In quantum field theory we often hear that particles sometimes behave like waves and sometimes behave like particles.
In quantum field theory we say particles are fundamentally fields.
Is it correct to say that particles and waves are limits of quantum fields? For example, if we take a quantum field and apply the limits appropriate for a double-slit set-up, would the field come out a wave?
Distinction between particles and waves is meaningful in classical physics, but doe snot exist in quantum world. What we can meaningfully talk about is particle-like or wave-like properties. E.g., electrons can interfere after passing through slits, i.e., they exhibit wave-like properties. Photons can be counted, which is a particle-like property.
Mathematically both are described by the second quantization formalism, which is second quantization only for what is classically a particle, but first quantization for anything that is classically a wave.
See also:
Particle- and wave-like properties
How does quantization arise in quantum mechanics?
I think your intuition is right, but I wouldn't use the word "limit" since there is not really a parameter that smoothly interpolates from particle-like to wave-like behavior. It's more that in certain situations a quantum field will exhibit particle behavior, and in other situations a quantum field will exhibit classical wave behavior, and in yet other situations may have no classical analogue.
One way to phrase wave-particle duality in field theory is that the number and phase operators don't commute. (Strictly speaking, there are technical obstacles to defining a phase operator in relativistic field theory, but this construction works in condensed matter physics and I think is useful to build intuition.) In other words, if you choose to measure the number of particles associated with the field, then the phase of the wavefronts of the field are not defined by the uncertainty principle. Conversely, if you measure wavefronts in the field with a definite phase, then by the uncertainty principle you cannot say how many particles are present.
Meanwhile, when quantum effects are strong, you can get phenomena which are not easily interpreted as being approximately classical particles or waves. For example, in QCD, at low energies there is confinement, where it's impossible to isolate a single particle (a quark or gluon). In this context, the natural observables are more abstract objects called Wilson lines, which don't really have an easy classical description in terms of particles or waves.
