We are hitting close to the mark. How is a particle in QFT point-like? I have heard this claim several times today, but cannot see it in the mathematics, which is a field theory.

FYI - I have a doctorate in mathematics and have studied and particularly like quantum field theory. I know what "observe a particle" can be interpreted to mean. But "point-like", it seems this phrase is being used in a jargon sense. A position that seems to be supported by your references, following up on that I got that a "point particle" means the lagrangian is a function of the fields at a single point.

@PonderStibbons The questions I link should answer that, e.g. this answer by Bosoneando.

I read that. It seems to support my view - that "point particle" is a jargon term and does not refer to any kind of physical point or physical particle - other than that I could say that if I integrate density I am adding up an infinite number of point particles. But, that is just a way of phrasing the Steiltjes sum - not a physical concept of something that can be observed.

@PonderStibbons I have one meaning of a point particle, which I'm not sure is the mainstream definition. It has to do with Feynman diagrams. In QFT diagrams, you always draw lines and point vertices. This has the significance that the QFT propagator can be calculated as the path integral of worldline action.You can contrast this with string theory, where propagator is the path integral of the worldsheet action, so string theory has extended particles in this sense.

@RyderRude comment is appreciated, and I think that this is something like the attitude a number of people I have met have. I would feel that these cannot be observed directly and are more clearly a computational tool.

@PonderStibbons I agree. Composite particles can experimentally be distinguished from elementary ones. So, that terminology is justified. But I don't see how the "point particle" terminology is justified other than if we're trying to categorise the computational techniques of QFT and string theory. I'm a beginner, so take this with a grain of salt.

For what it's worth, the shape of an object does not appear explicitly in at least classical presentations of kinematics of a rigid body either, and only appears when interactions (collisions) are to be considered. The motion of a free object is basically the motion of its center of mass. So in this regard, saying it is "only" relevant for scattering isn't really as much of a "diminuition" as one might think it is.

Of course, we certainly can make a mathematical description of a moving shape, it just isn't usually "covered". And this leads to an important but missable point about quantum mechanics: when we talk of the wave function, $\psi$, of a particle in even the first quantization, it actually doesn't stand in the theory at the place where the "shape" would be if we included it, it stands in the theory at the place where the kinematic attributes would be, namely position (and momentum).

Then QFT, or the second quantization, simply layers on top of this a mechanism for dynamically creating and destroying objects, which again does not change any of that preceding discussion (if anything, relativity is the one that really problematizes things more because it dinks in a lot of fraught ways with the idea of ascribing a "position" to a field quantum that is not present in non-relativistic QM-as-QFT).

@hft Different people have a very different idea about what a particle is. While your density of N point particles is definitely what I would agree is a density of N point particles - that is not what most people seem to be meaning by the term "particle". In the article I was referring to, a field was said to be for a point particle if the Lagrangian density was a function of the fields at each given point. That use definitely feels like a jargon term. And it is different from the one that you present.

To me "jargon term" sounds derogatory and seems to imply purposeful obfuscation. Of course the same phrase can mean different things to different people in different situations.

@hft: The big problem with RQFT specifically, though, is it doesn't appear to have such a thing as a position operator. (Non-relativistic multiparticle QM-as-QFT does not suffer from this problem, so the problem really is relativity, not the "field" concept; QM, by itself, thus effectively provides a duality between fields and particles).

@The_Sympathizer that's a good point.

@hft yes, I acknowledge that I have seen both definitions. But, I entirely meant "jargon" here to mean specialized language. The implication not being derogatory, only that it was not applied to particle in the classical sense. I have met several people recently who claim to be not followers of Bohm, and yet also claim with strong commitment that there is somehow a small lumpish particle in the field somewhere.

@hft I read Dirac's paper introducing the Dirac equation as stating that no particles other than spin 1/2 particles can have a position operator. Could you comment on that? Or should this be another question?

"I read Dirac's paper introducing the Dirac equation as stating that no particles other than spin 1/2 particles can have a position operator. Could you comment on that?" I am not familiar with this statement and I'm not sure if it is true. Might be better as a separate question with additional context such as the actual quote and a specific citations (Dirac wrote a lot of different papers, I'm sure).

@hft thanks, I will organize the references.