# Does the Pauli Exclusion Principle still apply if $c=\infty$?

The Pauli Exclusion Principle comes from the fact that the wave functions of particles with half integer spin are antisymmetric under particle exchange. From how I understand it, this relationship arises from combining relativity with quantum mechanics. Does the Pauli Exclusion Principle still hold if $c=\infty$ or does it require a finite value for $c$?

• "The Pauli Exclusion Principle comes from the fact that the wave functions of particles with half integer spin tend to produce destructive interference." I'm not at all sure that this is a good way to approach the matter: bosons interfere exactly as well as fermions, only they are not subject to exclusion. Count_to_10's answer below covers the proper difference. – dmckee Aug 4 '16 at 23:19
• The existing answer misses the point of the question. This question is basically asking whether half-integer spin obey Fermi-Dirac statistics without SR. In relativistic QFT, this is a consequence of the spin-statistics theorem, and SR plays a crucial role. To say that "spin [...] has nothing to do with special relativity" in this matter is extremely misleading. I'm not sure if the spin-statistics theorem still works for Galilean relativity. – knzhou Aug 6 '16 at 5:55
• @knzhou I read the question and because of it's remarks regarding c = infinity, and c being involved in the PEP, at my (admittedly lower) knowledge level, these seemed incorrect. I would be delighted, as I would learn something and the OP would receive an alternative, more advanced answer, if you would write an answer, that would explain to me how the points raised in the answer have a more advanced interpretation. Thanks very much. – user108787 Aug 6 '16 at 9:40
• @count_to_10 I don't know though, this is beyond my knowledge level! This resource seems to say that spin-statistics still holds as long as you make reasonable other assumptions, like stability (the Hamiltonian being bounded below). But I'm not confident enough to write an answer here. – knzhou Aug 6 '16 at 19:17
• @knzhou thanks for coming back. That's the trouble with posts, I have to guess the level and whether I know enough to answer. I did post on meta asking for level of knowledge to be indicated on the post but got nowhere. Anyway, I learned something writing it. Regards. – user108787 Aug 6 '16 at 20:22

Spin comes from combining relativity with quantum mechanics.

I can't stress enough that this is the wrong idea about spin. Spin is an intrinsic property of the particles that have it, and has nothing to do with special relativity in any way.

You might be confused because the Dirac Equation incorporates quantum mechanics with special relatively and the spin of an electron can be derived from it, but a particle at rest (which is not really possible) has the same spin as an identical particle travelling at 99.9999% c.

Spin Wikipedia

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.

Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus). The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.[5]

In some ways, spin is like a vector quantity; it has a definite magnitude, and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.

One way of looking at the Pauli exclusion principle is to think of it in terms of wavefunctions: half-integer spin particles must be described by antisymmetric wavefunctions, and particles of integer spin are required to have symmetric wavefunctions. The minus sign in the equation below implies the wavefunction must vanish identically if both states are "a" or "b", leading to the PEP, the law of nature that states that it is impossible for both electrons to occupy the same state in a bound system, that is, they cannot have the same 4 quantum numbers.

Does the Pauli Exclusion Principle still apply if c= ∞ or does it require a finite value for c?

The speed of light is not infinity, it is 299 792 458 m/s and it does not have any connection with the form of the wavefunction that describes the PEP.

• I interpret the question as asking whether the spin-statistics theorem can be proven in non-relativistic QM, and your answer simply asserts the connection between spin and statistics. – Javier Aug 6 '16 at 20:05