# Modern State of the Pauli Exclusion Principle and the Problems of Its Theoretical Foundation

## Abstract

**:**

## 1. Introduction

**is**the new order of quantum numbers ${m}_{1}{m}_{2}\dots {m}_{n}$ after the application of the permutation ${P}_{k}$. This function cannot have two particles in the same state and Heisenberg concluded that it satisfies the exclusion principle formulated by Pauli.

“An antisymmetric eigenfunction vanishes identically when two of the electrons are in the same orbit. This means that in the solution of the problem with antisymmetric eigenfunctions there can be no stationary states with two or more electrons in the same orbit, which is just Pauli’s exclusion principle.”

**s**with components s

_{x}, s

_{y}

_{,}s

_{z}obeying the same commutation relations as the components of the orbital angular moment

**l**. The obtained results were in a full accordance with experiment.

**s**,

_{x}**s**. acting on wave function. The latter was depending on the three spatial coordinates q and a spin coordinate. As a spin coordinate, the spin projection s

_{y,}s_{z}_{z}was used. The latter is discrete with only two values. Therefore the wave function ψ(q, s

_{z}) has only two components ψ

_{α}(q) and ψ

_{β}(q) corresponding to s

_{z}= ½ and s

_{z}= −1/2, respectively. The operator, acting on the two-component functions, is presented as a matrix of the second order. The explicit form of the spin operators was obtained. They were represented as

**s**1/2

_{x = }**σ**

_{x},

**s**1/2

_{y = }**σ**

_{y}, and

**s**1/2

_{z = }**σ**

_{z}, where

**σ**

_{τ}are the well-known Pauli matrices. Applying this formalism to the anomalous Zeeman effect, Pauli obtained the same results that were obtained by Heisenberg and Jordan [9] using the matrix approach.

**,**classical physics may not contain quantum terms. The inclusion of the zero-point radiation in classical electrodynamics provides it by the quantum properties. The zero-point radiation is a quantum phenomenon, its energy equal to $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\hslash {\omega}_{0}$. In the classical limit when ħ → 0, it does not exist.

“The only possible states of a system of identical particles possessing spin s are those for which the total wave function transforms upon interchange of any two particles as$${P}_{ij}\mathsf{\Psi}(1,\dots ,i,\dots ,j,\dots ,N)=(-1{)}^{2s}\mathsf{\Psi}(1,\dots ,i,\dots ,j,\dots ,N)$$That is, it is symmetric for integer values of s (the Bose-Einstein statistics) and antisymmetric for half-integer values of s (the Fermi-Dirac statistics)”.

^{16}O

_{2}molecule the validity of PEP was precisely checked in experiment, see Ref. [35] where a detailed discussion was presented.

## 2. Spin-Statistics Connection

“Already in my initial paper, I especially emphasized the fact that I could not find a logical substantiation for the exclusion principle nor derive it from more general assumptions. I always had a feeling, which remains until this day, that this is the fault of some flaw in the theory.”

**,**the modified parafermi statistics introduced by Kaplan [42], is valid for different types of quasiparticles in a periodical lattice among them: polaritons [43,44], defectons [45], delocalized holes in crystals [46], and some others [47,48].

“Why is it that particles with half-integral spin are Fermi particles whose amplitudes add with the minus sign, whereas particles with integral spin are Bose particles whose amplitudes add with the positive sign? We apologize for the fact that we cannot give you an elementary explanation… It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics”.

## 3. Theoretical Foundations of PEP

#### 3.1. Indistinguishability of Identical Particles and the Symmetrization Postulate

“whether the PEP limitation on the solution of the Schrödinger equation follows from the fundamental principles of quantum mechanics or it is an independent principle?”

“The exclusion principle could not be deduced from the new quantum mechanics but remains an independent principle which excludes a class of mathematically possible solutions of the wave equation. This excess of mathematical possibilities of the present-day theory, as compared with reality, is indications that in the region where it touches on relativity, quantum theory has not yet found its final form.”

_{12}gives

_{ki}(P

_{12}) form a square matrix of the order f.

_{12}to both sides of Equation (8) leads to the identity. In contrary with Equation (6), we cannot arrive at any information about the permutation symmetry of the wave function. The requirement that under permutation the wavefunction should be multiplied on an insignificant wave-factor, one actually postulates that the representation of the permutation group is one-dimensional. Thus, this proof is based on the initial statement, which is proved as a final result.

_{λ}-dimensional representation Γ

^{[λ]}of the group ${\mathit{\pi}}_{N}$ and the wave functions ${\mathsf{\Psi}}_{rt}^{\left[\mathsf{\lambda}\right]}$ are constructed by the Young operators ${\mathsf{\omega}}_{rt}^{\left[\mathsf{\lambda}\right]}$,see Appendix A, Equation (A2).

^{[λ]}of the permutation group ${\mathit{\pi}}_{N}$, the probability density, Equation (11), is a group invariant, that is, it is invariant upon action of an arbitrary permutation. For an arbitrary finite group, it was proved in Ref. [80]. Thus, for every permutation of the group ${\mathit{\pi}}_{N}$

^{[λ]}of the permutation group ${\mathit{\pi}}_{N}$, the full density matrix (and all reduced densities matrices) transforms according to the totally symmetric one-dimensional representation of ${\mathit{\pi}}_{N}$. In this regard one cannot distinguish between degenerate and nondegenerate permutation states. The expression for the probability density (12) obeys the indistinguishability principle even in the case of multi-dimensional representations. Therefore, the indistinguishability principle is insensitive to the symmetry of wavefunction. So, it cannot be used as the criterion for selecting of its correct symmetry.

^{16}O

_{2}molecule it was confirmed in experiment. Another example is the allowed multiplets in atomic spectroscopy. For example, in the (np)

^{2}electronic shell only

^{1}S,

^{3}P, and

^{1}D states are realized. The latter were used for the classification of the Cooper pair states in the theory of superconductivity.

#### 3.2. Analysis of the Properties of Identical Particle System Not Obeying PEP

^{[λ]}can be constructed by applying the Young operators ${\omega}_{rt}^{\left[\lambda \right]}$, see Equation (A2) in Appendix A, to the non-symmetrized product of one-electron orthonormal functions (spin-orbitals) ϕ

_{a}(i)

^{[λ]}

^{N}], are realized in nature, other irreducible representations are forbidden. In this subsection we examine the situation that arises when no symmetry constraints are imposed.

_{ab}: ${\mathsf{\Gamma}}^{\left[N\right]}\left({P}_{ab}\right)=1$ and ${\mathsf{\Gamma}}^{\left[{1}^{N}\right]}\left({P}_{ab}\right)=-1$ for all ${P}_{ab}$ and N. For multi-dimensional representations, the matrix elements ${\mathsf{\Gamma}}_{tt}^{\left[\mathsf{\lambda}\right]}\left({P}_{ab}\right)$ depend on [λ] and ${P}_{ab}$; in general, they are different for different pairs of identical particles. The matrices of transpositions of all irreducible representations for groups ${\mathit{\pi}}_{2}-{\mathit{\pi}}_{6}$ are presented in book [89], Appendix 5.

- (1)
- transitions between states with different symmetry [λ
_{N}] are strictly forbidden, - (2)
- N-particle states with different [λ
_{N}] have a different analytical formula for its energy.

_{N}] corresponds to a definite kind of particles with statistics determined by this permutation symmetry.

_{N}] corresponds to different statistics, the system of particles with the definite permutation symmetry [λ

_{N}] must have some additional inherent particle characteristics that establishes why N-particles system is characterized by this permutation symmetry, like half-integer and integer values of particle spin for fermions and bosons. This inherent characteristic has to be different for different [λ

_{N}]. Thus, the particles belonging to different types of permutation symmetry [λ

_{N}] are not identical. Just this, takes place in the particular cases of fermions, [1

^{N}], and bosons, [N], that characterized by odd and even values of the total spin S, respectively.

_{N}] with N = 2 to 4 is presented.

_{N}] describe particles with different statistics. The number of different statistics depends upon the number of particles and rapidly increases with $N$. In the case of the multi-dimensional representations, we cannot select non-intersecting chains, as it is in the boson and fermion cases.

_{N}], in the N-th generation can originate from particles with different kinds $\left[{\lambda}_{N-1}\right]$ in the $\left(\mathrm{N}-1\right)\mathrm{th}$ generation, even from fermions or bosons. Thus, the N-particle state $\left[{\lambda}_{N}\right]$ stems from the particles in the $\left(\mathrm{N}-1\right)\mathrm{th}$ generation with wave function, which must be in general described by a linear combination of wave functions with different permutations symmetry $\left[{\lambda}_{N-1}\right]$.

^{[λ]}of the permutation group, but, not by their linear combinations.

_{3}] = [21].

## 4. Concluding Remarks

“We may not expect that in future some unknown elementary particles can be discovered that are not fermions or bosons”.

_{2}molecule, in which PEP was certainly taken into account. The quantum mechanical calculations of the dissociation energy and the first ionization potential [93,94] are in a complete agreement with experimental values, see Table 1.1 in [95]. From this follows not only an additional confirmation of PEP, but also a rather general conclusion that molecules obey the same quantum-mechanical laws that obey traditionally physical objects: atoms and solids; at nanoscale we should not distinguish between chemical and physical systems.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Necessary Minimum Knowledge on the Permutation Group

_{i}is represented by a row of λ

_{i}cells. The presence of several rows of identical length λ

_{i}are denoted by a power of λ

_{i}. For example,

_{λ}is its dimension of the irreducible representation ${\mathsf{\Gamma}}^{\left[{\lambda}_{N}\right]}$, the summation over P runs over all N! permutations of the group ${\mathit{\pi}}_{N}$. The application of operator (20) to a nonsymmetrized product of orthonormal one-electron functions ${\phi}_{a}$

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Kaplan, I.G.
Modern State of the Pauli Exclusion Principle and the Problems of Its Theoretical Foundation. *Symmetry* **2021**, *13*, 21.
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Kaplan IG.
Modern State of the Pauli Exclusion Principle and the Problems of Its Theoretical Foundation. *Symmetry*. 2021; 13(1):21.
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2021. "Modern State of the Pauli Exclusion Principle and the Problems of Its Theoretical Foundation" *Symmetry* 13, no. 1: 21.
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