The Pauli Exclusion Principle. Can It Be Proved?

The modern state of the Pauli exclusion principle studies is discussed. The Pauli exclusion principle can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is a so-called spin-statistics connection. The reasons why the spin-statistics connection exists are still unknown, see discussion in text. On the other hand, according to the Pauli exclusion principle, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, all other types of permutation symmetry are forbidden; although the solutions of the Schrödinger equation may belong to any representation of the permutation group, including the multi-dimensional ones. It is demonstrated that the proofs of the Pauli exclusion principle in some textbooks on quantum mechanics are incorrect and, in general, the indistinguishability principle is insensitive to the permutation symmetry of the wave function and cannot be used as a criterion for the verification of the Pauli exclusion principle. Heuristic arguments are given in favor that the existence in nature only the one-dimensional permutation representations (symmetric and antisymmetric) are not accidental. As follows from the analysis of possible scenarios, the permission of multi-dimensional representations of the permutation group leads to contradictions with the concept of particle identity and their independence. Thus, the prohibition of the degenerate permutation states by the Pauli exclusion principle follows from the general physical assumptions underlying quantum theory.