$\def\rr{{\bf r}}\def\ii{{\rm i}}\newcommand{\ket}[1]{\lvert#1\rangle}$I do mostly atomic physics (where, for example, the scattering of atoms is very pertinent) so I have a slightly different take on what counts as "useful" in QFT. It is often very useful indeed to define an "atomic field operator" $\Psi(\rr)$. The excitations of this field are entire atoms, which may be either bosonic or fermionic, in which case the fields satisfy appropriate (anti-)commutation relations $$\Psi(\rr)\Psi^\dagger(\rr') \pm \Psi^\dagger(\rr')\Psi(\rr) = \delta(\rr-\rr'),$$
where the minus (plus) sign is for bosons (fermions). Thus, if $\ket{0}$ is the vacuum state containing no particles, then a general $N$-particle state corresponds to
$$\ket{\phi} = \int\prod_{j=1}^N\mathrm{d}\rr_j\; \phi(\rr_1,\rr_2,\ldots,\rr_N) \Psi^\dagger(\rr_1)\Psi^\dagger(\rr_2)\cdots \Psi^\dagger(\rr_N)\ket{0},$$
where $\phi(\rr_1,\ldots,\rr_N)$ is the $N$-body wave function in coordinate space.

So far we have considered the field operator in the Schroedinger picture. In the Heisenberg picture, the time evolution of the field is generated by the Heisenberg equation
$$ \ii\hbar\dot{\Psi}(\rr,t) = -\frac{\hbar^2 \nabla^2}{2m} \Psi(\rr,t) + V[\Psi;\rr]\Psi(\rr,t),$$
where the (non-linear) potential reads as
$$ V[\Psi;\rr] = V_1(\rr) + \int{\rm d}\rr'\; V_2(\rr-\rr') \Psi^\dagger(\rr',t)\Psi(\rr',t).$$
The first term $V_1(\rr)$ describes a one-particle external potential, the second term $V_2(\rr)$ describes a two-body interaction potential between the atoms. Higher-order (i.e. $n$-body potential) terms are possible as well. It is straightforward to generalise the field $\Psi(\rr)$ to a spinor having a component for each possible internal state of the atom. In this case the potential terms generalise to matrices or higher order tensors coupling together different internal states, while derivative terms in the potential could appear due to artificial gauge fields, for example.

As always in QFT, this is a low-energy effective theory. The non-relativistic approximation for the kinetic energy assumes that the centre-of-mass motion of the atoms is much slower than $c$. The field description of the atoms breaks down at scales where their internal structure becomes important, e.g. length scales comparable to the Bohr radius or energies comparable to the Rydberg energy.

If you want, you can view the above as the quantisation of a complex classical "Schroedinger" field $\Psi(\rr)$ (replace $\Psi^\dagger(\rr) \to \Psi^*(\rr)$). This classical Schroedinger field obeys a (non-linear) Schroedinger equation, with an associated Hamiltonian and Lagrangian etc. We usually do not do this, because the classical field does not describe anything particularly interesting. This is because 1) Nature is quantum and 2) no quantum states exist for which the classical field description is even close to a good approximation. The situation is different in, e.g., electrodynamics, where the observable dynamics of coherent states of the electromagnetic field can be well approximated by the classical Maxwell equations in many cases. However, in atomic field theories, the fundamental excitations are composed of fermions and their number is strictly conserved. This means that coherent states cannot be prepared* and therefore the classical field theory is largely useless (same reasoning applies, e.g. to the classical Dirac equation).

*In Bose-Einstein condensates, the off-diagonal long-range order makes a coherent state description a good approximation for many properties of interest. In this case the classical field theory is very useful; it goes by the name of the Gross-Pitaevskii equation.

This post imported from StackExchange Physics at 2017-11-23 19:29 (UTC), posted by SE-user Mark Mitchison