General relativity - covariant superconductivity, Meissner effect

[Maniac_XOX]

I am doing a project where the final scope is to find an extra operator to include in the proca lagrangian. When finding the new version of this lagrangian i'll be able to use the Euler-Lagrange equation to find the laws of motion for a photon accounting for that particular extra operator. I have no idea how to approach this though. I assume it is a similar process to finding the mass operator which turned the lagrangian from maxwell's to proca's, but how is this done? Any help is appreciated :)

As for now i have only derived the laws of motions starting from a) Maxwell's lagrangian and b) Proca lagrangian. Step c) would comprise finding a lagrangian with an extra operator to repeat the process.
Any links to useful external sources is appreciated as well, thank you.

 

[vanhees71]

I'm not sure, whether I understand your question right, but if it is about superconductivity, you are (nearly) on the right track, because from an effective-field theory point of view in a superconducting material the electromagnetic gauge symmetry is "Higgsed", i.e., the photon get massive via the Anderson-Higgs mechanism, which is mostly known in the context of the electroweak sector of the Standard Model, but in fact the Higgs mechanism was first discovered by Anderson in the context of superconductivity.

For a relativistic formulation of superconductivity, see

https://www.springer.com/de/book/9783319079462

 

[Maniac_XOX]

I'm not sure, whether I understand your question right, but if it is about superconductivity, you are (nearly) on the right track, because from an effective-field theory point of view in a superconducting material the electromagnetic gauge symmetry is "Higgsed", i.e., the photon get massive via the Anderson-Higgs mechanism, which is mostly known in the context of the electroweak sector of the Standard Model, but in fact the Higgs mechanism was first discovered by Anderson in the context of superconductivity.

For a relativistic formulation of superconductivity, see

https://www.springer.com/de/book/9783319079462

Basically, I've been made to practice with covariant classical field theory to derive the laws of motion from a maxwell's lagrangian using the euler-lagrange equation, and then from proca lagrangian accounting for the mass term. The scope now is to find an extra operator which modifies the lagrangian to a new form and having to find the laws of motion for that. Where do I start? I was thinking if I know how to derive the mass operator that modifed the maxwell lagrangian into the proca lagrangian then i would have the basis set. I suppose in fewer words the question is: how do i modify the proca lagrangian to account for an extra term?

 

[vanhees71]

What do you want to achieve? If it's superconductivity, you need to "Higgs" the em. local gauge invariance. Another very good reference is by Weinberg:

https://inspirehep.net/literature/18067

The KEK preprint is freely accessible.

 

[Demystifier]

As for now i have only derived the laws of motions starting from a) Maxwell's lagrangian and b) Proca lagrangian. Step c) would comprise finding a lagrangian with an extra operator to repeat the process.

First, the Proca Lagrangian already contains an extra term in comparison to the Maxwell Lagrangian. It seems that you want an additional extra term, but it's not clear what is its purpose.
Second, it's not clear what superconductivity has to do with all this.

 

[Maniac_XOX]

First, the Proca Lagrangian already contains an extra term in comparison to the Maxwell Lagrangian. It seems that you want an additional extra term, but it's not clear what is its purpose.
Second, it's not clear what superconductivity has to do with all this.

Superconductivity is the subject, i am studying meissner effect at low temperatures but specifically the photons laws of motion within the superconductor and we're doing this using the em field and potentials. The additional term is to find what other characteristics apply to the photons when in a superconductor, i was not told specifically how to approach it or what characteristics i am looking for as that was the question i needed to answer myself

 

[Maniac_XOX]

What do you want to achieve? If it's superconductivity, you need to "Higgs" the em. local gauge invariance. Another very good reference is by Weinberg:

https://inspirehep.net/literature/18067

The KEK preprint is freely accessible.

That i don't even know yet lol, i was only told to use the same process to find another operator besides the mass term, the proca lagrangian has the local gauge invariance included so i suppose it would be smarter to start from that

 

[vanhees71]

In electrodynamics there's no mass term for the photons of course, but you get one in superconductors from the Higgs mechanism. I really recommend Weinberg's paper quoted above. There's also a section with the same content in his Quantum Theory of Fields vol. 2.

A simpler version, using the non-relativistic Schrödinger equation is found in the Feynman Lectures vol 3:

https://www.feynmanlectures.caltech.edu/III_21.html