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Is it possible to explain the Ultraviolet Catastrophe as a manifestation of the Riemann-Lebesgue Lemma?

I don't fully understand any of both topics, but reading about the Ultraviolet Catastrophe on Wikipedia, it says that the problem is explaining the behavior of the black-body radiation on the higher frequencies (shorter wavelengths). since the Fourier Transforms also happen to be in the Frequency Domain, and the Riemann–Lebesgue lemma states that the Fourier Transform of Lebesgue Integrable functions falls to zero for higher frequencies, I am wondering if both phenomena are related.

Joako
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I do not think there is a particularly interesting relationship between the UV catastrophe and the Riemann-Lebesgue lemma.

The UV catastrophe is the observation that the spectral density of black body radiation diverges at high frequencies, when computed with classical physics. This is a problem, since the total power emitted by the black body is the integral over frequencies of the spectral density, which diverges. Quantum mechanics solves the UV catastrophe by introducing a "cost" to exciting high energy modes by proposing that a mode with frequency $\omega$ has a minimum energy excitation $\hbar \omega$, and the resulting spectral distribution goes to zero at infinite frequencies.

You can apply the Riemann-Lebesgue lemma to the correct, quantum-mechanical black body spectrum, known as the Planck distribution. Note that as physicists we would tend to call a move from the frequency to the time domain an inverse Fourier transform, but this is just a convention, and mathematically we can also think of this process as a Fourier transform, so we should be able to apply the Riemann-Lebesgue lemma.

The spectral density is an example of a power spectrum. By the Wiener-Khinchin theorem, the inverse Fourier transform of a power spectrum is an autocorrelation function. Then, the Riemann-Lebesgues lemma implies that since the Planck distribution is $L^1$ integrable (which it is since it gives a finite power), the autocorrelation function vanishes at infinity. I don't immediately see any particular physical significance to this statement. But, that would be the implication of the lemma applied to the black-body spectrum.

Perhaps you could try to use the converse of the Riemann-Lebesgues lemma to rephrase the original UV catastrophe in terms of the autocorrelation function instead of the spectral density. It's not obvious to me that this would work, though, since the Wiener-Kninchin theorem assumes the Fourier transform exists, which it may not if the spectral density diverges at infinity. Even if you could do this, I don't see why stating the UV catastrophe in terms of the autocorrelation function would give you any additional insight over the normal formulation in terms of the spectral density.

Andrew
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  • Thanks you for your detailed answer. Maybe I misunderstood what I read on Wiki, but since if Energy is considered to be absolute integrable in time, their frequency spectra will go to zero for higher frequencies, so I am trying to understand Why physicists arrives on first place to a law that goes against with something that is proved by math (the R-L lema), by assuming that energy will be distributed on frequencies diverging as they grow... it looks like a contradiction (surely I am mistaken, but I am trying to figure out Why). – Joako Mar 10 '22 at 20:06
  • @Joako The correct spectrum (the Planck distribution) does vanish at high frequencies. However, to get the correct spectrum, you need to account for quantum mechanics. The ultraviolet catastrophe (ie, a bad thing!) was the observation that the classical calculation led to a frequency spectrum that diverged at large frequencies. Everyone knew this was the wrong result (basically, for the reason you are saying, but you don't need to use language as fancy as you are), but before Planck proposed $E=\hbar \omega$ no one knew what was wrong with the calculation. – Andrew Mar 10 '22 at 21:34
  • For what I read, neither Max Planck like the idea of the quantization and said that was like a desperate alternative to fit results (A. Einstein dislike it, even nowadays R. Penrose said one of the biggest issues in physics is give an intuition to QM), and if you think on the R-L lemma on a "continuum" (instead of quantum), no distribution should follow a divergence path on frequencies (actually for a Fourier transform is strange don´t having its maximum value at the DC component), that is why I found this weird, not the Law of the B-Body radiation, but Why the made a model that diverges. – Joako Mar 11 '22 at 02:37
  • @Joako It was obvious to Rayleigh and Jeans that their spectrum (which exhibited the ultraviolet catastrophe) could not be correct. They did not make the spectrum up; it was the result of a rigorous calculation starting from classical electrodynamics. The problem was that no one could understand why this calculation would give a wrong answer; as far as anyone knew, the assumptions and steps in the calculation were correct. Planck resolved this issue by introducing his quantum hypothesis, which allowed him to calculate the correct spectrum that does not diverge. – Andrew Mar 11 '22 at 03:41
  • On a different note, you mentioned in your question that you don't fully understand the R-L lemma or the UV catastrophe. This is of course fine. But I'd like to suggest that based on that, you use this as an opportunity to learn. In that spirit: it does not matter what personal feelings Planck, Einstein, or Penrose have about quantum mechanics, this is not a scientific argument. QM is foundational to all of modern physics; every day its predictions are confirmed. Maybe there is something deeper, but that won't replace QM's many, many successes, including the Planck distribution. – Andrew Mar 11 '22 at 03:46
  • You are totally right, by QM was "built to fit", an as any other model has limitations (that is why the are also another theories trying to explain the same things), being true it explanatory power is awesome, is still a model, and not an "absolute true". As example, think of this: every solution in QM is given by Power Series or Perturbation Theory (in non-linear cases), which can only stand analytical solutions, which are necessarily non-local in space and time, at best decaying at infinity.... – Joako Mar 12 '22 at 00:07
  • Photons and electrons due are existing since the very beginning, But how is possible to explain then phenomena that have finite duration? Why the same laws will applicable since are much more restricted? I have another question related to solutions of finite-duration to ODEs here, which recently I found they exists... surely QM don´t take them into account, as example, a finite-duration solution will have an unlimited bandwidth described with an Analytical Fourier transform. – Joako Mar 12 '22 at 00:13
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One should not confuse physics (description of nature) and the math used in physics:
The ultraviolet catastrophe has to do with the divergence of an integral of a function. But the problem is the physical theory producing such a function, not the mathematical divergence itself (which is a mathematically trivial matter in this case).

Roger V.
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  • Thanks for answering. For what I read in Wikipedia, I saw there is a reason why it is supposed to diverge for higher frequencies (and actually by trying to figure out why it doesn't happen, Max Planck found the distribution that finally leads to quantization). But is exactly here Why I am making the question: if energy in time is considered an absolute integrable function, it frequency spectra will never rise to infinity because of the Riemann-Lebesgue Lema, so I am trying to understand Why the found a law in first-place, that violates something that is proven by math it don´t going to happen. – Joako Mar 10 '22 at 20:00
  • @Joako > If energy in time is considered an absolute integrable function -- Where is this assumption coming from? Electric field of equilibrium radiation has constant expected average square for all times and integral of its absolute value over all infinite time can be infinite. – Ján Lalinský Apr 19 '22 at 18:17
  • @Joako one can't use regular Fourier integral to express equilibrium radiation for all times, one must use either Fourier series or generalized Fourier integral with singular frequency distribution. – Ján Lalinský Apr 19 '22 at 18:20
  • @JánLalinský I am using the assumption of absolute integrable energy since I want its description done with an absolute continuous function, so its differential equation could be an ODE or PDE without having things like stochastic SDE or pathological solutions as Weierstrass functions... besides this, as personal opinion, knowing are widely used, modeling phenomena with never-ending solutions just don't seen realistic to me, right now I have a bounty looking for an example of a phenomena described with a solution of finite duration here – Joako Apr 19 '22 at 20:20
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    @Joako but beware those preferences are mathematical restrictions that may limit applicability of implied results to physics questions such as properties of equilibrium radiation, which is not commonly described nor seems describable as L1 function or finite support function. – Ján Lalinský Apr 19 '22 at 21:20
  • @JánLalinský Thanks for your interesting comments. Part of what I am trying to understand is why, conversely, is always used to model physics Lipschitz differential equation when they never have a finite ending time (at least for autonomous ODEs). Probably I am wrong given the tremendous success of EM theory, but also is kind of weird that nobody sees uncomfortable with the fact that its solutions exits for all time, when intuition tells it could mess with causality, so I was thinking if studying more restricted scenarios could lead to a description with a more intuitive explanation. – Joako Apr 20 '22 at 00:42
  • @JánLalinský Actually I started here asking for any example of solution to the electromagnetic wave equation which were compact-supported in the time variable, and looks like there are not known answers: now, I understand there is none, since the wave equation is Lipschitz, so It shouldn't stand finite duration solutions - but I am not 100% sure since the papers were about autonomous ODEs, and it don't directly extend to PDEs, even so, is there the issue of modeling time as R^n (spacetime) or R^{n+1} parametrization over space variables. – Joako Apr 20 '22 at 00:47
  • @ So far, through other questions, assuming the system is continuous (so there is no teleportation - jump discontinuities), and assuming that have an starting and an ending time, will imply the system is bounded (so no singularities), then if is absolute continuous (so described by classic diff. equations at least class $C^1{(t_0,,t_f)}$), it will be absolute integrable (I think, I found it on a question in MSE), then since is absolute integrable and bounded, it will be necessarily of finite energy. So I am thinking in things that are continuous with a beginning and an end, nothing exotic. – Joako Apr 20 '22 at 00:57
  • @JánLalinský And if it has a starting and an ending time, I have for sure that its Fourier Transform is Analytics fulfilling the Kramer-Kronig relations, so the process is causal... so having a compact-support in time indeed made a lot of restrictions to the functions (I didn't know, neither expecting to happen before I start asking on MSE), so I have the idea that maybe the same restriction was the responsible for making the spectra to fall at high frequencies due the Riemann-Lebesgue Lemma - that is the spirit of my question (obviously not meaning is right, is what I am trying to understand) – Joako Apr 20 '22 at 01:10