How the concept of "wave function of atom" in Bose-Einstein condensate should be interpreted from perspective of quantum field theory?

Question

A typical description of Bose-Einstein condensate goes along the line of "multiple atoms in the same ground state can be described by the same wave function". But hold on. Atoms are not elementary particles. They consists of electrons and quarks, and according to a usual way to present Standard Model and QED/QCD, different elementary particles correspond to wave solutions in different quantum fields. Those fields are orthogonal and cannot interact with each other directly. They interact by exchanging virtual force particles. Thus, there is no and cannot be a "quantum field of atom" where "wave function of atom" is an excitation.

I suppose that just as in the case of cooper pairs, when authors talk about "wave function of atom" what they really mean is rather a process where different quantum fields interact via virtual particles, and result of this process behaves as if this was a quantum wave solution for a particle, even though really it is not.

Is this correct? Is there any publication that explicitly construct an equation compatible with QED that gives a solution which looks-like-and-behaves-like a "wave function of an atom" even though there is no special field for its value? What is the value domain of such a wave function if it cannot be any one quantum field? Or is there a completely different method to assign meaning to a "wave function of atom"?

Answer 1

Such a theory would be an effective field theory. It is often the case that different degrees of freedom are separated by large differences in energy. For example when solving the wave equation for a hydrogen atom we can, to a good approximation, treat the nucleus as a point charge and ignore its internal structure. We need only worry about the nuclear structure at energies above a GeV or so when the non-zero size of the nucleus becomes important.

In the context of quantum field theories an example is the effective theory describing the strong nuclear force as a force between hadrons due to the exchange of virtual pions (there is an interesting review here). Neither the hadrons nor the pions are fundamental particles, but as long as we use the theory only at energies well below a GeV it works fine.

In principle I guess we could construct an effective field theory to describe the interactions between atoms in a BEC, but it isn't obvious that this would be a useful thing to do. QFT becomes important when the energies are high enough to make particle production and annihilation possible, but BECs can only be created at very, very low energies i.e. temperatures only a hair's breadth from absolute zero. In this regime it's hard to see what a QFT would add to regular quantum mechanics.