Difference between 'Quantum theories'

[A. Neumaier]

Sorry, I don't get it. I thought that this is the standard QFT method for building symmetrized and normalized N-particle states. See, for example, (4.2.2) in Weinberg's book. The only difference in my formula is that I am using position representation instead of Weinberg's momentum representation.

This was a minor remark only (and only partially correct, since, indeed, your states are symmetrized. Sometimes I am a bit too fas and then make small mistakes.) Fock space is a Hilbert space and does not contain unnormalizable states. Your states live in a corresponding rigged Hilbert space.

This is true. A satisfactory position operator for photons has not been constructed yet. Perhaps, it cannot be defined at all.

There are theorems that it cannot be defined in a unique way. (If one assumes all the properties that Newton and Wigner assume, there is none; if one relaxes the requirement on rotations, there are infinitely many, so none of them could be distinguished as ''the physical one'' - moreover, these wouldn't lead to a good position representation.) See the entry ''Particle positions and the position operator'' of Chapter B2 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#position)

Perhaps, photons cannot be exactly localized in the position space. I don't have a certain opinion about that. However, they can be perfectly localized and counted in the momentum space. So, they still look like discrete particles.

The momentum representation is completely nonlocal, in any physical sense of localization.

Yes, QFT applications to condensed matter processes use many of the same techniques as the fundamental relativistic QFT of elementary particles. However, at the fundamental level these two frameworks are quite different. Condensed matter QFT is inherently approximate as it ignores the discrete atomistic structure of matter. For example, the continuous phonon field is not a good approximation at distances comparable with interatomic separations in the crystal. On the other hand, relativistic QFT is supposed to be exact at all distances.

In the nonequilibrium statistical mechanics of a weakly interacting) boson and fermion gas, nothing is approximated, except for the usual approximations of perturbation theory.
The electron field in condensed matter theory and the photon field in quantum optics are also the same as in (relativistic or nonrelativistic) QED; only the nuclear structure is approximated.

Both approximation sare also used in the application of QED to the hydrogen atom; so there is no difference in principle.

Of course, there is also an "effective field theory" school of thought in relativistic QFT

I was not referring to that.

Another difference between the condensed matter and fundamental field theories is that the former does not use the concept of (Galilean) relativity, because there is always one distinguished frame of reference - the one connected with the underlying crystal lattice.

This is a minor difference only. The QED treatment of the hydrogen atom also does not use the concept of Poincare invariance.