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?