# Is Pauli's exclusion principle a postulate of Quantum Mechanics?

Is Pauli's exclusion principle a direct consequence of the postulates of Quantum Mechanics or is it an independent postulate?

Why are there particles that follow Pauli Exclusion Principle? (why fermions exist? / why does Pauli Priciple exist?)

Would the answer be a simple "cause that's how it is" or is there a more fundamental reason? (which can be derived or explained from QM or other theories like String theory etc)

The Pauli principle was introduced as an ad hoc remedy to explain some features of many-electron spectra. Within the framework of non-relativistic quantum mechanics, it is introduced without foundation as an additional assumption, which has been shown by experiment to apply to all particles with half-integer spin.

Kaplan in

Kaplan IG. The Pauli exclusion principle. Can it be proved?. Foundations of Physics. 2013 Oct;43(10):1233-51.

presents "heuristic arguments" suggesting " that the existence in nature only the one-dimensional permutation representations (symmetric and antisymmetric) are not accidental." The physics behind the argument is that, if particles are indistinguishable, average values of one-body operators in many-body states must be independent of the state-labelling. The mathematics of the argument are found in

Kaplan IG. The exclusion principle and indistinguishability of identical particles in quantum mechanics. Soviet Physics Uspekhi. 1975 Dec 31;18(12):988.

Thus, there is some heuristic evidence that there can only be two types of statistics - bosonic and fermionic - because there are only two one-dimensional representation of the permutation group $$S_n$$ of $$n$$ particles.

There is no suggestion in Kaplan's argument that this is in any way tied to the spin of the particle, i.e. there is nothing to suggest that many-electrons states should be antisymmetric rather than symmetric, or that many-body states of - say - indistinguishable protons need also be antisymmetric; only that the many-body states must be one or the other. In this sense, introducing the Pauli principles, which follows from the antisymmetry of many-electrons states, was just an expedient for solving a particular problem in atomic spectra.

Moreover, the argument is completely rooted in the representation theory of the permutation group and cannot accommodate anyonic statistics (which can occur in 2d as quasiparticles, and where the permutation group is replaced by a braid group).

It is only within quantum field theory that the possible symmetry properties of many particle states were definitely linked to the spin of particles.

• And relativistically the spin naturally comes from transforming into a frame which is relativistic and quantum mechanical. I read this somewhere ...is this correct?
– Lost
Jul 22 at 8:12

I have to disagree with @JunSeoHe's answer; the exclusion principle is not an independent postulate. It follows from fermions' wavefunctions being antisymmetric under the exchange of identical particles. The wavefunction has to be either symmetric or antisymmetric under such an exchange (but only because $$\Bbb R^3\setminus\{0\}$$ is simply connected). The spin-statistics theorem identifies which particle species have each exchange response, albeit only in QFT, since for that we need its insight into the relativistic origin of spin.

• Agree! It is presented as a postulate to undergrads because proving the spin-statistics theorem is a pretty difficult task. Jul 20 at 19:07
• I agree too, it is furthermore pretty easy to demonstrate that the spectrum of a fermionic theory is $n=1$. Jul 20 at 19:22
• @JeanbaptisteRoux What do you mean by the spectrum? Do you mean the exchange induces a $(-1)^n$ factor?
– J.G.
Jul 20 at 19:41
• The proof of the spin-statistics theorem is a field theory result: it requires Lorentz invariance, which is certainly not part of the QM requirement. So whereas it’s certainly not an independent postulate in field theory, the situation is not so clear in non-relativistic QM. Jul 20 at 21:00
• @ZeroTheHero I've edited to address that. We only need such material to connect spin to statistics, not to determine which statistics are legal.
– J.G.
Jul 20 at 21:05

As far as non relativistic quantum mechanics goes, the Pauli Exclusion Principle is inserted as an empirical observation. This is how it worked historically. It was only during the advent of formulations of relativistic quantum mechanics that we came to realize that there is a more fundamental reason electrons (and other half integer spin particles) obey the Pauli Exclusion Principle.

Without going into too many technical details, the reason is that if the electron did not follow these statistics, you would be able to extract as much energy as you wanted from the vacuum. This is something we cannot have happen in our physical laws, since this would imply there are no stable states, and atoms, molecules, or humans would not exist. So we must conclude that two electrons cannot be in the same state.

This is essentially half of the so called spin-statistics theorem. The other half is regarding bosons, which have particular statistics for a different reason.

Paul's exclusion principle is a ndependent postulate. It was made to explain why only 2 electrons could occupy the same spatial wavefunction in a atomic orbital.