In one of my papers, I debunked the myth that the Lamb shift would require a new theory: classical quantum mechanics (think of Richard Feynman’s treatment of the interaction between the magnetic moments of the electron and the proton in a hydrogen atom, for example) will do.
I will, therefore, not waste too much time and space on this, and refer you to the paper I did on this. I will just copy the introduction to it to wet your appetite. 🙂
According to the Wikipedia article on the Lamb shift (which is only useful because it parrots the rather fantastical mainstream explanation of this tiny split in spectral lines), we should think of the Lamb shift as a small difference (in energy) between the 2S1/2 and 2P1/2 orbitals in a hydrogen atom which can only be explained in terms of quantum field theory and which, therefore, basically confirms this theory.
How small? It is usually expressed as a fraction of α5·mec2 = α5·Ee but in absolute terms, it’s equal to about 4.372 millionths of an eV. Hence, the order of magnitude (in eV energy units) is 10-6. So that is very small, indeed. The Planck-Einstein relation tells us that corresponds to a frequency of a bit more than 1000 megaherz. That is much below the frequency of visible light (430 to 770 teraherz) but still corresponds to an easily detectable radiowave frequency.
Can we compare it with anything? We can. The order of magnitude is about the same as the energy difference between the spectral lines that are separated by the spin of an electron (the splitting that is referred to as the Zeeman effect) or the fine structure of the hydrogen spectrum. To be precise, it is one order of magnitude smaller: the Lamb shift is measured in terms of 10-6 eV, while the Zeeman effect or the energy difference between the fine lines of the spectrum is about 10 times larger. The order of magnitude of the Lamb shift is, in fact, the same as that of the hyperfine structure, which is associated with the 1420 MHz radio hiss coming from outer space. This radiation comes from spin flips of the proton and the electron inside of the hydrogen atom.
We are all very familiar with proton spin flips nowadays because of their use in magnetic resonance imagining. Indeed, I must assume that – if you have ever had one – you also googled and watched one or more YouTube videos explaining the physics underpinning this amazing technology. If not, I warmly recommend you do so because they are often better than Wikipedia explanations.
The point is this: for the novice in physics, these mainstream explanations in terms of quantum field theory comes across as somewhat bizarre. We explain the Zeeman effect and the fine and hyperfine structure of the hydrogen spectrum in terms of the orbital and spin angular moment of the electron and – in case of the hyperfine structure – of the proton (or, more generally speaking, the nucleus of the atom that we are looking at). For the Lamb shift, however, we are fed a very different story line. It goes like this:
Dirac’s equation – for a bound electron – does not predict this tiny energy difference. Dirac’s equation must, therefore, be totally wrong. We can only explain this in terms of “interaction between vacuum energy fluctuations.”
Let me quote Wikipedia in full here:
“This particular difference is a one-loop effect of quantum electrodynamics, and can be interpreted as the influence of virtual photons that have been emitted and re-absorbed by the atom. In quantum electrodynamics the electromagnetic field is quantized and, like the harmonic oscillator in quantum mechanics, its lowest state is not zero. Thus, there exist small zero-point oscillations that cause the electron to execute rapid oscillatory motions.”
I will let you digest this for a second. […]
It sounds fantastic, doesn’t it? Willis Eugene Lamb Jr. got a Nobel Prize in Physics for his discovery in 1955. He had to share it with Polykarp Kusch. To be precise, Lamb got his half of the prize “for his discoveries concerning the fine structure of the hydrogen spectrum” (the Lamb shift), while Polykarp Kusch got it “for his precision determination of the magnetic moment of the electron” (the so-called anomaly in the magnetic moment).
You should note that Lamb did not get it for the above-mentioned explanation which, judging from some of the later publications of Lamb, he found rather fantastical as well. Likewise, Kusch had measured the anomaly but left the explaining of it to (other) physicists¾some more famous names you probably are more acquainted with, such as Julian Schwinger and Richard Feynman. The latter, together with Sin-Itiro Tomonaga, effectively got the Nobel Prize – almost 20 years after Lamb’s discovery and Bethe’s first work on it – for explaining these seemingly strange measurements using even stranger theories (renormalization and quantum field theories).
[As mentioned above, this is just the introduction. The paper itself shows why and how we can interpret the nature of the Lamb shift as being exactly the same as that of the hyperfine structure.]
 The Hyperphysics site (http://hyperphysics.phy-astr.gsu.edu/) gives an energy difference of 4.5´10-5 eV between the two (fine) lines for the 2P level. The same site also calculates an energy difference between lines split by the Zeeman effect equal to 5.79´10-5 eV. These calculations are quite tricky because they depend on the strength of the magnetic field that is being applied to create these splits. The 5.79´10-5 eV energy difference, for example, is calculated for a magnetic field of 1 T (tesla). We mentioned the α5mec2 unit: it’s equal to about 1´10-5 eV so its order of magnitude effectively seems to connect all of the mentioned finer divisions of the hydrogen spectrum.
 The reader should not confuse this with the cosmic microwave background radiation, which is understood to be a remnant from the Big Bang. The 21 cm line was discovered in the 1930s, and was confirmed to be a hydrogen spectrum line in 1951. In contrast, cosmic background radiation was accurately measured in the 1950s and 1960s only and – as mentioned – it is associated with a temperature (about 2.725 degrees Kelvin, to be precise). It is, therefore, not related to any specific spectral line. For all practical purposes, one might say the cosmic background radiation reflects the temperature of the Universe, which is close to but above zero.
 Dirac first developed a wave equation for a free particle (Principles of Quantum Mechanics, section 30), which is a particle free of any forces. Section 39 of the Principles then further use this theory to deal with what Dirac refers to as the electron’s ‘motion in a central field of force’, based on which he then develops a wave equation that gives us the energy levels of the hydrogen atom (section 40). This is, basically, a modified version of Schrödinger’s wave equation for the hydrogen atom. We note that Dirac consistently describes his equations as the ‘equations of motion’ of the electric charge. We also think all of physics can and should be expressed in terms of the equations of the motion of charges. We, therefore, like the conclusion of his Principles very much: “It is to be hoped that with increasing knowledge a way will eventually be found for adapting the high-energy theories into a scheme based on equations of motion, and so unifying them with those of low-energy physics.” We could not agree more.
 See: https://www.nobelprize.org/prizes/physics/1955/summary/, accessed on 29 March 2020. We write the ‘so-called’ anomaly because we do not think of the anomaly as an anomaly. We think it is a perfectly normal deviation from some theoretical value. Indeed, one would always expect the measurement to be slightly different from its theoretical value. Why? Because a theoretical value is always based on mathematical idealizations that do not really exist. In this particular case, we should just acknowledge that zero-dimensional charges do not really exist: they must have some (spatial) dimension. Once one accepts that hypothesis, there is no longer any mystery in quantum mechanics. Moreover, because the deviation is systematic, one should learn from it so as to detail the model—which is exactly what we have been trying to do.
 See: W.E. Lamb, Anti-photon, Applied Physics B volume 60, pp. 77–84 (1995). We offer some comments on this remarkable paper – which Lamb wrote when he was over 80 years old – at the end of our paper.
 Unlike what you might think, Tomonaga was not working with Schwinger, Feynman or any of the other American scientists. He apparently discovered the renormalization method independently of Julian Schwinger and calculated physical quantities such as the Lamb shift at the same time. See: https://en.wikipedia.org/wiki/Shin%27ichir%C5%8D_Tomonaga
 The Lamb shift was measured in the Columbia Radiation Laboratory in 1947. From W.E. Lamb’s Nobel Prize Lecture, I gather the heavy lifting was actually done by one of his graduate students, Robert Curtis Retherford, whom, sadly, is only mentioned once in Lamb’s Nobel Prize lecture, and who did not share in the Nobel Prize.
 For a good overview of the rather ‘dirty work’ that Bethe seems to have done, see: Oliver Consa, Something is rotten in the state of QED (https://www.researchgate.net/publication/338980602_Something_is_rotten_in_the_state_of_QED). As for the theorists getting the prize only 20 years after the experimental discovery, it should be noted this is not unusual: the Nobel Prize Committee has tended to favor new experimental results above new theories. An exception was probably made in regard to the Higgs hypothesis – as theorists received their prize (for theoretical work that was actually done in the 1960s) almost immediately after the CERN ‘discovery’ of the Higgs field—or Higgs particle or whatever it was they claim to have measured.