I Is Maxwell's electromagnetism an effective (field) theory?

[Husserliana97]

Is it possible, and fruitful, to use certain conceptual and technical tools from effective field theory (coarse-graining/integrating-out, power-counting, matching, RG) to think about the relationship between the fundamental (quantum) and the emergent (classical), both to account for the quasi-autonomy of the classical level and to quantify residual quantum corrections?

By “emergent,” I mean the following: after integrating out fast/irrelevant quantum degrees of freedom (high-energy modes, environment, etc.), one obtains an effective action for the slow or collective variables. The stationary condition of this effective action then yields the familiar classical laws — Maxwell equations for the averaged electromagnetic field, or Newtonian trajectories for macroscopic bodies — as dominant terms, while all residual quantum effects appear as small, systematically organized corrections (Euler–Heisenberg terms for photons, friction/noise terms for macroscopic bodies in a bath).

My provisional thesis is that, even if Maxwell and Newton are not EFTs in the strict textbook Wilsonian sense, adopting this broader, methodological perspective is very useful. It allows us to:

*Formally show why classical laws dominate in their domain.

*Quantify residual quantum effects in a controlled, hierarchical way.

*Specify the range of validity of classical approximations.

Decoherence alone explains qualitatively (and quantitatively) why we do not observe macroscopic superpositions, but it does not provide a systematic framework to calculate corrections, estimate their size, or justify the quasi-autonomy of the classical level. Wilsonian tools — integrating out, coarse-graining, power-counting, RG — supply exactly this.

Do you think this broader, methodological use of EFT ideas to describe the quantum → classical transition is valid and useful, or does it introduce conceptual confusion?

 

[Dale]

I see where you are coming from, but I think that usually EFT's are considered in the context of renormalization groups etc. I don't believe that those would be fruitful in EM, at least I haven't worked on an EM problem where I wanted to use such techniques. Maybe they would be useful for point charges in some way, but I am not aware of that.

 

[Husserliana97]

Hello, and thank you for your reply.
Indeed, I haven't seen much on the application of Wilson's tools in electromagnetism. I have already come across texts saying that Maxwell's and Newton's theories are EFTs ‘for the classical world’ (I am attaching an example defending this), but nothing that makes use of RG, etc.

Perhaps we need to distinguish between two uses of the concept of EFT, or even ‘effective theory’. The first, very broad, is that of Howard Georgi in his 1993 article (from which I quote the first page):
" One of the most astonishing things about the world in which we live is that there seems to be interesting physics at all scales . Whenever we look in a previously unexplored regime of distance, time, or energy, we find new physical phenomena. From the age of universe, about 1018 sec, to the lifetime of a W or Z, a few times 10-25 sec, in almost every regime we can identify physical phenomena worthy of study. To do physics amid this remarkable richness, it is convenient to be able to isolate a set of phenomena from all the rest, so that we can describe it without having to understand everything. Fortunately, this is often possible. We can divide the parameter space of the world into different regions, in each of which there is a different appropriate description of the important physics. Such an appropriate description of the important physics is an "effective theory."
The two key words here are appropriate and important. The word important is key because the physical processes that are relevant differ from one place in parameter space to another. The word appropriate is key because there is no single description of physics that is useful everywhere in parameter space.

The common idea is this: if there are parameters that are very large or very small compared to the physical quantities (with the same dimension) we are interested in, we may get a simpler approximate description of the physics by setting the small parameters to zero and the large parameters to infinity . Then the finite effects of the parameters can be included as small perturbations about this simple approximate starting point.

This is an old trick, without which much of our current understanding of physics would have been impossible. We use it without thinking about it. For example, we still teach Newtonian mechanics as a separate discipline, not as the limit of relativistic mechanics for small velocities. In the (familiar) region of parameter space in which all velocities are much smaller than the speed of light, we can ignore relativity altogether. It is not that there is anything wrong with treating mechanics in a fully relativistic fashion. It is simply easier not to include relativity if you don't have to. This simple example is typical.

It is not necessary to use an effective theory, if you think that you know the full theory of everything. You can always compute anything in the full theory if you are sufficiently clever. It is, however, very convenient to use the effective theory. It makes calculations easier, because you are forced to concentrate on the important physics. In the particle physics application of effective theories, the relevant parameter is distance scale. In the extreme relativistic and quantum mechanical limit of interest in particle physics, this is the only relevant parameter. The strategy is to take any features of the physics that are small compared to the distance scale of interest and shrink them down to zero size. This gives a useful and simple picture of the important physics. The finite size effects that you have ignored are small and can be included as perturbations.

Again, this process is very familiar. We use it, for example, in the multipole t!xpansion in electrodynamics , or in replacing a physical dielectric with a uniform one. However, in a relativistic, quantum mechanical theory, in which particles are created and destroyed, the construction of an effective theory (now an effective quantum field theory-EQFT) is particularly interesting and useful."

Understood in this sense, I believe we can all agree that Maxwell's electromagnetism is an effective theory with respect to QED ! It is possible and useful to treat Maxwell as an effective description valid in a certain parameter range (low energies/weak fields): we start with the simple description (Maxwell), set the small parameter to zero (E/m_e≪ 1,) and then incorporate the finite effects (vacuum polarisation, Euler–Heisenberg non-linearities) as systematic corrections.


But I believe we can go even further, towards an even more rigorous interpretation of EFT, which makes direct use of Wilson's tools (even if, once again, I can find nothing in the literature). Because these tools make it possible to transform intuition (‘Maxwell dominates, QED corrects’) into a rigorous construction: defining a scale cutoff, integrating modes/heavy particles, organising corrections by dimension/operator, calculating coefficients numerically (matching), and tracking scale dependence (RG). And this is not a purely formal exercise, I believe, since it allows us to highlight an explicit effective action 𝑆_eff[𝐴] = Maxwell + Euler–Heisenberg + 𝑂(1/𝑚_𝑒⁸), effective equations, error estimates, and predictions (Euler–Heisenberg, vacuum polarisation, charge renormalisation, etc.), which also measure the imprint of the fundamental level on the emergent, etc.