What are the sources of infinities in (unrenormalized) quantum field theory calculations? The PSE questions and answers about this question I've found don't answer it to my   satisfaction, so I am asking my own version, with the principal options I am aware of   listed. Although one comment about this subject that I saw somewhere, and didn't fully   understand, said that there were no infinities in QFT calculations, multiple sources say   that renormalization is needed, and is used, to remove, in a systematic but not   physically or mathematically theoretically justified way, the physically unrealistic   infinities in such calculations to somehow get finite (and sometimes very accurate) final   answers. Richard Feynman, one of the creators of such renormalization techniques,   said, in the Feynman Lectures on Physics, that the problem of, specifically, infinite   self-energy (in their electrostatic field) of charged particles hadn't been solved. P. A. M.   Dirac, the originator of relativistic quantum field theory, said that theories with only   renormalization ad hoc solutions for removing the infinities were unsatisfactory.
In this question I am not asking directly about the details of renormalization, just about   the sources of the infinities but thus, indirectly, about what the claimed justifications, or   at least motivations, for these renormalization procedures might be. (I do argue against   what I think is one of the incorrect proposed solutions to the infinite self-energy  problem.) Is there any consensus in the physics community about what at least some   of the infinities sources are? Regardless of whether there is, which of the following do   PSE readers think are among the causes of the infinities?

*

*The infinite EM field self-energies of point electrically charged particles, such as the   electron and the quarks.
I understand that this posed a problem for Lorentz even in classical, non-quantum EM theory, because, for example, the infinite electrostatic field self-energy of a point charged particle would have infinite mass, and so it was not clear how it could be  accelerated, as it would have to be, at least piecemeal, when the particle was accelerated. I don't know how this problem was resolved in classical EM theory, if it was resolved. I have read that this problem isn't solved, in both classical E&M theory and quantum electrodynamics, by assigning a non-zero radius to the particles, for example the so-called "classical" radius of an electron to it, which is the radius of a uniform sphere, or maybe shell, of charge equal to the electron charge which would have a field energy/mass equal to the (measured) electron energy/mass. This is because measurements have shown that the electron has a radius (can be localized to) much less than the classical radius, in fact no non-zero radius has yet been shown for it using available accelerator particle energies. Also, I don't know whether this problem is widely believed to be solved by the polarization of the vacuum, whereby the effective, measured field of an electron is less (in some places) than the field of a "bare" electron, due to the partial shielding provided by the virtual(?) positrons of the electron-positron pairs created from the vacuum, the virtual(?) electrons of which are farther from the real electron than are the virtual(?) positrons. An argument against this proposed solution is as follows:

All this is done assuming classical field energy calculations, except for the quantumpolarization of the vacuum.
It can easily be shown that the fields of a finite number of positrons cannot reduce the field of a point electron enough to make the resulting field energy finite (unless
one of the positrons is located exactly at the position of the electron). If there are an
infinite number of positrons created from the vacuum by the electron (and within
some common finite distance of the electron, as is necessary for them to reduce the
total electric field energy of the electron to a finite value), their total mass, and so
their combined gravitational field, would be infinite. This infinite gravitational field is
not observed. It can be argued in reply that they are virtual positrons, so their mass
and its gravitational field are not observable. However, if they reduce the effective
electrostatic field of the electron, their own electrostatic fields have this observable
effect. Is it coherent to consider them as having electric fields which have
observable effects but (infinite) gravitational fields whose effects are not
observable? Is this related to the problem of the 10^60 to 10^120 times too large
predicted mass of the vacuum? The predicted mass described, due to the infinite  number of positron-electron pairs created from the vacuum, would be infinitely too
large. Sabine Hossenfelder, in her 2018 book Lost in Math, says (p. 78, paperback
ed.) that the electron self-energy problem is cured by the existence of these virtual
electron-positron pairs, but Sabine is sometimes wrong. One of the virtues of string   theory is said to be that the field singularities of strings of charge are less
troublesome than the singularities of equal-charge point charges. However, they still
would have infinite self-energies. I don't know how string theory avoids this problem.


*Internal loops in the Feynman diagrams used to calculate values of physical
observables. These loops involve quantities which are not determined by the loops'   input and output quantities, and this causes divergence problems. Is this believed to be   a problem caused just by the perturbation approximation procedure involved, rather   than something more physical, such as that in 1 above?


*The fact that fields have an infinite number of variables- the field values at all the   points in space- and so an infinite number of variables that can be uncertain ( is "vary" here instead of "be uncertain" actually correct?) according to the Heisenberg   Uncertainty Principle, which can lead to infinite uncertainty, or infinite average   uncertainty (variation), and so infinite average values of the squares, and so energy, of   various field observables.


*The problems of "ultraviolet" divergences, involving arbitrarily small wavelengths of   particle wavefunctions, so arbitrarily high frequencies, so, in some theories, infinite   contributions to system energies by arbitrarily high (virtual?) particle energies; also   "infrared" divergences, involving arbitrarily long range interactions.
Some of the above may be related to, or even aspects of, others listed. Also, are there   any physical or mathematical effects other than these 4 listed above which some   physicists or PSE readers think contribute to the infinities?
 A: 
Is there any consensus in the physics community about what at least some of the infinities sources are?

There is a consensus in the broad sense, that the best theory we have is incomplete. This has two aspects.
We can't derive all the physics facts from mathematics alone. We have to adopt lots of assumptions, e.g. lots of various numbers that we can't explain (fine structure constant, masses of electron, muon, quarks, etc.). This is acceptable in physics, it is not a principial problem. It is expected that some questions are hard to answer, such as why certain physics law is law and we can't derive it from something simpler.
However, as you mention, we encounter also problems in calculations, where using the mathematics naively in the most direct way leads to infinite values of quantities we know or want to be finite. So people invented tweaking the calculations with additional concepts and steps to avoid those infinities in results. Many got accustomed to this as acceptable too, because it brings some results, and there is no other magic solution that solves all the problems. However, some people do not accept this because they suspect we do not have a consistent theory.
Dirac believed that some infinities are really bad, in the sense we can't be content with renormalization, even though it apparently sometimes brings interesting results. Motivated by desire for internal consistency, he suggested we really have the wrong assumptions in the mathematical model (maybe we have the wrong Hamiltonian). Feynman questioned even the idea that the theory has to have a Hamiltonian.
The present status quo is that they and other critics may very well be right in some of their criticism and some suggestions are interesting, but it was not a great contribution, because they didn't directly lead to "The solution".
One societal aspect of theoretical physics is that to change minds, it is not enough to show failings of the standard theory or even to fix one partial problematic aspect of it (with infinities, or some inconsistency) by changing the assumptions or using some new idea, when all the other similar problems are left unsolved. The benchmark of acceptance of radical new ideas is, traditionally, to solve most of the related problems in some revealing way. Except for the renormalization group, this did not happen with the problem of infinities in QFT, so we are stuck with the standard schemes and the ideology of "it's just an effective theory, so it's reasonable to ignore some infinities, let's not kid ourselves we should expect much more from this".


*

*The infinite EM field self-energies of point electrically charged particles, such as the electron and the quarks.
I understand that this posed a problem for Lorentz even in classical, non-quantum EM theory, because, for example, the infinite electrostatic field self-energy of a point charged particle would have infinite mass, and so it was not clear how it could be accelerated, as it would have to be, at least piecemeal, when the particle was accelerated.



I am not asking directly about the details of renormalization, just about the sources of the infinities

For example, the solution to the problem of infinite energy of point charges in classical theory was published by J. Frenkel in 1925 (also by others like Fokker and Tetrode before him, but I like Frenkel's paper).
You asked about the sources of the infinities. In this particular case, the source of infinity is the questionable assumption that in a theory of point particles, Poynting formula can be interpreted in the standard way as density of EM energy.
Frenkel's solution postulates point charged particles, rejects energy interpretation of the Poynting formulae and formulates its own law of local energy conservation where all energies are finite (unless two particles are at the same point of space). This solution is beautifully simple, and consistent with Maxwell's equations and the Lorentz force formula. No other assumption is needed to make the theory self-consistent.
But almost nobody cared, since quantum theory was the hype of the day, and the fix to the obscure marginal problem of the classical theory did not seem to provide much help in solving the other problems with the EM theory; such as the stability of atoms and molecules, the black body radiation, and other phenomena that later got a quantum theoretical explanation.
When quantum theory was first applied to EM theory in 20s and 30s, infinite energies once again popped up and people got very worried, since they were not used to renormalization tricks yet. Some of the infinities have the same origin as in the classical theory; infinite Poynting energy. Feynman and Wheeler tried to revive the Fokker/Tetrode/Frenkel ideas and come up with quantum theory that would be free of these infinities just as the Frenkel's classical theory was, but they only published papers on the classical variants with some cosmological ideas, and never published how to apply these ideas to  quantum theory.
So we see, one particular problem with infinities has been resolved, in a beautiful way, in classical theory, but nobody was able to apply this kind of solution to quantum field theory. Maybe it can happen in the future, but it's also quite possible this won't solve all the problems the theory has.
A: Infinities are due to the internal loops of Feynman diagrams. When one talks about Feynman diagram one implies that the perturbative method is used. Thus the infinities come from the perturbative method. Nowadays we have a better understanding of these infinities than Feynman and Dirac had because we have the normalization group which, roughly speaking, links the amplitudes at different energetic scales. At the first order, the infinities come from the fact that we are dealing with loops at a 0 energetic scale, which is used to describe the theory at the zeroth order (no quantum effects so energeic scale of 0). Thus the quantum effects (our loops) have to be infinite to exist, this is why they are indeed infinite. Now to supress these infinities one deals with renormalization, which is a procedure used to be at the «right energetic scale». So at the first order of the perturbation theory one has infinite renormalization coefficients, corresponding to the gap between the energetic scale at the zeroth order (0) and at the first order (depending on the renormalization scheme). Thus our coefficients are infinite and this is the «same infinite» as the one from the loops. Same resoning for the other order of the perturbation theory.
