# Distance can be zero

Can a distance between two particles can be exactly zero? Example: If I have two opposite charged particles having uniform charge. They will attract each other and the distance between them will reduce. How can I find the minimum distance between them? Can the distance be zero, like both charges stuck togther?

Similary, if a book is placed on a table, it seems that the distance between two is zero but microscopically, it is not. So is there any possibility of distance being zero?

Distance between classical particles can be zero on paper, but that's fairly meaningless. A "classical particle" is the assumption that one can approximate the real movements of an extended body by nothing but the dynamics of its center of mass. This doesn't imply that one can bring two of these bodies together and that the approximation will still be useful. Consider planetary movement: one can approximate all the planets in the solar system as "particles" and the equations of motion will describe the solar system very well. This assumption wouldn't make any sense if we wanted to know what happened to the Mars size body that struck the proto-Earth in the collision that is assumed to have formed the Moon.

If you are talking about quantum mechanical "particles", then the answer is that the expectation value of their distance can never be zero because of the uncertainty principle.

If we approach this problem mathematically, we see that the distance between two point is zero only when they are the same points. This should mean that if we consider the particles to be point objects, the distance between them will be zero only if they coincide. If we are to consider the particles to be extended objects, then at least two points on their surfaces should coincide with each other for the distance between the two objects to be zero. So, for the distance between two objects to be zero, their surfaces actually need to overlap each other so that there is a common point between them.

If the distance between two particles are $0$ they coincide...so you have no way to say that they are two different particles.

Consider two classical particles, like the electron, and consider what it means for them to have exactly zero distance between them.

First the gravitational attraction between them is infinite; as is the electrical repulsion; but infinities, in physical theories, are usually a sign that something, somewhere, is not quite right.

This suggests that either:

(a) extensionless particles are not possible - so possibly, instead, strings or membranes;

(b) or if they are possible, then they cannot approach each other infinitesimally, or coincide.

One might even say that this leads to a principle of preservation of particle number; for if particles can't coincide, then they can't 'merge', so dropping the particle number by one, or 'demerge' so increasing the particle number by one.

It's only in QFT that we see this principle being violated: when particles are both annihilated/created; but of course QFT is not a classical theory.