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I was reading about the Casimir effect in an optical cavity and I came across the following paper by Casimir and Polder:

https://doi.org/10.1103/PhysRev.73.360

Which, if I am not wrong, states that the Casimir effect most textbooks describe in terms of the vacuum energy of the quantized electromagnetic field can be equally well described taking into accounts the electrons of the metal plates of a cavity and letting them interact using the London-van der Waals formulae. I wanted to ask if the previous statement is actually true or if I misinterpreted what I read.

Moreover, if the Casimir effect can actually be described entirely in terms of the London-van der Waals interactions I wanted to ask whether polariton formation inside cavities can also be described without using the field description but instead using solely a microscopic point of view.

Edit

I am thinking about this with the second part of the question:

If I was to insert all the interaction between electrons, protons, and neutrons from the plates in the Hamiltonian describing the cavity but I did not use any electromagnetic field that is not directly caused by the charges on the surfaces (so Coulomb or the interaction between currents, etc are allowed) would I be able to describe the situation where I have 1 one photon inside the cavity? I would guess not but I am not sure about this.

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    $\begingroup$ related: physics.stackexchange.com/questions/11544/… $\endgroup$
    – Mauricio
    Jul 2 at 19:00
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    $\begingroup$ The first part of your question might be partly addressed in the answers to Why do we not observe a greater Casimir force than we do? Regarding the second part of your question: When you say "without using the field description but instead using solely a microscopic point of view," do you mean something like "without using idealized boundary conditions on the EM field but instead using EM interactions between physical surfaces made of molecules"? $\endgroup$ Jul 2 at 20:18
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    $\begingroup$ Yes @Chiral Anomaly I exactly meant "without using idealized boundary conditions on the EM field but instead using EM interactions between physical surfaces made of molecules". Also, thank you for the link $\endgroup$
    – Yepman
    Jul 2 at 20:32
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In principle, one can use any of the pictures described in the OP. One could even start from the standard model and try to take it from there... in principle.

Practically, already the most simplified models are infeasible to solve. For something like cavity QED, starting from basic QED including all the atom interactions in the mirror is way too hard. Luckily, it is also unnecessary, since we can construct effective theories capturing the macroscopic effect of the cavity.

A nice outline of the hierarchy of approximation which leads to what is typically employed in cavity QED is given in this review (https://arxiv.org/abs/0902.3586) on macroscopic QED. It essentially discusses how the microscopic laws or non-relativistic microscopic models can be expressed in terms of macroscopic Maxwell equations an their associated field theory, where the mirrors etc. simply enter via the material response functions such as the refractive index.

This argument in principle applies to dispersion forces in cavities (Casimir, Casimir-Polder, van-der-Waals...) and also to quantum optical effects such as polariton formation. Note, however, that the effective theories do include assumptions and can break down in certain regimes.

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  • $\begingroup$ Thank you very much for your answer, it is exactly what I was looking for!! Thank you in particular for the arxiv reference. $\endgroup$
    – Yepman
    Jul 3 at 15:00
  • $\begingroup$ @Yepman Thanks! Glad it was helpful, let me know if you have any follow-up questions! $\endgroup$ Jul 4 at 11:35

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