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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:

  1. 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.
  2. 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.

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The Heisenberg uncertainty principle requires there to be some error in our measuring of dimension. As a result, there would be no way to ever determine if a particle was in fact a point particle, as even if you found a particle to be a point particle, there would be some non-zero error in the measurement of length. However, many particles are so incredibly small that approximating them to be points does not change the result (within the error of experiment).

The singularity of a black hole is a difficult issue because, as far as I know, it is derived from an incorrect view. The space around a non-rotating black hole is homogenous on spherical surfaces. However, this coordinate system breaks at the event horizon. There is a coordinate system that can be derived to work within the event horizon, but in this system, time outside of the event horizon changes into a spacial coordinate. So if two free particles (particles which cannot affect the spacetime metric) are following behind one another with exactly the same velocity, one would expect them to hit the same spot. But, they actually hit different spots inside of the black hole, because time switches into a spacial dimension inside, which means that their separation in time results in a separation in space once they've entered the black hole. In that sense, the entire interior of the event horizon is the singularity. It is a difficult concept to understand, but so far as this goes the singularity is a misnomer.

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