# Rethinking Electron Statistics Rules

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## Abstract

**:**

## 1. The Historic Origin of Electron Statistics Rules

**Phenomenological formulation of the exclusion principle, proposed by Pauli in the 1920s**: only one electron may occupy any quantum mechanical state, where the electron states are defined using four quantum numbers: principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (m), and spin quantum number (s).

## 2. Coherent versus Incoherent Electron States

- N electrons are said to be in a
**coherent**state if all quantum numbers of each electron are the same, i.e., they are all in the same quantum mechanical state. - N electrons are said to be in a
**incoherent**state if each electron is in a different quantum mechanical state.

## 3. Spin Correlations between Particles Occupying Different Orbitals

**ISC always occurs pair-wise**, i.e., the ISC of N > 2 particles is never observed. Table 1 illustrates this effect for the simplest atoms: an electron is either spin-correlated to an other electron or to a nucleus but never to both at the same time. Whether we look at particles sharing the same orbital or particles occupying different orbitals, we thus observe exactly the same phenomenology of strictly pair-wise ISC coupling. This suggests the same origin of the ISC coupling limit, and we therefore look for a unifying principle. Taking the example of hydrogen spin isomers, it is obvious that Pauli’s microcausality arguments do not apply to well-separated nuclei, and it is also obvious that there would be nothing anti-symmetric about the exchange of two separated nuclei.

## 4. Isotropic Spin-Coupling Limit for Incoherent Electron States

_{2}and D

_{2}under linearly polarized incident light, employing 33.66 eV photon energy. They measured the angular correlation function of electromagnetic Lyman-alpha radiation produced by the resulting atom pair in order to find out whether the atom pair is entangled or not. The authors of [4] conclude that an entangled electron pair is produced through the photo dissociation of a hydrogen molecule, and this entanglement originates from their molecular state. The results of [4] thus demonstrate that it is principally possible to photo-dissociate a bonding orbital occupying electron pair, without breaking their entanglement, and then measure their individual spin state.

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 5. Spin Statistics for Coherent Electron States

#### 5.1. A Review of the Darwin Lagrangian

#### 5.2. A Brief Review of Electron Zitterbewegung

^{2}/h oscillation frequency named after him, which was then directly described as a light-speed oscillation by Schrödinger. References [2,6] present an experimentally validated Zitterbewegung model of the electron structure. As shown in [2,6], the electron spin is generated by its circular Zitterbewegung oscillation. This idea of the electron spin being generated by circular Zitterbewegung oscillation has a long history; reference [7] presents a thorough discussion of this topic. The Thomson scattering phenomenon is electron–light interaction in the low-photon-frequency limit: it measures the electron’s “reduced Compton radius” size, which corresponds to the radius of light-speed charge circulation at the mc

^{2}/h frequency.

^{2}/h.

#### 5.3. The Stable Equilibrium of Coherent Electron States

**c**= 1 natural units notation for convenience. Calling ${r}_{ec}$ the electron charge radius, ${r}_{e}$ the Zitterbewegung radius, ${m}_{e}$ the electron mass, ${\omega}_{e}$ the Zitterbewegung angular speed, and $\mathit{c}$ the charge velocity vector in a vacuum, we relate these values to each other in natural units notation:

**Theorem**

**2.**

**Proof.**

**coherent electrons obey Bose–Einstein statistics.**

## 6. Is Superconductivity the Realization of Coherent Electron States?

#### 6.1. The Bose-Einstein condensed state of superconducting electrons

#### 6.2. Preceding Meissner Effect Models

#### 6.3. The London Equation

**Theorem**

**3.**

**Proof.**

#### 6.4. The Microscopic Structure of Meissner Flows

## 7. Discussion

^{87}Rb nuclei is observed [16], as anticipated. The temperature difference between electron versus

^{87}Rb condensation relates to the inverse proportionality between the particle mass and its Bose–Einstein condensation temperature.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. A Useful Vector Field Identity

## References

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**Figure 1.**An illustration of two protons’ Larmor spin precession in a hydrogen molecule. Each proton perceives the other proton’s magnetic field (directed red curves) as an externally applied magnetic field and Larmor spin-precesses (cones with arrow) around the external magnetic field line.

**Figure 2.**An illustration of electrons’ coherent state. The dotted line represents the shared Zitterbewegung axis, the ellipses represent the 0.386 pm radius Zitterbewegung trajectories, and the blue spheres represent the electron charges. Each electron has the same momentum, and their kinetic speed vectors point along the Zitterbewegung axis.

**Figure 3.**The correlation between ${T}_{c}$ and spin fluctuation temperature ${T}_{SF}$ in various superconductors. Reproduced from [11].

**Figure 4.**An illustration of vortices that comprise the Meissner flow. The externally applied magnetic field (B) is represented by the arrow.

${\mathit{p}}^{+}+{\mathit{e}}^{-}$ (H) | ${\mathit{p}}^{+}+2{\mathit{e}}^{-}$ (H ^{−}) | ${}^{3}\mathit{He}^{2+}+{\mathit{e}}^{-}$ (He ^{+}) | ${}^{3}\mathit{He}^{2+}+2{\mathit{e}}^{-}$ (He) | |
---|---|---|---|---|

Hyperfine split | yes | no | yes | no |

ISC electrons | - | yes | - | yes |

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**MDPI and ACS Style**

Kovacs, A.; Vassallo, G.
Rethinking Electron Statistics Rules. *Symmetry* **2024**, *16*, 1185.
https://doi.org/10.3390/sym16091185

**AMA Style**

Kovacs A, Vassallo G.
Rethinking Electron Statistics Rules. *Symmetry*. 2024; 16(9):1185.
https://doi.org/10.3390/sym16091185

**Chicago/Turabian Style**

Kovacs, Andras, and Giorgio Vassallo.
2024. "Rethinking Electron Statistics Rules" *Symmetry* 16, no. 9: 1185.
https://doi.org/10.3390/sym16091185