A particle is a dimensionless point travelling on a world-line … at least according to Eugene Wigner (cannot remember where I read that). A dimensionless point is a problematic thing if one wants to give it properties such as mass and charge. The mass density and charge density of the particle will have to be infinite, which is part of the reason for the infinities that one finds in quantum field theory.
Yet when subjected to high energy elastic scattering these particles produce a scale invariant behaviour to arbitrary high energies — a phenomenon called Bjorken scaling. This seems to suggest that they are point-like and dimensionless.
Let’s consider this carefully and be brutally honest with ourselves. What is it really that we are observing here? We don’t see any particles directly. We deduce their existence based on observations of scattering, absorption or some other type of interaction. If fact, without some form of interaction we simply won’t be able to make any observations of particles.
Now for the brutal honesty: what we really see is a localised interaction. Instead of observing a dimensionless point particle, what we in actual fact observation is a dimensionless interaction point. The notion of a dimensionless point particle is simple a way to explain why interactions are localised at dimensionless points. But is this the only possible explanation? Perhaps there are no particles, only fields. Perhaps the fields interact with each other (or with themselves) at these dimensionless points.
Why would these fields interact at dimensionless points, if there are no particles? Well we know that these fields often have internal degrees of freedom (like phase or spin). These internal degrees of freedom usually allow topological defects, such as vortices and monopoles, to exist in these fields. These topological may actually mediate these interactions.
That would explain why we always observe localised point-like interactions without using the notion of dimensionless point particle.