So a thought occurred to a friend of mine the other day: in a neutron star, neutrons are prevented from sitting directly on top of one another due to the Pauli exclusion principle, what with neutrons being spin-1/2 particles. This manifests itself as a pressure that keeps the star from collapsing in on itself.

However, composite objects of half-integer spin can have integer spin. For instance, a 'dineutron' has either spin-0 or spin-1, and is hence a boson. This fact is what allows, for instance, helium-4 to undergo Bose-Einstein condensation. So merely by considering the neutrons in a neutron star in pairs, we ought to conclude that the star should collapse in on itself, on account that it is composed of bosons. Obviously this is not right, not just because it is not observed, but because how we group the particles into composites in our heads cannot affect what we actually observe...

The same holds for helium-4: if I want to think of my fundamental particle as the helium-4 atom, I conclude that helium-4 can undergo Bose-Einstein condensation. But if I think of my mixture as being a 1:1:1 collection of electrons, protons and neutrons, then the Pauli exclusion principle applied to each species prevents this condensation --- there is a 'Fermi degeneracy pressure'.

What's going on here?


1 Answer 1


Commutation relations for creation/annihilation operators of composite particles only approximately coincide with such relationships for elementary bosons/fermions (see, e.g., Lipkin, Quantum Mechanics: New Approach to Selected Topics). The differences become essential for high densities, when the wave functions of constituent particles overlap.

Note, by the way, that neutrons are also composite particles.

  • $\begingroup$ Does the treatment of a neutron star interior as a dense packing of neutrons really make sense? Wouldn't it be more appropriate to treat it as a quark-gluon plasma? $\endgroup$
    – CuriousOne
    Dec 28, 2014 at 16:36
  • $\begingroup$ @CuriousOne: I am not sure, but I guess it depends on the density. Furthermore, even the composition of quark-gluon plasma depends on the density: at higher densities you can have, say, strange quark matter, consisting of up, down, and strange quarks (en.wikipedia.org/wiki/Strange_matter ) $\endgroup$
    – akhmeteli
    Dec 28, 2014 at 16:54
  • $\begingroup$ I am asking out of relative ignorance. I have next to no physical intuition at what densities the neutron model fails and a quark based picture is needed to make predictions about the equation of state. $\endgroup$
    – CuriousOne
    Dec 28, 2014 at 16:59
  • $\begingroup$ @CuriousOne: I guess this question does not have a clear answer so far (en.wikipedia.org/wiki/Quark_star ). $\endgroup$
    – akhmeteli
    Dec 28, 2014 at 17:10
  • $\begingroup$ I'm not quite sure how this resolves my problem. If the commutation relations for the He-4 creation and annihilation operators are approximately bosonic, then we should observe them behaving approximately as bosons. But by considering instead the system to be a collection of protons, neutrons and electrons (or, if you prefer, quarks and electrons), then we ought to see distinctly fermionic behaviour of all particles involved. Low densities, high densities, whatever... My argument says that the system should have vastly different qualitative properties depending on how we consider it. $\endgroup$
    – gj255
    Dec 28, 2014 at 23:18

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