the "awkward" renormalization procedure would be the "method of self-adjoint extension".)

Are these reasons different sides of the same coin? And if yes, how can we understand the connection between them?


Edit: The "answer" by Luboš Motl is not an answer at all. He simply states: Yeah there are infiniteies, but you can get rid of them, so you shouldn't care about them. Basically "Stop trying to understand things; shut up and calculate". Everyone is free to take this attitude, but it certainly doesn't explain why we encounter infinities in the first place and hence we the renormalization procedure is needed. Because his answers gets a surprisingly large number of upvotes and no one else want's to write another answer, because Motl's answer appears as the "accepted answer", I feel the need to add some more thoughts here:

In contrast, a much more interesting perspective is mentioned by Batterman in "Emergence, Singularities, and Symmetry Breaking":

Many physicists and philosophers apparently believe that singularities appearing in our theories are indications of modeling failures. In particular, they hold that such blowups are signatures of some kind of unphysical assumption in the underlying model. Singularities are, on this view, information sinks. We cannot learn anything about the physical system until we rid the theory of such monstrosities. (This attitude is still quite prevalent in the literature on quantum field theory, a little bit of which will be discussed in the next section.) On the contrary, I’m suggesting that an important lesson from the renormalization group successes is that we rethink the use of models in physics. If we include mathematical features as essential parts of physical modeling then we will see that blowups or singularities are often sources of information. Mark Wilson has noted something akin to this in his discussion of determinism in classical physics. He notes that while from a modeling point of view we are inclined to object to the appearance of singularities . . . , from a mathematical point of view we often greatly value these same breakdowns; for as Riemann and Cauchy demonstrated long ago, the singularities of a problem commonly represent the precise features of the mathematical landscape we should seek in our efforts to understand how the qualitative mathematics of a set of equations unfolds. Insofar as the project of achieving mathe-matical understanding goes, singularities frequently prove our best friends, not our enemies [24, pp. 184–185].

and in a similar spirit here's a quote from Roman Jackiew:

“the divergences of quantum field theory must not be viewed as unmitigated defects; on the contrary they convey crucially important information about the physical situation, without which most of our theories would not be physically acceptable.”


Moreover, some background on the origin of my question: Some of these reasons are mentioned in the form of questions at page 6 in "An introduction to the nonperturbative renormalization group" by Bertrand Delamotte. However, after stating the questions he only mentions that

"Some of these questions will be answered directly in the following. For some others the reader will have to build his own answer. I hope that these notes will help him finding answers."

Unfortunately, I wasn't able to find conclusive answers in his paper.


I found a nice discussion of "The mystery of renormalization" in "Phase Transitions and Renormalization Group" by Zinn-Justin:

Though it was now obvious that QED was the correct theory to describe electromagnetic interactions, the renormalization procedure itself, allowing the extraction of finite results from initial infinite quantities, had remained a matter of some concern for theorists: the meaning of the renormalization ‘recipe’ and, thus, of the bare parameters remained obscure. Much effort was devoted to try to overcome this initial conceptual weakness of the theory. Several types of solutions were proposed:

(i) The problem came from the use of an unjustified perturbative expansion and a correct summation of the expansion in powers of α would solve it. Somewhat related, in spirit, was the development of the so-called axiomatic QFT, which tried to derive rigorous, non-perturbative, results from the general principles on which QFT was based.

(ii) The principle of QFT had to be modified: only renormalized perturbation theory was meaningful. The initial bare theory with its bare parameters had no physical meaning. This line of thought led to the BPHZ (Bogoliubov, Parasiuk, Hepp, Zimmerman) formalism and, finally, to the work of Epstein and Glaser, where the problem of divergences in position space (instead of momentum space) was reduced to a mathematical problem of a correct definition of singular products of distributions. The corresponding efforts much clarified the principles of perturbation theory, but disguised the problem of divergences in such a way that it seemed never having existed in the first place.

(iii) Finally, the cut-off had a physical meaning and was generated by additional interactions, non-describable by QFT. In the 1960s some physicists thought that strong interactions could play this role (the cut-off then being provided by the range of nuclear forces). Renormalizable theories could then be thought as theories some- what insensitive to this additional unknown short-distance structure, a property that obviously required some better understanding.

This latter point of view is in some sense close to our modern understanding, even though the cut-off is no longer provided by strong interactions.


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    $\begingroup$ 2 and (in rare cases) 4 are reasons for not being able to formulate a well-defined finite quantum-mechanical theory. 3 is the reason for the perturbative approach to give untractable infinities. 1 is not even true, any finite model of QFT in 2d spacetime would suffice as a counterexample. $\endgroup$ – Prof. Legolasov Jun 14 '17 at 7:25
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    $\begingroup$ @SolenodonParadoxus: nice to have mentioned that 4 can even plague quantum mechanical rather than QFT models. The recent PRL paper by Bender, Brody and Muller comes to mind as an example. However, I don't know what you mean by 2d QFT models being finite. They need to be renormalized like everbody else. It is just that these renormalizations are easy in 2d. $\endgroup$ – Abdelmalek Abdesselam Sep 21 '17 at 18:16
  • $\begingroup$ +1: But I don't understand your approach. For the first, second, and fourth proposed reasons to be compelling 1. Classical Electrodynamics in vacuum should have had infinities; 2. Non-relativistic QM should have had infinities; and 4. Hamiltonians would need to have been non-Hermitian in QFT (which they aren't). Also, the texts you quoted were all enjoyable reads but they seem to describe how people have reacted to renormalization historically--and have not discussed the physics as to why its need arises (which was your question). $\endgroup$ – Dvij Mankad May 12 at 18:10
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None of the four explanations is quite right.

  1. We know theories with finite observables but an infinite number of degrees of freedom
  2. the delta-functions and distributions appear already at the tree level but the divergences that need to be renormalized only appear in loops, so they have nothing to do with each other
  3. it is really equivalent to (2)
  4. Hamiltonians in QFTs are Hermitian

Instead, there are old-fashioned correct reasons to explain why renormalization is needed, and the modern explanation based on the renormalization group paradigm.

The old-fashioned need for renormalization is simple: when one calculates the loop corrections to Green's functions or scattering amplitudes, and assumes that the bare coupling constants in the Lagrangian are finite, then he gets infinite contributions from the integrals over very high (or very low, IR divergences) momenta of virtual particles. That means that the total result is infinite as well which can't match finite observations. So the solution is to assume that the bare constants aren't really finite, they have infinite terms in them as well, and the infinities cancel between those infinite parts of the coupling constants (or between counterterms) on one side and the loop divergences on the other.

The modern explanation why renormalization is needed is the renormalization group paradigm. It says that all quantum field theories we use are really "effective", i.e. good for answering questions about low-energy enough processes. It means that one must really clarify the theory and express the "typical energy scale" where we still want the theory to be valid. When this dependence on the scale is studied carefully, we find out that the dimensionless constants such as the fine-structure constant aren't really dimensionless, or aren't really constant. They run – depend on the energy – in a slower way and this running is the reason why the limit is singular if we want the theory to work at arbitrarily high energies or short distance scales.

I would recommend you to focus on the question how the quantities are actually calculated, and not just why they need infinite subtractions. Once you understand how a calculation actually proceeds, you will know about a more useful and important thing – how to calculate – and you will also understand why your guesses about the "causes of divergences and renormalization" differ from the real ones.

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    $\begingroup$ I agree with 1, but your argument against 2. is certainly not correct. See, for example, en.wikipedia.org/wiki/Causal_perturbation_theory. Also I don't agree with your comment about 3. It is not trivially the same as 2. (If you think otherwise, a comment why you think so would be great. ) The point of view stated in 3. comes from studies of quantum mechanical problems as non-relativistic limits of QFTs, see e.g. projecteuclid.org/euclid.cmp/1103901240 . This is a quite different and very pictorial perspective on renormalization, in contrast with the formal argument in 2. $\endgroup$ – jak Jun 14 '17 at 8:06
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    $\begingroup$ Yeah... as always, as soon as you notice that you're wrong you start insulting. QCD may be "UV complete" according to some definition (and not "UV complete" according to another sciencedirect.com/science/article/pii/0370269382911170 ). The point is that this has nothing to do with my argument, which is that the infinities arise because in QFT we model interactions as contact interactions, i.e. with delta potentials. $\endgroup$ – jak Jun 15 '17 at 6:17
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    $\begingroup$ so... according to your logic, no person here is allowed to disagree with an answer that is given to a question he has asked? If you would have spent only 5 minutes reading the question and thinking about it, you would notice that none of the bullet points in the OP is my idea. Again you don't give any arguments, but instead insult me and start patronizing. $\endgroup$ – jak Jun 19 '17 at 5:22
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    $\begingroup$ Your answer is basically "just ignore everything" $\endgroup$ – user41826 Sep 13 '17 at 5:53
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    $\begingroup$ No, my answer is "learn everything", and when you do, you will see that what is being done in QFT is right and unavoidable. And the need for renormalization etc. follows from deeper facts and ideas that simply can't be fully comprehended before you learn QFT properly. And there are different sources of infinities and new things and it's just wrong to assume that all of those new or divergent things are the same. $\endgroup$ – Luboš Motl Sep 13 '17 at 5:58
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The question and its answers seem to be controversial, even among experts:

The claim that the problem of UV divergences originates from the continuum assumption is in fact controversial, and physicists have come up with three main types of responses which I will respectively call the “continuum”, the “effective” and the “axiomatic” approach to the problem of UV divergences.

  • According to the continuum approach, the problem arises because we are not working with the correct type of QFT or because we have not appropriately parametrized the QFT at hand in the first place. The hope is that the continuum assumption holds for a specific class of QFTs and that all that needs to be doneis to sensibly fine-tune their parameters with the tools of renormalization theory.
  • According to the effective approach, the problem arises because the continuum assumption is false. The solution is to impose explicit restrictions on the domain of energy scales of QFTs and adjust the sensitivity to high energy phenomena with the tools of renormalization theory.
  • According to the axiomatic approach, the problem arises because the mathematical structure of the QFT framework is ill-defined in the first place. The solution is to develop a rigorous mathematical formulation of QFTs with explicitly stated axioms—so that, if anything goes wrong, we can at least clearly identify the origin of the problem. (In this context, the problem of UV divergences is usually associated with the fact that the product of distributions at a point is ill-defined.)

(source: Renormalization Scrutinized by Sebastien Rivat)

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