# The Pauli Exclusion Principle and the Problems of Its Experimental Verification

## Abstract

**:**

## 1. Introduction: Discovery of the Pauli Exclusion Principle and Its Generalized Formulation for all Elementary Particles

“According to this point of view, the doublet structure of alkali spectra… is due to a particular two-valuedness of the quantum theoretic properties of the electron, which cannot be described from the classical point of view.”

_{j}(in the modern notations); by n and l, he denoted the well-known (at that time) principal and orbital angular momentum quantum numbers, where j and m

_{j}are the total angular momentum and its projection, respectively. Thus, Pauli characterized the electron by some additional quantum number j, which in the case of l = 0 was equal to ±1/2. For this new quantum number j, Pauli did not give any physical interpretations, since he was sure that it could not be described in the terms of classical physics.

“Physicists found it difficult to understand the exclusion principle, since no meaning in terms of a model was given to the fourth degree of freedom of the electron.”

“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.”

^{6}times greater than the average density of the sun; it is approximately 10

^{6}g/sm

^{3}. Fowler [13] resolved the paradox of why dense objects, such as white dwarfs, do not collapse at low temperature. He applied the Fermi–Dirac statistics to the electron gas in the white dwarfs, also introduced in 1926, and showed that, even at very low temperatures, the electron gas, called degenerate in these conditions, still possesses a high energy. Compression of a white dwarf leads to an increase of the inner electron pressure, which is due, according to Fowler, to the exclusion principle suggested by Pauli. Thus, the repulsion from PEP prevents the white dwarfs from undergoing gravitational collapse.

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

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 (see a detailed discussion in Reference [20]).

## 2. Experimental Verifications of the Pauli Exclusion Principle

#### 2.1. Motivation: Theoretical Conceptions on the PEP Violations

^{2}atomic energy level will induce the radiative transition 2p

^{6}→ 1s

^{1}allowed in quantum mechanics (see Figure 2), while the transition to the filled 1s

^{2}K-shell can only occur if PEP is violated. As was stressed in References [26,27], in contrast with the spontaneous violation of parity conservation in β-decay or violation of CP invariance, the non-conservation of electric charge requires a global reconstruction of the contemporary theory of electromagnetic processes. Nevertheless, in some publications, the authors discussed the conditions in which charge non-conservation can take place (see References [28,29,30,31]).

#### 2.2. Experimental Tests of the PEP Violations: What Is Measured

^{−15}). The search for non-Pauli atoms of

^{20}$\tilde{\mathrm{Ne}}$ and

^{36}$\tilde{\mathrm{Ar}}$ was performed. Negative F (fluorine) and Cl (chlorine) ions were extracted from natural fluorine and chlorine samples, respectively. The results of processing these measurements yielded the following upper limits:

^{+}and β

^{−}decays [59,63], and nuclear reactions, for instance, on

^{12}C [64]. It should be mentioned that the prohibition of non-Pauli transitions [37], following from the superselection rules is not valid for transitions changing the number of identical fermions and in some other cases (see References [65,66]).

^{12}C nuclei for non-Pauli transitions were reported:

^{7}energy level to the 1s

^{2}level, forming the 1s

^{3}level forbidden by PEP. For the probability of PEP-violating transitions, which is usually denoted by the parameter 1/2 β

^{2}[40,70], the following quite impressive low value was obtained [69]:

^{7}shell to the closed 1s

^{2}shell of Cu atoms was as follows [72]:

#### 2.3. Comments on the Experimental Tests of the Pauli-Forbidden Electron Transitions

_{N}, which are labeled by the symbol [λ] of the corresponding Young diagram with N boxes and denoted by Γ

^{[λ]}or simply by [λ] (see Appendix A). The construction of the basis functions of irreducible representations of the permutation group π

_{N}can be performed by applying the Young operators ${\omega}_{rt}^{\left[\lambda \right]}$ (Equation (A5)) to the non-symmetrized product of one-electron orthonormal functions (Equation (A2)). The obtained function ${\mathrm{\Psi}}_{rt}^{\left[\lambda \right]}$ (Equation (A6)) is transformed under permutations according to the representation Γ

^{[λ]}. From the generalized formulation of PEP, see p.3, it follows that only irreducible one-dimensional representations, either ${\mathrm{\Gamma}}^{\left[N\right]}$ or ${\mathrm{\Gamma}}^{[{1}^{N}]}$, are realized in our nature; all other irreducible representations are forbidden. However, for non-Pauli electrons, this restriction is not valid. It is instructive to analyze properties of such unusual electrons.

_{ab}) is the diagonal matrix element of the transposition P

_{ab}of functions ψ

_{a}and ψ

_{b}in Equation (A2), and h and g are one- and two-particle operators, respectively. Only exchange terms in Equation (11) depend upon the symmetry of the state. For one-dimensional representations, ${\mathrm{\Gamma}}_{tt}^{\left[\lambda \right]}$(P

_{ab}) does not depend on the number of particles and permutations, i.e., ${\mathrm{\Gamma}}^{[N]}({P}_{ab})=1$ and ${\mathrm{\Gamma}}^{[{1}^{N}]}({P}_{ab})=-1$ for all ${P}_{ab}$ and N. For multi-dimensional representations, the matrix elements ${\mathrm{\Gamma}}_{tt}^{\left[\mathsf{\lambda}\right]}\left({P}_{ab}\right)$ depend on [λ] and ${P}_{ab}$; in general, they are different for different pairs of identical particles (the matrices of transpositions for all irreducible representations of groups π

_{2}–π

_{6}are presented in Reference [81], Appendix 5).

_{N}] have different analytical formulae for its energy, and (b) transitions between states with different symmetry [λ

_{N}] are strictly forbidden (superselection rule), then we must conclude that each type of symmetry [λ

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

_{N}] is connected exclusively to the identity of particles. Therefore, there must be some additional inherent particle characteristics, which establishes the N particle system in a state with definite permutation symmetry, such as integer and half-integer values of particle spin for bosons and fermions, and this inherent characteristic must be different for different [λ

_{N}].

_{N}], are not identical and may not be mixed not only with “normal” electrons with symmetry [1

^{N}], but also among themselves.

_{N}] and, for N ≥ 3, several possible symmetries, corresponding to different energies, must be considered. However, at present, we do not know any quantum-mechanical phenomena where PEP is not satisfied. Therefore, the probability of the existence of non-Pauli electrons must be very small, and it is reasonable to assume that, in the electronic systems studied in experiments, no more than one non-Pauli electron can exist. This was accepted in the VIP experiments.

- (a)
- Mallon et al. [83] applied the Desclaux method for heavy atoms. For these atoms, the relativistic approach is appropriate; however, in this case, the j–j coupling should be used (as they did), and spin ceases to be a good quantum number. On the other hand, the elements of the fourth period of the periodical table, with 3d-electrons, can be calculated with good precision at the non-relativistic level (see very precise MR CI calculations in References [84,85]). Thus, Cu can be calculated at the non-relativistic level using the Russel-Saunders (L–S) coupling and without the Breit–Pauli, Lamb shift, and others relativistic corrections. It is worthwhile to mention that, in this case, the non-relativistic Greenberg quon model [46] is quite justified.
- (b)
- If, in analogy with muons, it can be assumed that a non-Pauli electron can be attached to atoms in solids, it is quite doubtful that it will be located on the same atomic shells as the “normal” Pauli electrons and then to the K-shell. This is a crucial point in the treatment of experiments for detecting the Pauli forbidden transition, and it must be argued more carefully.

## 3. Concluding Remarks

^{12}C (see Bellini et al. [65] and Equation (6) in Section 2). It is important to mention that the prohibition of non-Pauli transitions following from the superselection rules is not valid for transitions with a changing number of identical fermions.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Some Necessary 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}is convenient to indicate by the power of λ

_{i}. For example,

[λ] = [2^{2} 1^{2}]. |

_{2}, one can form only two Young diagrams from two cells.

[2] | [1^{2}] |

_{3}, one can form three Young diagrams from three cells.

[3] | [2.1] | [1^{3}] |

_{4}has five Young diagrams.

[4] | [3 1] | [2^{2}] | [2 1^{2}] | [1^{4}] |

^{[λ]}of the group π

_{N}. The assignment of a Young diagram determines the permutation symmetry of the basis functions for an irreducible representation, i.e., it determines the behavior of the basis functions under permutations of their arguments. The diagram with only one row corresponds to a symmetric function in all its arguments, and with one column corresponds to a completely antisymmetric function, both correspond to one-dimensional representations. All other types of diagrams correspond to intermediate types of symmetry and describe multi-dimensional representations.

_{0}(Equation (2)) by the antisymmetrization operator

**, Γ**

_{N}^{[λ]}the matrix elements, and f

_{λ}is the dimension of the irreducible representation Γ

^{[λ]}. For [λ] = [1

^{N}], the Young operator coincides with the presented above antisymmetrization operator.

^{[λ]},

_{λ}functions ${\mathrm{\Psi}}_{rt}^{\left[\lambda \right]}$ with fixed second index t forms a basis for the representation Γ

^{[λ]}, and t enumerates f

_{λ}different bases for Γ

^{[λ]}. The second index t describes the symmetry under permutation of the one-electron functions in the non-symmetrized product (Equation (A2)).

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**Figure 2.**Scheme of permitted radiative transition to K-shell, if the 1s electron undergoes spontaneous decay.

**Figure 3.**The schematic representation of the formation of the Pauli-forbidden K-shell population in experiments of the Ramberg and Snow type.

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Kaplan, I.G.
The Pauli Exclusion Principle and the Problems of Its Experimental Verification. *Symmetry* **2020**, *12*, 320.
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Kaplan IG.
The Pauli Exclusion Principle and the Problems of Its Experimental Verification. *Symmetry*. 2020; 12(2):320.
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