When can a classical field theory be quantized? Given a classical field theory can it be always quantized? Put in another way, Does there necessarily need to exist a particle excitation given a generic classical field theory? By generic I mean all the field theory variants, specially Higher Derivative QFT(particularly Lee-Wick field theories).
 I ask this question because, several times in QFT we come across non physical particles when we try to quantize some field theory for e.g. ghost fields and ghost particles. These fields have opposite sign in front of the kinetic energy term. Such terms are common in higher derivative field theories. Hence we have to ask should we talk about particles in such situations.
  Now if the answer is No! Then we have to ask what is more fundamental in nature, particles or fields?
Schwinger disliked Feynman diagrams because he felt that they made the student focus on the particles and forget about local fields, which in his view inhibited understanding. -Source Wikipedia
 A: Fields are the fundamental objects, and observable particles are their irreducible excitations. The particle content of a field theory can be inferred only from closer analysis. The bare particles which go into the description of the Feynman diagrams (and must already be renormalized to even make sense) only tell part of the story.
In the sense of an effective field theory, every classical theory can be quantized. See arXiv:hep-ph/0308266 for a recent survey on effective field theories.
But for a ''fundamental'' theory one usually requires renormalizability, which drastically restricts the allowed theories. (But see also :
J. Gomis and S. Weinberg,
Are Nonrenormalizable Gauge Theories Renormalizable?
https://arxiv.org/abs/hep-th/9510087 )
General relativity is one of the classical theories that can be successfully quantized as an effective field theory; see, e.g., 
P. Burgess,
Quantum Gravity in Everyday Life:
General Relativity as an Effective Field Theory
Living Reviews in Relativity 7 (2004), 5
https://doi.org/10.12942/lrr-2004-5
But it is not perturbatively renormalizable, which makes many people search for a more fundamental way of quantizing gravity.
