# 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

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–  Qmechanic Aug 31 '12 at 10:15

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 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? http://arxiv.org/pdf/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 http://www.livingreviews.org/lrr-2004-5
But it is not perturbatively renormalizable, which makes many people search for a more fundamental way of quantizing gravity.

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Why is renormalizability the most important criterion to quantize a classical field? This may be a simple question, but I don't have a good understanding of field quantization. –  Antillar Maximus Aug 31 '12 at 11:49
Because for a relativistic QFT, renormalizability guarantees finite results at large energies without having to introduce more and more additional constants (higher order deviations from the classical equations) as the approximation order is increased. –  Arnold Neumaier Aug 31 '12 at 12:53
@ArnoldNeumaier Independently of renormalizability, can you truly quantize arbitrary theories? For instance, can you quantize the theory given by the Lagrangian $L=e^\dot{x}$?. Its classical limit is the free particle in one dimension. –  drake Aug 31 '12 at 17:13
@drake: doesn't your remark already answer your question? The quantum free particle will be a quantization. - On the level of perturbation theory, one only needs the Taylor expansion of the Lagrangian up to a particular order, and as many counterterms. The problem is how to fix the counterterms to get a ''unique'' quantization. This is not expected to be possible in general; already in QM one has free parameters due to ordeing ambiguities. Renormalization just ensures that one doesn't have an infinite-dimensional manifold of theories, but only a few-dimensional one. –  Arnold Neumaier Aug 31 '12 at 17:25
If the symplectic form of the phase space is non-integral, there is no hope for a single-valued wavefunction. This as well as anomalies don't seem to be addressed by this answer. Or maybe I am misunderstanding? –  Ryan Thorngren Aug 31 '12 at 17:43