When the Biort-Savart law was discovered, at the beginning of the XIX century, nobody knew what was a fermion or a boson concepts that need the spin and quantum mechanical context in general. That's to say, both concepts are not possible to relate and the Biot-Savart law was not applied to fermions too. The electron is a much older concept and also its spin.

Statistics of particles is relevant only when there is more than one particle. From this perspective, a single boson of charge e and mass m is electrodynamically the same as a single fermion of charge e and mass m. Further, in non-relativistic quantum mechanics the relationship between spin and statistics is one that is postulated; it does not arise from the fundamental requirements of the theory (this is in contrast to the relativistic quantum mechanics).

Dear Behnam,

I agree with your answer this time and perhaps we could add that the difference between boson and fermion is the Pauli exclusion principle. This concept is very important to enable stability of white dwarfs would be possible under so big gravitational pressure or the Fermi level in an electronic gas for condensed matter. But there are no relation at all with the Biot-Savart or Ampere's laws for the magnetic field produced by a density of current. Notice that even these laws do not define particles which produce the currents, that could be ions as in atomic plasma.

Thank you Daniel. In non-relativistic quantum mechanics, we elect to call a particle fermion or boson and for that rely on such strictly relativistic concept as spin (non-relativistic quantum mechanics is a patchwork of various ideas). Now, so long as there is no interaction between the particles, construction of femionic and bosonic many-particle states comes down to how we occupy the single-particle states -- for fermions, no two identical particles are assigned to a single-particle state and for bosons no such restriction applies. An assembly of non-interacting particles, whether fermions or bosons, can therefore be treated just like a set of isolated particles (and this applies as regards the Biot-Savart law), the only issue that is different from the case of having a single isolated particle is the availability of single-particle states to the particle in question. The situation changes dramatically as soon as we switch on some interaction, no matter how weak, between the particles. In such case, the physics can (and often does) dramatically change, with individual particles not having any sharp meaning any longer: if we have N interacting particles, there are only N-particle states to be considered. Naturally, when interaction is "weak" in some sense, mean-field theory may in an effective way take account of the inter-particle interaction, providing use with the simple picture of individual non-interacting particles, all interacting with some non-trivial effective potential. With reference to our earlier discussion on another post by Arbab, which you state not to agree with my views over ther, I believe that if you had truly carefully considered the issues that I raise there, you would have agreed with me. Key point in that discussion is that a superconducting state, even within the BCS framework, is a strongly-correlated state.

Dear Benham,

You mix many things that they are not wrong but they are far of the issue, from my humble point of view. Let me to try to answer in the most clear and simplest form:

1. Biot-Savart law has not anything do to with fermion or boson concepts. The Cooper currents are thought as a current of bosons but from the electromagnetic point of view it is only a number of particles with minus two electric charges. Thanks to Biot-Savart law it is possible to calculate the magnetic induction B and for Cooper current you obtain exactly the London equation which explains you the Meissner effect. B=0 in the inner part, i.e. at distance less than the Cooper length which of some angstroms.

2. The fact that the Cooper pairs are bosons allow them to be in the same energy level and therefore they can move without having losses. This is the explanation of the superconductivity.

3. Obviously for having attracting electrons, even if they are a relative long distance each other, it is necessary to explain how the Coulomb law seems to fail. This was the idea of Frölich that was employed in BCS. BUT ELECTRODYNAMICS is the usual.

4. It is true that the electrodynamics employed in superconductivity can be within the gauge theories that in fact we could have an internal symmetry larger than the usual U(1) and in fact we have a torus U(1)xU(1) which takes into account the electromagnetic gauge symmetry of the potentials and the quantum symmetries associated to the macroscopic phases of the Cooper currents.

5. Dirac equation (relativistic quantum fields) taking into account of the spin is not necessary at all in superconductivity and less in relation with the Biot-Savart law. That is to say, to employ a book based in QFT for superconductivity may be formally interesting but physically unnecessary from my humble point of view. But perhaps you could explain it and I would be very glad to learn it. Let me to tell you that I prefer to use the Mahan for working with Many-Body and QFT than your used book, but I understand that there are interesting mathematical chapters.

Excuse me Daniel, but I am beginning to feel uneasy about your ad hominem responses which attempt to rephrase my responses in forms not recognisable to me. For instance, you begin with "Biot-Savart law has not anything do to with fermion or boson concepts.". But actually, where did I say what you are attempting to ascribe to me and subsequently to negate?! Your next statement, formally addressed to me, is "The fact that the Cooper pairs are bosons ...", is actually contrary to what I have written in my earlier comment on a different page, the link to which I give below (note your "The fact"). There also you maintained to insist on certain issues with which I strongly disagree. Etc.

Several points are in order. Firstly, I have personally no problem, whatever, with Biot-Savart! Secondly, your first statement, which I have quoted above, is in a way contained in my first response on this very page. I wrote explicitly: "Statistics of particles is relevant only when there is more than one particle. From this perspective, a single boson of charge e and mass m is electrodynamically the same as a single fermion of charge e and mass m." Thirdly and lastly, I did not come to this page to be told how I should understand certain basic concepts (generally, I do not respond to any question on RG over which I do not consider myself sufficiently qualified); I was explicitly invited by Arbab to this very page to give my view on his question, and I wrote what I considered to be relevant to the question. Please in future present your views as you see fit, without involving me personally into your arguments (in principle I have no problem with being involved in arguments, provided that my views are presented accurately), and leave the rest to the judgement of individual readers as to what response is the most appropriate. Thank you.

https://www.researchgate.net/post/What_are_electric_and_magnetic_fields_produced_by_Coope-pairs_in_superconductors/1