Superconductivity and Quantum Field Theory

[vkc102]

Why do theories of superconductivity need to be quantum field theories with 2nd quantisation? Isn't first quantisation enough?

 

[ZapperZ]

Why do theories of superconductivity need to be quantum field theories with 2nd quantisation? Isn't first quantisation enough?

That question doesn't make sense. It's like asking why we use the Lagrangian/Hamiltonian when Newton's Force law approach should be "enough".

If you wish to use the variational approach to BCS Theory, then do it. There's nothing wrong with that. However, you'll be missing out on the powerful technique from the field theoretic approach that allows for a visualization of all the relevant interactions and mechanism of the phenomena. Furthermore, in most cases, the only thing you can know and write are just the Hamiltonian of the system (this if often the case in condensed matter when you're dealing with a gazillion interacting particles). You have ZERO ability to even "guess" at the wavefunction. Thus, you start with the necessary creation-destruction operators at the Hamiltonian that describes all the necessary interaction of the system. Such a description will be transparent to know what is involved in the system.

Zz.

 

[vkc102]

Oops I think didn't phrase my question well enough. I meant to ask why doesn't a classical field theory like the Kronig Penny model work for superconductivity. Is it because without quantisation of the field, you can't get the phonons electron interaction to form Cooper pairs? Can you even get phonons at all for that matter with a classical field?

 

[dextercioby]

"Kronig Penny classical field theory"??

Please do some reading on QM and Solid State Physics.

Phonons,like all possible "on-s" are particles which appear from the second quantization,i.e.using the operators of creating and annihilating uniparticle states in a Fock space.

Daniel.

 

[ZapperZ]

Oops I think didn't phrase my question well enough. I meant to ask why doesn't a classical field theory like the Kronig Penny model work for superconductivity. Is it because without quantisation of the field, you can't get the phonons electron interaction to form Cooper pairs? Can you even get phonons at all for that matter with a classical field?

This doesn't make it any better. In fact, I am even more confused now.

Why would "Kronig-Penny" potential work for, of all things, superconductivity? I mean, if you really look at the BCS Theory, the "mechanism" for it is actually quite general. They just HAPPENED to use phonons as the mechanism for the pairing because that was what was known and accepted at that time. However, there's nothing to say that spin-fluctuation, etc. cannot be used. Thus, phonons are quite irrelevant in such a phenomena in general.

Zz.

 

[julian_brown]

Originally Posted by vkc102
"Why do theories of superconductivity need to be quantum field theories with 2nd quantisation? Isn't first quantisation enough"


I think this is a good question and the meaning is perfectly clear.

Yes, in principle a many-body wavefunction solution of the instantaneous Coulumb interactions between the lattice and electron particles would yield Cooper pair type solutions without a phonon or retarded interaction in sight. In fact you really can write a fairly realistic Gaussian product type wfn that solves for harmonic potential interactions and which encapsulates a lot of the basic physics.

Conventional 2nd quantization treatments are intrinsically messy and limited, because they invoke a hybrid semi-classical picture of the crystal ie. classical lattice points with vibration modes plus quantum fermi liquid. My impression is that nobody in the field likes Fröhlich / BCS much.