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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?

  1. 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.

  1. 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?

  2. 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.

  3. 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?

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  • $\begingroup$ The email notification I received yesterday of the suggested format edits showed a post that indeed had weird line breaks, as the notification said. When I left the post on the 14th, it was not nearly that way, with only a few slightly infelicitous breaks. What happened in the meantime I don't know. Anyway, thanks to those who suggested edits to fix this strange problem. $\endgroup$ Jan 16, 2021 at 16:56

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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 still 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 major problem. It is expected that some questions are hard to answer, such as why certain fundamental physics law is so and that 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 tweaks of the calculations, with additional concepts and steps (including renormalization), to avoid infinities hampering the calculations and to get finite results. Many got accustomed to this as acceptable too, because it brings some results, and there is no other good solution that would solve all those problems and allow us to get the results. However, some people do not think this is acceptable theory, because they suspect we do not have a consistent theory and the tweaks are just a crutch to get around that.

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 and getting rid of infinities, he suggested that we have wrong assumptions in the mathematical model (maybe we have the wrong Hamiltonian). Later Feynman questioned even the idea that the theory has to have an Hamiltonian and developed other formulations (Lagrangian, path integrals).

The present status quo is that those and other critics may very well be right in some of their criticism and some of their suggestions are interesting, but this criticism was not a great contribution to physics, because it 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, if still 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 novel, revealing way. Except for the renormalization group, this did not happen with the problem of infinities in QFT, so we are still using the regularization/renormalization procedures and the ideology of "it's just an effective theory, so it's reasonable to not worry about some infinities, let's not kid ourselves we should expect much more from this effective theory".

  1. 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 assumption that in a theory of point particles, the Poynting expression $\frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0}B^2$ can be interpreted in the standard way as density of EM energy (actually very dubious).

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 charged 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 free of self-interaction infinities.

But almost nobody cared, since in 20's quantum theory was the exciting new thing in physics, and a fix to the obscure marginal problem of the classical theory did not get attention. It also did not resolve immediately the other problems with the EM theory; such as the stability of atoms and molecules, the black body radiation, and other phenomena that were getting a quantum-theoretical explanation.

When quantum theory was first applied to EM field in 20s and 30s, infinite energies once again popped up and people got very worried, since they did not know how to solve them already in classical theory, and they did not know the Feynman/Schwinger/Tomonaga methods yet. Some of the infinities have the same origin as in the classical theory; infinite Poynting energy of point particles. Feynman and Wheeler tried to revive the Fokker/Tetrode/Frenkel ideas of no self-interaction and tried to come up with a quantum theory that would be free of these infinities just as Frenkel's classical theory was. But they only published papers on the classical variant of Frenkel's theory with some novel cosmological ideas (absorber theory), and never published how to apply these ideas to quantum theory.

So, one particular problem with infinities has been resolved in a beautiful way in classical theory, but nobody since then was able to apply this kind of thinking (no self-interaction, different expression for energy) to quantum theory. Maybe it can happen in the future, but it's also quite possible this won't solve all the consistency problems the theory has.

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  • $\begingroup$ Frenkel's model may be mathematically consistent, but it models a point charge orbiting an opposite charge of infinite mass as radiating without loss of kinetic energy and not spiralling inwards. Hence it comes across to me as a failed physical theory compared to models that have self interaction, treating CED as an effective field theory down to the Compton wavelength. $\endgroup$ May 4, 2023 at 23:14
  • $\begingroup$ It's a model showing how point charges can be consistent with EM theory, not a theory of all observed behaviour of electrons. However, I do not think loss of energy of electron (which afaik according to current knowledge still could be a point particle) in perfectly static field is an established fact. In real experiments with accelerating electrons, the source body does not have infinite mass. Thus it accelerates due to the electron field, and produces its own field acting on the electron, which makes it spiral down (this was calculated by Synge). $\endgroup$ May 5, 2023 at 0:39
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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.

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In my view, barring "physical metaphysics", infinities are just a symptom of the fact that the used mathematical model is wrong. Focusing attention of that symptom takes us away from the possible real causes.

There is a well-known way to stop computations before reaching the infinities in ultraviolet renormalization, but keeping all (ultraviolet) renormalization ambiguities (i.e. all the terms of finite renormalization). This is the Epstein-Glaser procedure. It is so powerful that its use extends to curved spacetime, where there is no translational invariance and one cannot use the momentum representation.

Here it is evident that the problem arises when, for instance, one attempts to multiply distributions. This mathematical procedure sometime makes sense but it is affected by ambiguities and these ambiguities are exactly the finite ultraviolet renormalization ambiguities.

This is a safe viewpoint, in my view, that illustrates the physical problem (without cumbersome metaphysical objectes as infinities, which only make more obscure a difficult issue): the use of these distributions to describe physics has something wrong, even if this description seems to be suggested by other apparently sound physical ideas (the spacetime is a continuum, interactions are described by local Lagrangians, and all that).

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