(Not a physics person, but) As I read it, he is just arguing against the implication that Countability implies Particles. So he is saying that adding a quantum that is monochromatic has an effect across all space (which is clearly a non-particle like effect). Then he admits that you can get localization through superpositions of quanta.
Since he is arguing against a logical implication he only needs to show a counterexample exists.
The problematic part in choosing a single mode like this is that it can lead to a non-normalizable integral at some point (ie, it won't be a member of the Hilbert space).
QFT always made more since to me. Everything flows more naturally than in QM, where out of the blue, they introduce wave packets.
The paper does feel a bit shaky though especially as it is essentially taking a pedagogical point (start teaching from a QFT viewpoint) and making it into an argument about the foundations of physics. The non-relativistic QM view point is taught because it represents a smooth departure point from classical thinking without requiring the full mathematical complexity of field theory.
I think the general assumption is that QFT is true, it is just whether you can call the physics it creates as being particles. At least that is how I interpret it. QFT is QM, but it just has a lot more degrees of freedom (infinite) than what you are talking here about when you say QM. You can write a Schroedinger's equation for a field. Rather than the wave function being a function of X, the wave function is a function of F(X) where F(x) is a field configuration over space.
I would say some of his argument comes down to semantics because by his definition in order for something to be a particle it can't have spatial extent. I think a lot of people don't include this in the definition of a particle. They do associate the number of "Quanta" with the number of particles, at least in a field theory.
What I was saying above is that I believe you can create an object by the "Quanta" definition in QFT that has no spatial extent. His argument is based on the fact that this is not possible.
Adding these Quanta together to form a zero extent object is similar to saying in Fourier analysis you can take a number of different frequencies (like the quanta at different wave lengths) and create a delta function (which has no spatial extent). The thing is, even though we are dealing with adding a bunch of frequencies together, it is not exactly the same as a Fourier series because we are adding them together in a different way.
You make a good point about people first teaching the simple Schroedinger's equation of particles because it is a lot easier. I think what is so hard about quantum mechanics is relating it to the real world, and not the axioms themselves. The simple form is much easier to relate to the real world.
Yeah, I just meant QM as meaning the semi-classical version. Making QM work with SR requires moving to QFT/"second quantization".
You can presumably create particle-like quanta with a creation operator that creates particles in position space vs momentum space. The particle then won't have a definite momentum in that case, so there's still an extent problem at least in configuration space. If there is an ultraviolet energy cut-off then you end up with problems as well.
I think we perceive living in a position space more than we feel we live in a momentum space. We would prefer things to have a definite position more than we care about them having a definite momentum. The fact that they are conjugate observables is the troubling part.
Since he is arguing against a logical implication he only needs to show a counterexample exists.
The problematic part in choosing a single mode like this is that it can lead to a non-normalizable integral at some point (ie, it won't be a member of the Hilbert space).
QFT always made more since to me. Everything flows more naturally than in QM, where out of the blue, they introduce wave packets.
The paper does feel a bit shaky though especially as it is essentially taking a pedagogical point (start teaching from a QFT viewpoint) and making it into an argument about the foundations of physics. The non-relativistic QM view point is taught because it represents a smooth departure point from classical thinking without requiring the full mathematical complexity of field theory.