atomic physics - Are there fields corresponding to the composite particles (e.g. hydrogen atom field)?

In classical physics, particles and fields are completely different stuff. However, when a field is quantized, the particles appear as its excitations (e.g. photon appears as a field excitation in the quantization of electromagnetic field). In fact, for all the elementary particles, there is a corresponding field.

I am interested whether this is also true for any composite particles. Could we define, for any given composite particle, a field for which, upon quantization, that composite particle appears as its excitation? Is there, for example, anything like "hydrogen atom field"?

Answer

It depends on the exact circumstance whether or not such an idea is a good approximation for the physics you want to describe.

For the hydrogen atom, you're usually not interesting in it's "scattering behaviour", you're interested in its internal energy states, how it behaves in external electromagnetic fields, etc. Such internal energy states are not well-modelled by QFT. In particular, you'll usually want to consider the proton as "fixed" and the electron as able to jump between its different energy levels. Considering the "hydrogen atom" as an indivisible (or atomic, as it were...) object is not particularly useful.

But there are composite particles where associating a field is perfectly sensible, for example the pion, whose effective field theory describes the nuclear force between hadrons - and the hadrons are also composite particles that are treated with a single field here, for instance by means of chiral perturbation theory.

There are, besides an interest in scattering behaviour (which you also might legitimately have for the hydrogen atom or other atoms, I'm not implying you should never treat the hydrogen atom this way), other reasons to model certain objects as the particles of a field:

Modern many body physics as in condensed matter theory is essentially quantum field theory, too, and it is very frequent there to have fields for composite particles, or even pseudo-particles like phonons. For instance, a simple but powerful model for superconductivity, the Landau model, just treats a conductor as a bunch of charged bosonic particles, thought of as the quanta of a field, coupled to the electromagnetic field, and superconductivity is then another instance of the Higgs mechanism of quantum field theory.