The vast theories of physics (and mathematics) lay on the notion of material point. However, relativity and quantum mechanics cast doubts about the ultimate existence of zero-dimensional points:
- General relativity includes points with infinite density where it is not valid (the known singularity issue). Furthermore, due to diffeomorphism invariance, we get problems considering the nature of single points.
- Quantum mechanics, due to the uncertainty principle, implies that a pure point is like something "fuzzy". The same thing applies in Quantum Field Theory, pure pointlike interactions are just a bit "fuzzy" even with local field theory defined on a manifold with material points...
Question: can the existence of pure points of matter be said to be unattainable due to the foundations of the two pilars of physics or is there some loophole that makes pure points, yet useful mathematical models, physically REAL?
Related: the brane revolution also would cast doubt on the nature of the different p-dimensional objects, since they can be dually related to each other, so there is also the questions of what is the real p-dimensional object. And note that for real, I mean something positivist from a philosophical viewpoint, something measurable or testable.