It seems QFT and QM contradict with each other?

[fxdung]

In QFT a photon with arbitrary energy can interact with an electron.But in QM and Solid State Physics, if energy of photon smaller than energy gap of electron then photon can not interact the electron.So it seems to me there is a contradiction?

 

[atyy]

There is also an interaction in condensed matter, but it can often be neglected as the magnitude is small. However, you can look up things like two-photon effects, second harmonic effects which involve higher order power series terms.

Raman scattering might also interest you.

 

[vanhees71]

I've no clue what the OP is after. Where have you read that electromagnetic fields don't interact with a bound electron if it's energy is too small to excite the bound electron? It's of course wrong. There is elastic scattering of photons with bound electrons. In solids the analysis leads to the usual theory of dispersion, i.e., the derivation of the dielectric function from quantum (field) theory.

QFT and QM cannot be contradictory to each other, because QM is a special case of (non-relativistic) QFT in situations where the particle number is conserved in all considered processes.

 

[fxdung]

But in Superconductor when energy of phonon is smaller than energy gap of Cooper pair then there is not phonon-electron interaction?

 

[Vanadium 50]

Why are you bringing phonons into this?

 

[fxdung]

By the way I would like to know whether phonon interacts electron or not when its energy smaller than the energy gap?Because one could apply QFT in Condensed Matter Physics.

 

[PeterDonis]

in Superconductor when energy of phonon is smaller than energy gap of Cooper pair then there is not phonon-electron interaction?

Phonons are not photons. Your OP talked about photons. Which do you want to talk about?

one could apply QFT in Condensed Matter Physics

Yes, indeed one can. Quantum field theory is not limited to theories of fundamental particles like the Standard Model. You can have effective field theories at various levels.

However, none of this shows any contradiction between QFT and QM. It just shows that you can have theoretical descriptions at various levels.

 

[fxdung]

I like to know both: phonon and photon

 

[PeterDonis]

I like to know both: phonon and photon

You are basically asking for a course in both quantum field theory and condensed matter physics. That's way beyond the scope of a PF thread.

Even if we limit the scope to one particular question for each, we can't really answer them as you ask them:

In QFT a photon with arbitrary energy can interact with an electron.

It depends on the states of the photon and electron. If the electron is bound in an atom, your statement is not true. So you are starting from a false premise.

in Superconductor when energy of phonon is smaller than energy gap of Cooper pair then there is not phonon-electron interaction?

Phonons don't interact with single electrons to begin with. To the extent they can be said to interact with anything else, it would be Cooper pairs. But Cooper pairs are not electrons; they're not even "pairs of electrons" in the usual sense of the term "electrons". Phonons and Cooper pairs are simply elements of a different quantum field theory at a different level from the QFT of electrons and photons. So again you are starting from a false premise.

The one thing we can definitely say is that there is no contradiction between QFT and QM in any of these cases.

 

[Demystifier]

In QFT a photon with arbitrary energy can interact with an electron.But in QM and Solid State Physics, if energy of photon smaller than energy gap of electron then photon can not interact the electron.So it seems to me there is a contradiction?

A photon with arbitrary energy can interact with a "free" electron (that is, electron that was free before the photon arrived), because the energy spectrum of the free electron is continuous. In atoms or crystal lattices, where the electrons are not free, the energy spectrum is often not continuous. But it has nothing to do with a difference between QFT and QM. In principle, both can study either free or non-free electrons. In practice, however, QFT is often applied to study scattering of elementary particles, where the asymptotic scattering states are free.

 

[vanhees71]

But in Superconductor when energy of phonon is smaller than energy gap of Cooper pair then there is not phonon-electron interaction?

Of course there is phonon-electron interaction, first of all forming the Cooper pairs, generating a mass gap and "Anderson-Higgsing" the photon.

 

[vanhees71]

A photon with arbitrary energy can interact with a "free" electron (that is, electron that was free before the photon arrived), because the energy spectrum of the free electron is continuous. In atoms or crystal lattices, where the electrons are not free, the energy spectrum is often not continuous. But it has nothing to do with a difference between QFT and QM. In principle, both can study either free or non-free electrons. In practice, however, QFT is often applied to study scattering of elementary particles, where the asymptotic scattering states are free.

I just wanted to make clear that photons and electrons always interact. For a bound electron, however you have a finite gap between the energy of this state and of the next energy eigenstate (which may be again a bound state or also a scattering state). A single photon with an energy less than this energy difference is necessarily elastically scattered on this bound electron, i.e., it leaves the atom in the same intrinsic state as it was before but still scatters elastically. Energy-momentum conservation is then fulfilled by the scattered photon and the atom as a whole. The free center-mass motion of the atom never has a energy gap and thus can always change in an interaction with a photon of any energy.

 

[fxdung]

What is Anderson-Higgsing photon?

 

[vanhees71]

It's the same mechanism that is at work in the electroweak standard model! As the "Higgs condensate" makes the W- and Z-gauge-bosons massive without violating local gauge invariance, the photon becomes massive within a superconductor through the Cooper pairs.

This always happens, when a spontaneously broken global symmetry is "gauged", i.e., made local. In this process the massless Goldstone modes of the spontaneously broken symmetry group get "absorbed" (in the unitary gauge) into the gauge-boson fields, making them massive and providing the additional 3rd polarization degree of freedom of a massive vector field compared to the only two polarization states of a massless one.

Note that a local gauge symmetry cannot be spontaneously broken (Elitzur's theorem). Only the corresponding global symmetry before "gauging" it is spontaneously broken. Thus I avoid to call the "Higgsing" of a gauge theory "spontaneous symmetry breaking".

Now, particularly in the case of the description of a superconductor by "Higgsing" electromagnetic gauge theory to call it only "the Higgs mechanism" is historically unjust, because it was in fact Anderson in this context who discovered the "Higgs mechanism" first. That's why I tend to call it the "Anderon-Higgs mechanism" at least in the context of superconductivity. In electroweak theory the outstanding addition by Higgs as well as Englert and Brout compared to the many more fathers of this idea is that they predicted the existence of an additional physical particle in this theory due to the Higgs field needed to get the W's and Z's massive without violating gauge symmetry. Unfortunately for all the other discoverers (there are in addition to Higgs at least, Brout, Englert, Guralnik, and Kibble) somehow the name "Higgs boson" stuck. Admittedly the "Anderson-Higgse-Brout-Englert-Guralnik-Kibble-et-al mechanism" would be not such a convenient name as simply "Higgs mechanism".

 

[fxdung]

Which books say about Anderson-Higgsing photon?

 

[vanhees71]

The best exposition I know is in Weinberg's textbook

S. Weinberg, The Quantum Theory of Fields, Vol. II, Cambridge University Press

The same content can be found in his paper

S. Weinberg, Superconductivity for Particular Theorists
Progress of Theoretical Physics Supplement, No. 86, pp. 43-53
https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.86.43/5329534/86-43.pdf