Is it possible to overcome pauli exclusion principle by smashing together 2 fermions really really hard? The neutron star avoids further collapses under it's own weight due to degeneracy pressure, I was wondering beside gravity is it possible to overwhelm the pauli exclusion principle by other means such as colliding 2 electron together?
 A: At any attainable energy, no it's not possible. The Pauli exclusion principle is just that, a principle. In physics, principles are usually stronger than laws, they can't be broken. That said, all of physics is just a collection of effective theories. There is a non-zero probability that at insanely high, unachievable energy levels, there exists properties of physics that we haven't yet theorized that allow the collision of electrons to overcome this principle. But the more likely thing is that such high energies isolated in a small volume would form a black hole from particle production alone (albeit, a short lived and not dangerous black hole).
A: Nice question---it explores what the Pauli exclusion principle is. The answer is no: you can't push identical fermions even a little bit into the same state, no matter how hard you try.
So, then, why not?
There are two ways of getting at the exclusion principle in the fundamentals of physics. In one way one adopts an earlier style in which we write states of particles, and then we find the pair states that are available to pairs of identical fermions simply don't include cases where two identical fermions are in the same state.
A further (and ultimately better) insight comes from field theory where we say the particles such as electrons are excitations of fields such as the Dirac field, and to write a state we can adopt a notation which gives the degree of excitation of each mode of the field. Then a fermionic field has just two states for each mode (or, to be more mathematical about it, a two-dimensional Hilbert space); these correspond to either zero or one electron. There just aren't any other states (i.e. any other dimensions in the Hilbert space). Your question then becomes "can I think of a way to make a mode of the Dirac field go to a third state?" but there isn't a third state for it to go to. And the mathematical reason for this is quite strong: it follows immediately from certain anti-commutation relations that are very basic to the whole nature of these fields. You can't tinker with it without changing the whole structure of the field theory.
Well that is the situation as far as contemporary physics goes. But if you smash two anythings together at sufficiently high energy density then you will reach a parameter regime where all our knowledge runs out. So who knows what will happen? Let's try it and find out!
