According to quantum mechanics, bosons can occupy the same quantum state in the same system but fermions cannot because the wave function for the interchange of two identical fermions would be asymmetric, according to the Pauli exclusion principle.

I was wondering what is the definition of such a system and its reach, e.g. I can have a free fermion at LHC and another one at LANL and both of them can be at the same state (because there is no quantum entanglement?), but in a neutron star, fermion degeneracy, i.e. that all these neutrons cannot occupy the same state, is what is giving the neutron star its particular characteristics.

So the question is: what is the formal definition of a system? When are these fermions entangled and cannot occupy the same state and when aren't they?

Two identical fermions cannot be in the same state, period.

Location is part of the state, so two fermions in different locations are automatically not in the same state. "Gaussian wavepacket at $p$ momentum at the LHC" and "Gaussian wavepacket at $p$ momentum at LANL" are different states.

In principle all identical fermions have an exchange interaction regardless of distance, but in practice it's only significant at short distances.

  • and what is the formal difference in the wave function? is there an additional term corresponding to the location? – Juanjo May 11 at 6:41
  • @Juanjo the distance is within the wavefunction solutions the 1/r of the potential between charges. It is what zeroes the off diagonal terms in the density matrix. – anna v May 11 at 7:36
  • @Juanjo The expectation value of the position, $\left<x\right>=\left<\psi\middle|\vec r\middle|\psi\right>$. If the positions are different, it should be obvious the wavefunctions must be as well. – Chris May 11 at 8:23
  • Imagine a fermion on one side of a neutron star and another one on the opposite side, 20 km away, does it mean that these two neutrons can have the same wavefunction but for the $r$ term? Aren't anyway all particles in a system occupying different $r$ positions? I'm sorry, not trying to be annoying, just to fully get it. – Juanjo May 11 at 9:56
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    @NorbertSchuch The question in the title is based on a false premise- there is no "formal definition of a system" in the way the OP is asking. I am addressing the core of the question- "when can fermions be in the same state?" And the answer is never. – Chris May 13 at 8:51

I will stick with this definition for a physical system:

In physics, a physical system is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the system.

You ask for many body systems:

So the question is: what is the formal definition of a system? When are these fermions entangled and cannot occupy the same state and when aren't they?

As an experimentalist I have found the density matrix formalism as useful for developing intuition about many particle systems.

If the density matrix of the many body system has non zero off diagonal elements, it means that the particles in the chosen system are represented by one quantum mechanical wavefunction. The off diagonal elements show the degree of "entanglement" for the particular pair of particles.

This question and its answer might interest you for neutron stars.

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    Re that wikipedia definition for physical system, would you agree (or not) that in addition to simply "a portion of the physical universe", any such portion chosen as a physical system must be characterized by a state space satisfying some mathematical properties? I won't try (don't know) how to specify those properties for utmost generality (necessary and sufficient properties). But you can't just choose any old "portion of the universe" that suits your fancy at the moment. It has to have some "internal consistency" reflected by the structure of its state space. (P.S. I'm not the downvoter:) – John Forkosh May 11 at 6:54
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    @JohnForkosh well, one can always choose a portion of the physical universe, whether one can mathematically analyze it is a further step, imo, and whether it is a successful analysis one step further. System is a very general term that needs context. – anna v May 11 at 7:28
  • Thanks for your remarks. (And note that I have a recent ongoing interest in "system", as per my question physics.stackexchange.com/questions/403452 which got closed.) By the way, I agree "System is a very general term...", but note that the wikipedia entry defines "physical system", emphasize "physical", which I was suggesting permits additional constraints corresponding to the adjective (e.g., "ball" is very general, whereas "soccer ball" has additional constraints). – John Forkosh May 11 at 9:26

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