How are scalar fields = particles? Two such particles I'm thinking of are the inflaton & the Higgs. They are both scalar fields, but they're also both particles with well-defined masses.
How is it that scalar fields correspond to particles? The two don't seem related. We have wave-particle duality, but how do we also have scalar field-particle duality? There's also the converse question: does the existence of a proton mean there's a "proton scalar field"? If not, why not?
I'm looking for some explanation as to why these two scalar fields seem to have a corresponding particle, but others (such as the Newtonian gravitational potential) don't.
 A: Particle physics is modeled with a quantum field theory. In a quantum field theory, there exist the quantum fields, in your question the Higgs field and the inflaton field, and creation and  annihilation operators operating on these fields, which create the particles, in your case the particle Higgs and the particle inflaton. In general all particles in the standard model table are expected to be generated from the corresponding fields, the electrons from the electron field, the photons from the photon field etc. The hypothetical inflaton follows the same QFT rules.
For a long time , even though the data validated the electroweak QFTheory with the Higgs field, it was not sure that the Higgs particle existed, until a candidate was found at LHC. The story is here. .
The inflaton field was introduced in order to fit the data of the history of the universe, a hypothetical field at the beginning of the Big Bang, introducing an inflation in the size of the universe for a time interval. There is no expectation AFAIK that particle physics experiments are looking for the inflaton particle.
A: *

*The inflaton is a hypothetical particle. The evidence for its existence is far shakier than that for the existence of
the Higgs particle.


*For a general discussion of how particles arise from quantum fields, see this question and its answers (and read any introductory textbook on QFT if you really want to see how it works).


*Which particles "exist" depend on what you look at. A quantum field theory (QFT) is a model for a particular physical situation like any other physical theory - in some circumstances you might want to include certain effects, in others you don't. For instance, between nuclei, you might think of the strong force as the residual strong force mediated by pions, while inside a nucleus, you will have to think of it as being mediated by gluons. In the first case, there is a 'fundamental' pion field whose associated particles are the pions, in the second case, there is a 'fundamental' gluon field and the pion is a composite particle (interestingly not composed solely of gluons).
In principle, nothing forbids you from considering a quantum field theory involving a Newtonian gravitational potential, but that is not very useful in practice because generally QFT becomes more relevant (compared to classical mechanics or ordinary quantum mechanics) for particles interacting with each other at high energies (because then there is enough energy around to actually create new particles out of the energies of the interacting particles), and the scales at which QFT becomes relevant are such that we must consider it as a relativistic theory - and Newtonian gravity is evidently not a relativistic theory.
Since gravity is comparatively weak, even situations where QFT becomes relevant in non-relativistic situations will typically be dominated by other interactions - a large class of QFT applications lie in the theory of condensed matter systems, but it is not gravity that is responsible for the structure of matter as electromagnetism dominates over it by several orders of magnitude, so it is irrelevant there, too.


*Trying to combine general relativity - as the relativistic theory of gravity - and relativistic QFT as the relativistic theory of everything else is unfortunately not straightforward and we haven't figured out how to consistently do that yet.
