# Background on Meissner Kernel

I am reading Girvin and Yang's Modern Condensed Matter Physics on superconductivity. In chapter 19 on phenomenological theories, section 3, the "Meissner Kernel" is introduced with hardly any explanation. The defining equation is: $$J^\mu(\vec r,t) = -\frac{c}{4\pi}\int d^3\vec r' K^{\mu\nu}(\vec r-\vec r') A_\nu(\vec r',t)$$ $$J$$ is the current density, $$A$$ is the vector potential and $$K$$ is the "Meissner Kernel".

I've looked through several other texts, checked Wikipedia and searched the term here, but can find no information on this. Does anyone have any reference or resource that can give some background on this? I understand the form of the equation is a Green's function equation, but I'd like to see a derivation of it.

• This is a general expression in linear response theory. You apply a small perturbation to your system (magnetic field a.k.a. vector potential) and the answer of your system to this field is a current density. The function that connects these two quantities is the Kernel function (sometimes called response function, or for classical systems: Green's function). Take a look at linear response theory link Commented Dec 29, 2021 at 14:25
• @zltn.guba that seems like an answer Commented Dec 29, 2021 at 14:34
• @BySymmetry Thanks. I added an answer based on the comment. Commented Dec 29, 2021 at 14:36

This is a general expression in linear response theory. You apply a small perturbation to your system (magnetic field a.k.a. vector potential) and the answer of your system to this field is a current density. The function that connects these two quantities is the Kernel function (sometimes called response function, or for classical systems: Green's function). Take a look at linear response theory link.

• Thanks! So Kernels are integral parts of linear response theory formulas, and in this particular case, it's just that a name was given to the kernel since in superconductivity a relation between the current density and the vector potential exists.
– CGS
Commented Dec 29, 2021 at 14:46
• Yes. these Kernel functions are what we are interested in usually, because if you know what the Kernel function is then you can tell (in this case) the current density for any kind of applied magnetic field. If you can find the Kernel function then you know everything about your system (in terms of magnetic field and current density, of course). I am not sure about the reason why they call it "Kernel function" in SC. I think that "Kernel function" is a generally used name for these objects. Commented Dec 29, 2021 at 14:49

In this context the kernel is known as the the Pippard Kernel. You may also enjoy the account in Steven Weinberg's Superconductivity for Particular Theorists, (Progress in Theoretical Physics Suplement {86} (1986) 43-53.)

• Thank you! That is a nice paper to have and it contains an explanation of persistent currents that is also used by Girvin and Yang, but I had not seen anywhere else. I'll offer a paper to you which has an interesting explanation for superconductivity: the photon acquires a mass. This is point of view in Frank Wilczek's paper, "In Search of Symmetry Lost".
– CGS
Commented Dec 29, 2021 at 21:14
• @CGS I usually explain the Higgs effect by saying it's analogous to superconductivity and superconductivity by saying it's analogous to the Higgs effect. Not circular at all.... :) Commented Dec 29, 2021 at 22:49