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According to Wikipedia,

classical electromagnetism is scale-invariant.

I understand what it means mathematically as explained in Wikipedia. But what does it really imply physically?

Next, here it says that

The QFT describing the interactions of photons and charged particles is quantum electrodynamics (QED), and this theory is not scale-invariant.

What does that mean, physically?

Wikipedia explains what it means mathematically but does not explain what is it really saying, physically. Can anyone give a clear idea of what do these symmetries imply?

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    $\begingroup$ It means that scale invariant electromagnetic field (photon) interacts with a non-scale invariant field (e.g., electron), such that the whole theory becomes non-scale invariant. $\endgroup$
    – user142288
    Jul 3, 2021 at 4:46
  • $\begingroup$ Classical EM is scale-invariant. My first question is what does it imply physically. QED is not scale-invariant. My second question is what does that imply physically. $\endgroup$ Jul 3, 2021 at 7:09
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/258298 $\endgroup$ Jul 3, 2021 at 16:45
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    $\begingroup$ If the electron were massless (and therefore classically scale invariant), QED would still generate a scale at the quantum level. $\endgroup$ Jul 3, 2021 at 20:13

2 Answers 2

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Scale invariance of a theory means that if we notice one phenomenon on one scale (like an electromagnetic wave with the wavelength 400 nm), we can expect the similar phenomena to be possible at any other scale: 400 m, 400 km, etc. The magnetic field of a small magnet has exactly the same shape as the magnetic field of a larger magnet of the same shape, just rescaled. It should only be remembered that all relevant qualities are appropraitely rescaled. EM is rescaled under the transformation $x \mapsto b x$, $t\mapsto b t$, which means that when you rescale spacial dimensions, you also need to rescale time appropriately. That means that that if you consider a wave that is $b$ times shorter in space, it will also have $b$ times shorter period of oscilations in time, etc. So you not only know about the existence of waves of other length, but you can also predict their properties.

The lack of scale invariance means that observing one phenomenon does not guarantee the existance of similar phenomena at other scales. For example, all atoms have similar size (which can be calculated from the mass and charge of the electron, and there are no meter- or kilometer-big atoms. Another phenomenon is connected to the annihilation of elctrons with antielectrons: the photons created in the process have fixed energies (and wavelenngths). There are also more subtle effects, for example, through the creation of virtual electron-antielectron pairs, photons are able to interact with each other, and the strength of this interaction depends on their energy/wavelength.

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  • $\begingroup$ Thanks. So scale invariance of EM implies that if there is an EM wave of wavelength $\lambda$, there is also an EM wave of wavelength $b\lambda$ where $b$ is an arbitrary real constant. Is this right? Can you say slightly clearly what does the breakdown of scale invariance in a quantum theory of photons and electrons means and how do we make sense of that? $\endgroup$ Jul 3, 2021 at 13:30
  • $\begingroup$ @mithusengupta123 you can also say that wave $\lambda$ should interact with atom in the same way as wave $b\lambda$. But we know that is not true. However interaction with point charge has the same form $\endgroup$ Jul 3, 2021 at 14:19
  • $\begingroup$ @mithusengupta123 Added more explanations to the answer. $\endgroup$ Jul 3, 2021 at 19:19
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That a theory is scale invariant roughly means that there is no singled-out length scale, that the theory acts the same at all length scales.

Take for example classical electromagnetism. One of the well-known behaviours is the existence of plane waves solutions in which the electric field is of the form $$E(t,x) \sim E_0\mathrm{Re}~e^{i(kx -\omega t)}$$ and there is an associated magnetic field. The wavenumber $k$ is related to the wavelength $\lambda$ of the wave via $$k = \frac{2\pi}{\lambda}.$$ The wavelength is a characteristic lenghtscale for this kind of phenomenon. But in classical EM, there are plane waves solutions for all values of $\lambda$. All plane waves act the same, regardless of how long or short their wavelength. In this sense, classical EM is scale invariant.

Now in QED, things are different. The energy $\epsilon$ of a photon is related to its wavelength via: $$\epsilon = \hbar\omega = h f = \frac{hc}{\lambda}.$$ If the wavelength is short enough so that $$\epsilon>2m_ec^2,$$ where $m_e$ denotes the mass of the electron, this photon has enough energy to create an electron positron pair. If it collides with another photon with arbitrarily low energy, there is a chance that an electron and a positron will come out of the collision. So in QED, photons of different wavelengths do different things. In this sense, the theory is not scale-invariant.

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